![Page 1: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/1.jpg)
Logic meets number theory in o-minimalityThe work of PETERZIL, PILA, STARCHENKO, and WILKIE
Logic Colloquium, Vienna 2014
Matthias AschenbrennerUniversity of California, Los Angeles
![Page 2: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/2.jpg)
KOBI
PETERZIL
JONATHAN
PILA
SERGEI
STARCHENKOALEX WILKIE
![Page 3: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/3.jpg)
The citation for the Karp Prize 2014 mentions . . .
• J. PILA, O-minimality and the André-Oort conjecture for Cn
Ann. of Math. 173 (2011), 1779–1840.
• J. PILA and A. J. WILKIE, The rational points of a definable setDuke Math. J. 133 (2006), 591–616.
• Y. PETERZIL and S. STARCHENKO, Uniform definability of theWeierstrass ℘-functions and generalized tori of dimension oneSelecta Math. (N.S.) 10 (2004), 525–550.
• , Definability of restricted theta functions and families ofabelian varietiesDuke Math. J. 162 (2013), 731–765.
. . . plus four more papers!
![Page 4: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/4.jpg)
Logic meets Number Theory
The history of interactions between these two subjects datesback almost to the beginnings of mathematical logic:PRESBURGER, SKOLEM, GÖDEL (1920s/30s);DAVIS-PUTNAM-ROBINSON, MATIYASEVICH (1950–70).
In number theory, questions about Z are often studied byrelating them to the (more tractable) fields R (of real numbers)and Qp (of p-adic numbers). These structures turned out to bewell-behaved also from the point of view of mathematical logic:
• TARSKI (1940s) for R;ë
• AX-KOCHEN, ERŠOV (1960s) & MACINTYRE (1976) for Qp.
More recent applications of mathematical logic to problems of anumber-theoretic flavor, initiated by HRUSHOVSKI, involveddeep “pure” model-theoretic results (geometric stability theory)applied to fields enriched with extra operators.
![Page 5: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/5.jpg)
Logic meets Number Theory
The history of interactions between these two subjects datesback almost to the beginnings of mathematical logic:PRESBURGER, SKOLEM, GÖDEL (1920s/30s);DAVIS-PUTNAM-ROBINSON, MATIYASEVICH (1950–70).
In number theory, questions about Z are often studied byrelating them to the (more tractable) fields R (of real numbers)and Qp (of p-adic numbers). These structures turned out to bewell-behaved also from the point of view of mathematical logic:
• TARSKI (1940s) for R;ë
• AX-KOCHEN, ERŠOV (1960s) & MACINTYRE (1976) for Qp.
More recent applications of mathematical logic to problems of anumber-theoretic flavor, initiated by HRUSHOVSKI, involveddeep “pure” model-theoretic results (geometric stability theory)applied to fields enriched with extra operators.
![Page 6: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/6.jpg)
Logic meets Number Theory
The history of interactions between these two subjects datesback almost to the beginnings of mathematical logic:PRESBURGER, SKOLEM, GÖDEL (1920s/30s);DAVIS-PUTNAM-ROBINSON, MATIYASEVICH (1950–70).
In number theory, questions about Z are often studied byrelating them to the (more tractable) fields R (of real numbers)and Qp (of p-adic numbers). These structures turned out to bewell-behaved also from the point of view of mathematical logic:
• TARSKI (1940s) for R;
ë
• AX-KOCHEN, ERŠOV (1960s) & MACINTYRE (1976) for Qp.
More recent applications of mathematical logic to problems of anumber-theoretic flavor, initiated by HRUSHOVSKI, involveddeep “pure” model-theoretic results (geometric stability theory)applied to fields enriched with extra operators.
![Page 7: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/7.jpg)
Logic meets Number Theory
The history of interactions between these two subjects datesback almost to the beginnings of mathematical logic:PRESBURGER, SKOLEM, GÖDEL (1920s/30s);DAVIS-PUTNAM-ROBINSON, MATIYASEVICH (1950–70).
In number theory, questions about Z are often studied byrelating them to the (more tractable) fields R (of real numbers)and Qp (of p-adic numbers). These structures turned out to bewell-behaved also from the point of view of mathematical logic:
• TARSKI (1940s) for R;
ë
• AX-KOCHEN, ERŠOV (1960s) & MACINTYRE (1976) for Qp.
More recent applications of mathematical logic to problems of anumber-theoretic flavor, initiated by HRUSHOVSKI, involveddeep “pure” model-theoretic results (geometric stability theory)applied to fields enriched with extra operators.
![Page 8: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/8.jpg)
Logic meets Number Theory
The history of interactions between these two subjects datesback almost to the beginnings of mathematical logic:PRESBURGER, SKOLEM, GÖDEL (1920s/30s);DAVIS-PUTNAM-ROBINSON, MATIYASEVICH (1950–70).
In number theory, questions about Z are often studied byrelating them to the (more tractable) fields R (of real numbers)and Qp (of p-adic numbers). These structures turned out to bewell-behaved also from the point of view of mathematical logic:
• TARSKI (1940s) for R;
ë
• AX-KOCHEN, ERŠOV (1960s) & MACINTYRE (1976) for Qp.
More recent applications of mathematical logic to problems of anumber-theoretic flavor, initiated by HRUSHOVSKI, involveddeep “pure” model-theoretic results (geometric stability theory)applied to fields enriched with extra operators.
![Page 9: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/9.jpg)
Logic meets Number Theory
The history of interactions between these two subjects datesback almost to the beginnings of mathematical logic:PRESBURGER, SKOLEM, GÖDEL (1920s/30s);DAVIS-PUTNAM-ROBINSON, MATIYASEVICH (1950–70).
In number theory, questions about Z are often studied byrelating them to the (more tractable) fields R (of real numbers)and Qp (of p-adic numbers). These structures turned out to bewell-behaved also from the point of view of mathematical logic:
• TARSKI (1940s) for R;ë o-minimality;
• AX-KOCHEN, ERŠOV (1960s) & MACINTYRE (1976) for Qp.
More recent applications of mathematical logic to problems of anumber-theoretic flavor, initiated by HRUSHOVSKI, involveddeep “pure” model-theoretic results (geometric stability theory)applied to fields enriched with extra operators.
![Page 10: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/10.jpg)
The citation for the Karp Prize 2014 mentions . . .
• J. PILA, O-minimality and the André-Oort conjecture for Cn
Ann. of Math. 173 (2011), 1779–1840.
• J. PILA and A. J. WILKIE, The rational points of a definable setDuke Math. J. 133 (2006), 591–616.
• Y. PETERZIL and S. STARCHENKO, Uniform definability of theWeierstrass ℘-functions and generalized tori of dimension oneSelecta Math. (N.S.) 10 (2004), 525–550.
• , Definability of restricted theta functions and families ofabelian varietiesDuke Math. J. 162 (2013), 731–765.
. . . plus four more papers!
![Page 11: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/11.jpg)
The citation for the Karp Prize 2014 mentions . . .
• J. PILA, O-minimality and the André-Oort conjecture for Cn
Ann. of Math. 173 (2011), 1779–1840.
• J. PILA and A. J. WILKIE, The rational points of a definable setDuke Math. J. 133 (2006), 591–616.
• Y. PETERZIL and S. STARCHENKO, Uniform definability of theWeierstrass ℘-functions and generalized tori of dimension oneSelecta Math. (N.S.) 10 (2004), 525–550.
• , Definability of restricted theta functions and families ofabelian varietiesDuke Math. J. 162 (2013), 731–765.
. . . plus four more papers!
![Page 12: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/12.jpg)
An algebraic geometry primer
1 A variety V Ď Cn is the zero set of finitely manypolynomials P1, . . . , Pm P CrX1, . . . , Xns:
V “
x P Cn : P1pxq “ ¨ ¨ ¨ “ Pmpxq “ 0(
.
Hypersurface = zero set of a single polynomial.2 The varieties form the closed sets of a topology on Cn, the
ZARISKI topology. Below: dense = ZARISKI-dense.3 A variety V is irreducible if it is not the union of two
varieties properly contained in V .4 Every variety V is the union of a finite number of
irreducible varieties; this decomposition is unique if oneremoves those subsets that are contained in another one,and the elements of this unique decomposition are calledirreducible components of V .
![Page 13: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/13.jpg)
Diophantine Geometry
Diophantine Geometry studies how the geometric features of avariety V Ď Cn interact with its diophantine properties. Forexample, given an “interesting” set K Ď C, how does thegeometry of V influence the structure of V pKq :“ Kn X V ?
A general principle
If V is a special variety and X Ď V is a variety which contains adense set of special points, then X, too, has to be special.(Whatever “special” means.)
A conjecture, due to ANDRÉ (1989) and OORT (1995), statesthat so-called SHIMURA varieties V obey such a principle.
PILA-ZANNIER found a general procedure using o-minimality toprove instances of the “general principle.” This allowed PILA togive the first unconditional proof (not relying, e.g., on theRIEMANN Hypothesis) of ANDRÉ-OORT for the case V “ Cn.
![Page 14: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/14.jpg)
Diophantine Geometry
Diophantine Geometry studies how the geometric features of avariety V Ď Cn interact with its diophantine properties. Forexample, given an “interesting” set K Ď C, how does thegeometry of V influence the structure of V pKq :“ Kn X V ?
A general principle
If V is a special variety and X Ď V is a variety which contains adense set of special points, then X, too, has to be special.(Whatever “special” means.)
A conjecture, due to ANDRÉ (1989) and OORT (1995), statesthat so-called SHIMURA varieties V obey such a principle.
PILA-ZANNIER found a general procedure using o-minimality toprove instances of the “general principle.” This allowed PILA togive the first unconditional proof (not relying, e.g., on theRIEMANN Hypothesis) of ANDRÉ-OORT for the case V “ Cn.
![Page 15: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/15.jpg)
Diophantine Geometry
Diophantine Geometry studies how the geometric features of avariety V Ď Cn interact with its diophantine properties. Forexample, given an “interesting” set K Ď C, how does thegeometry of V influence the structure of V pKq :“ Kn X V ?
A general principle
If V is a special variety and X Ď V is a variety which contains adense set of special points, then X, too, has to be special.(Whatever “special” means.)
A conjecture, due to ANDRÉ (1989) and OORT (1995), statesthat so-called SHIMURA varieties V obey such a principle.
PILA-ZANNIER found a general procedure using o-minimality toprove instances of the “general principle.” This allowed PILA togive the first unconditional proof (not relying, e.g., on theRIEMANN Hypothesis) of ANDRÉ-OORT for the case V “ Cn.
![Page 16: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/16.jpg)
Diophantine Geometry
Diophantine Geometry studies how the geometric features of avariety V Ď Cn interact with its diophantine properties. Forexample, given an “interesting” set K Ď C, how does thegeometry of V influence the structure of V pKq :“ Kn X V ?
A general principle
If V is a special variety and X Ď V is a variety which contains adense set of special points, then X, too, has to be special.(Whatever “special” means.)
A conjecture, due to ANDRÉ (1989) and OORT (1995), statesthat so-called SHIMURA varieties V obey such a principle.
PILA-ZANNIER found a general procedure using o-minimality toprove instances of the “general principle.” This allowed PILA togive the first unconditional proof (not relying, e.g., on theRIEMANN Hypothesis) of ANDRÉ-OORT for the case V “ Cn.
![Page 17: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/17.jpg)
Diophantine Geometry
Here is an archetypical example of the “general principle.” Put
U :“
z P C : zn “ 1 for some n ě 1(
(roots of unity).
The elements of U are our special points.
Theorem (LAURENT, 1984)
Let X Ď pCˆqn be irreducible. If XpUq is dense in X, then X isdefined by equations
Xα11 ¨ ¨ ¨Xαn
n “ b pα1, . . . , αn P Z, b P Uq.
This is an instance of the MANIN-MUMFORD Conjecture(= RAYNAUD’s Theorem). The PILA-ZANNIER method (extendedby PETERZIL-STARCHENKO) gives (yet) another proof.
![Page 18: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/18.jpg)
Diophantine Geometry
Here is an archetypical example of the “general principle.” Put
U :“
z P C : zn “ 1 for some n ě 1(
(roots of unity).
The elements of U are our special points.
Theorem (LAURENT, 1984)
Let X Ď pCˆqn be irreducible. If XpUq is dense in X, then X isdefined by equations
Xα11 ¨ ¨ ¨Xαn
n “ b pα1, . . . , αn P Z, b P Uq.
This is an instance of the MANIN-MUMFORD Conjecture(= RAYNAUD’s Theorem). The PILA-ZANNIER method (extendedby PETERZIL-STARCHENKO) gives (yet) another proof.
![Page 19: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/19.jpg)
Diophantine Geometry
Here is an archetypical example of the “general principle.” Put
U :“
z P C : zn “ 1 for some n ě 1(
(roots of unity).
The elements of U are our special points.
Theorem (LAURENT, 1984)
Let X Ď pCˆqn be irreducible. If XpUq is dense in X, then X isdefined by equations
Xα11 ¨ ¨ ¨Xαn
n “ b pα1, . . . , αn P Z, b P Uq.
This is an instance of the MANIN-MUMFORD Conjecture(= RAYNAUD’s Theorem). The PILA-ZANNIER method (extendedby PETERZIL-STARCHENKO) gives (yet) another proof.
![Page 20: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/20.jpg)
The method of PILA-ZANNIER
The main idea
• We have an analytic surjection
e : Cn Ñ pCˆqn, epz1, . . . , znq “ pe2πiz1 , . . . , e2πiznq.
Note: ζ P Un ðñ ζ “ epzq for some z P Qn.• e has a fundamental domain:
D :“
pz1, . . . , znq P Cn : 0 ď Repziq ă 1 for each i(
.
Then with re :“ e æ D, we still have
ζ P Un ðñ ζ “ repzq for some z P D XQn.
![Page 21: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/21.jpg)
The method of PILA-ZANNIER
The main idea
• We have an analytic surjection
e : Cn Ñ pCˆqn, epz1, . . . , znq “ pe2πiz1 , . . . , e2πiznq.
Note: ζ P Un ðñ ζ “ epzq for some z P Qn.
• e has a fundamental domain:
D :“
pz1, . . . , znq P Cn : 0 ď Repziq ă 1 for each i(
.
Then with re :“ e æ D, we still have
ζ P Un ðñ ζ “ repzq for some z P D XQn.
![Page 22: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/22.jpg)
The method of PILA-ZANNIER
The main idea
• We have an analytic surjection
e : Cn Ñ pCˆqn, epz1, . . . , znq “ pe2πiz1 , . . . , e2πiznq.
Note: ζ P Un ðñ ζ “ epzq for some z P Qn.• e has a fundamental domain:
D :“
pz1, . . . , znq P Cn : 0 ď Repziq ă 1 for each i(
.
Then with re :“ e æ D, we still have
ζ P Un ðñ ζ “ repzq for some z P D XQn.
![Page 23: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/23.jpg)
The method of PILA-ZANNIER
The main idea
• We have an analytic surjection
e : Cn Ñ pCˆqn, epz1, . . . , znq “ pe2πiz1 , . . . , e2πiznq.
Note: ζ P Un ðñ ζ “ epzq for some z P Qn.• e has a fundamental domain:
D :“
pz1, . . . , znq P Cn : 0 ď Repziq ă 1 for each i(
.
Then with re :“ e æ D, we still have
ζ P Un ðñ ζ “ repzq for some z P D XQn.
![Page 24: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/24.jpg)
The method of PILA-ZANNIER
e is “logically” badly behaved (itskernel is Zn), but re and thus
rX :“ re´1pXq
are definable in the o-minimalstructure`
R;ă, 0, 1,`,ˆ, exp, sin æ r0, 2πs˘
,
with rXpQq “ re´1`
XpUq˘
.
0
i
1
D Cˆre
(Identify C with R2.)
ea`ib “ eapcos b` i sin bq
(Definability in an o-minimal structure is obvious in this case,but by far non-obvious in many other applications of thePILA-ZANNIER method Ñ PETERZIL-STARCHENKO.)
![Page 25: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/25.jpg)
The method of PILA-ZANNIER
e is “logically” badly behaved (itskernel is Zn), but re and thus
rX :“ re´1pXq
are definable in the o-minimalstructure`
R;ă, 0, 1,`,ˆ, exp, sin æ r0, 2πs˘
,
with rXpQq “ re´1`
XpUq˘
.
0
i
1
D Cˆre
(Identify C with R2.)
ea`ib “ eapcos b` i sin bq
(Definability in an o-minimal structure is obvious in this case,but by far non-obvious in many other applications of thePILA-ZANNIER method Ñ PETERZIL-STARCHENKO.)
![Page 26: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/26.jpg)
The citation for the Karp Prize 2014 mentions . . .
• J. PILA, O-minimality and the André-Oort conjecture for Cn
Ann. of Math. 173 (2011), 1779–1840.
• J. PILA and A. J. WILKIE, The rational points of a definable setDuke Math. J. 133 (2006), 591–616.
• Y. PETERZIL and S. STARCHENKO, Uniform definability of theWeierstrass ℘-functions and generalized tori of dimension oneSelecta Math. (N.S.) 10 (2004), 525–550.
• , Definability of restricted theta functions and families ofabelian varietiesDuke Math. J. 162 (2013), 731–765.
. . . plus four more papers!
![Page 27: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/27.jpg)
The method of PILA-ZANNIER
Split rX “ rXalgloomoon
algebraic part
9Y rXtransloomoon
transcendental part
(to be defined).
Strategy
1 The upper bound: Prove that rXtranspQq is “small.”
[Follows from definability of rX and a theorem of PILA-WILKIE.]
2 The lower bound: Suppose that X contains a dense set ofspecial points (here: that XpUq is dense in X). Show that thisimplies that rXtranspQq actually is finite.
[Involves an automorphism argument and some number theory;here, only simple properties of EULER’s ϕ-function.]
3 Analyze rXalgpQq: Let A be a variety contained in e´1pXq; takesuch A maximal and irreducible. Show that A is an affinesubspace of Cn defined over Q.
[Uses AX’ functional analogue of LINDEMANN-WEIERSTRASS.]
![Page 28: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/28.jpg)
The method of PILA-ZANNIER
Split rX “ rXalgloomoon
algebraic part
9Y rXtransloomoon
transcendental part
(to be defined).
Strategy
1 The upper bound: Prove that rXtranspQq is “small.”
[Follows from definability of rX and a theorem of PILA-WILKIE.]2 The lower bound: Suppose that X contains a dense set of
special points (here: that XpUq is dense in X). Show that thisimplies that rXtranspQq actually is finite.
[Involves an automorphism argument and some number theory;here, only simple properties of EULER’s ϕ-function.]
3 Analyze rXalgpQq: Let A be a variety contained in e´1pXq; takesuch A maximal and irreducible. Show that A is an affinesubspace of Cn defined over Q.
[Uses AX’ functional analogue of LINDEMANN-WEIERSTRASS.]
![Page 29: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/29.jpg)
The method of PILA-ZANNIER
Split rX “ rXalgloomoon
algebraic part
9Y rXtransloomoon
transcendental part
(to be defined).
Strategy
1 The upper bound: Prove that rXtranspQq is “small.”
[Follows from definability of rX and a theorem of PILA-WILKIE.]
2 The lower bound: Suppose that X contains a dense set ofspecial points (here: that XpUq is dense in X). Show that thisimplies that rXtranspQq actually is finite.
[Involves an automorphism argument and some number theory;here, only simple properties of EULER’s ϕ-function.]
3 Analyze rXalgpQq: Let A be a variety contained in e´1pXq; takesuch A maximal and irreducible. Show that A is an affinesubspace of Cn defined over Q.
[Uses AX’ functional analogue of LINDEMANN-WEIERSTRASS.]
![Page 30: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/30.jpg)
The method of PILA-ZANNIER
Split rX “ rXalgloomoon
algebraic part
9Y rXtransloomoon
transcendental part
(to be defined).
Strategy
1 The upper bound: Prove that rXtranspQq is “small.”
[Follows from definability of rX and a theorem of PILA-WILKIE.]
2 The lower bound: Suppose that X contains a dense set ofspecial points (here: that XpUq is dense in X). Show that thisimplies that rXtranspQq actually is finite.
[Involves an automorphism argument and some number theory;here, only simple properties of EULER’s ϕ-function.]
3 Analyze rXalgpQq: Let A be a variety contained in e´1pXq; takesuch A maximal and irreducible. Show that A is an affinesubspace of Cn defined over Q.
[Uses AX’ functional analogue of LINDEMANN-WEIERSTRASS.]
![Page 31: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/31.jpg)
The method of PILA-ZANNIER
Split rX “ rXalgloomoon
algebraic part
9Y rXtransloomoon
transcendental part
(to be defined).
Strategy
1 The upper bound: Prove that rXtranspQq is “small.”[Follows from definability of rX and a theorem of PILA-WILKIE.]
2 The lower bound: Suppose that X contains a dense set ofspecial points (here: that XpUq is dense in X). Show that thisimplies that rXtranspQq actually is finite.
[Involves an automorphism argument and some number theory;here, only simple properties of EULER’s ϕ-function.]
3 Analyze rXalgpQq: Let A be a variety contained in e´1pXq; takesuch A maximal and irreducible. Show that A is an affinesubspace of Cn defined over Q.
[Uses AX’ functional analogue of LINDEMANN-WEIERSTRASS.]
![Page 32: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/32.jpg)
The method of PILA-ZANNIER
Split rX “ rXalgloomoon
algebraic part
9Y rXtransloomoon
transcendental part
(to be defined).
Strategy
1 The upper bound: Prove that rXtranspQq is “small.”[Follows from definability of rX and a theorem of PILA-WILKIE.]
2 The lower bound: Suppose that X contains a dense set ofspecial points (here: that XpUq is dense in X). Show that thisimplies that rXtranspQq actually is finite.[Involves an automorphism argument and some number theory;here, only simple properties of EULER’s ϕ-function.]
3 Analyze rXalgpQq: Let A be a variety contained in e´1pXq; takesuch A maximal and irreducible. Show that A is an affinesubspace of Cn defined over Q.
[Uses AX’ functional analogue of LINDEMANN-WEIERSTRASS.]
![Page 33: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/33.jpg)
The method of PILA-ZANNIER
Split rX “ rXalgloomoon
algebraic part
9Y rXtransloomoon
transcendental part
(to be defined).
Strategy
1 The upper bound: Prove that rXtranspQq is “small.”[Follows from definability of rX and a theorem of PILA-WILKIE.]
2 The lower bound: Suppose that X contains a dense set ofspecial points (here: that XpUq is dense in X). Show that thisimplies that rXtranspQq actually is finite.[Involves an automorphism argument and some number theory;here, only simple properties of EULER’s ϕ-function.]
3 Analyze rXalgpQq: Let A be a variety contained in e´1pXq; takesuch A maximal and irreducible. Show that A is an affinesubspace of Cn defined over Q.[Uses AX’ functional analogue of LINDEMANN-WEIERSTRASS.]
![Page 34: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/34.jpg)
The method of PILA-ZANNIER
Split rX “ rXalgloomoon
algebraic part
9Y rXtransloomoon
transcendental part
(to be defined).
Strategy
1 The upper bound: Prove that rXtranspQq is “small.”[Follows from definability of rX and a theorem of PILA-WILKIE.]
2 The lower bound: Suppose that X contains a dense set ofspecial points (here: that XpUq is dense in X). Show that thisimplies that rXtranspQq actually is finite.[Involves an automorphism argument and some number theory;here, only simple properties of EULER’s ϕ-function.]
3 Analyze rXalgpQq: Let A be a variety contained in e´1pXq; takesuch A maximal and irreducible. Show that A is an affinesubspace of Cn defined over Q.[Uses AX’ functional analogue of LINDEMANN-WEIERSTRASS.]
![Page 35: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/35.jpg)
O-minimal structures
O-minimal structures were introduced 30 years ago (VAN DEN
DRIES, PILLAY-STEINHORN) in order to provide an analogue forthe model-theoretic tameness notion of strong minimality in anordered context (“o-minimal” = “order-minimal”).
Let R “ pR;ă, . . . q be an expansion of the ordered set of reals.
(O-minimality can be developed if instead of pR;ăq we take anylinearly ordered set pR;ăq without endpoints, and it is indeeduseful to have the extra flexibility.)
Below, “definable” means “definable in R, possibly withparameters.” A map f : S Ñ Rn, where S Ď Rm, is calleddefinable if its graph Γpfq Ď Rm`n is.
![Page 36: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/36.jpg)
O-minimal structures
O-minimal structures were introduced 30 years ago (VAN DEN
DRIES, PILLAY-STEINHORN) in order to provide an analogue forthe model-theoretic tameness notion of strong minimality in anordered context (“o-minimal” = “order-minimal”).
Let R “ pR;ă, . . . q be an expansion of the ordered set of reals.
(O-minimality can be developed if instead of pR;ăq we take anylinearly ordered set pR;ăq without endpoints, and it is indeeduseful to have the extra flexibility.)
Below, “definable” means “definable in R, possibly withparameters.” A map f : S Ñ Rn, where S Ď Rm, is calleddefinable if its graph Γpfq Ď Rm`n is.
![Page 37: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/37.jpg)
O-minimal structures
O-minimal structures were introduced 30 years ago (VAN DEN
DRIES, PILLAY-STEINHORN) in order to provide an analogue forthe model-theoretic tameness notion of strong minimality in anordered context (“o-minimal” = “order-minimal”).
Let R “ pR;ă, . . . q be an expansion of the ordered set of reals.
(O-minimality can be developed if instead of pR;ăq we take anylinearly ordered set pR;ăq without endpoints, and it is indeeduseful to have the extra flexibility.)
Below, “definable” means “definable in R, possibly withparameters.” A map f : S Ñ Rn, where S Ď Rm, is calleddefinable if its graph Γpfq Ď Rm`n is.
![Page 38: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/38.jpg)
O-minimal structures
O-minimal structures were introduced 30 years ago (VAN DEN
DRIES, PILLAY-STEINHORN) in order to provide an analogue forthe model-theoretic tameness notion of strong minimality in anordered context (“o-minimal” = “order-minimal”).
Let R “ pR;ă, . . . q be an expansion of the ordered set of reals.
(O-minimality can be developed if instead of pR;ăq we take anylinearly ordered set pR;ăq without endpoints, and it is indeeduseful to have the extra flexibility.)
Below, “definable” means “definable in R, possibly withparameters.” A map f : S Ñ Rn, where S Ď Rm, is calleddefinable if its graph Γpfq Ď Rm`n is.
![Page 39: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/39.jpg)
O-minimal structures
Definition
R is o-minimal :ðñ
# all definable subsets of R are finiteunions of singletons and (open)intervals
ðñ
# all definable subsets of R havefinitely many connected compo-nents
ðñ
$
&
%
the definable subsets of R arethose that are already definable inthe reduct pR;ăq of R
![Page 40: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/40.jpg)
O-minimal structures
Why do we care about o-minimality?
Although the o-minimality axiom only refers to definablesubsets of R, it implies finiteness properties for the definablesubsets of Rn for arbitrary n ě 1.
Some points in case
• Cell Decomposition Theorem ñ definable subsets of Rnhave only finitely many connected components.
• Definable maps S Ñ Rn (S Ď Rm) are very regular, e.g.,• piecewise differentiable up to some fixed finite order;• the finite fibers have uniformly bounded cardinality.
• Dimension of definable sets is very well-behaved, e.g.,invariant under definable bijections: no space-filling curves.
![Page 41: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/41.jpg)
O-minimal structures
Why do we care about o-minimality?
Although the o-minimality axiom only refers to definablesubsets of R, it implies finiteness properties for the definablesubsets of Rn for arbitrary n ě 1.
Some points in case
• Cell Decomposition Theorem ñ definable subsets of Rnhave only finitely many connected components.
• Definable maps S Ñ Rn (S Ď Rm) are very regular, e.g.,• piecewise differentiable up to some fixed finite order;• the finite fibers have uniformly bounded cardinality.
• Dimension of definable sets is very well-behaved, e.g.,invariant under definable bijections: no space-filling curves.
![Page 42: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/42.jpg)
O-minimal structures
Why do we care about o-minimality?
Although the o-minimality axiom only refers to definablesubsets of R, it implies finiteness properties for the definablesubsets of Rn for arbitrary n ě 1.
Some points in case
• Cell Decomposition Theorem ñ definable subsets of Rnhave only finitely many connected components.
• Definable maps S Ñ Rn (S Ď Rm) are very regular, e.g.,• piecewise differentiable up to some fixed finite order;
• the finite fibers have uniformly bounded cardinality.
• Dimension of definable sets is very well-behaved, e.g.,invariant under definable bijections: no space-filling curves.
![Page 43: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/43.jpg)
O-minimal structures
Why do we care about o-minimality?
Although the o-minimality axiom only refers to definablesubsets of R, it implies finiteness properties for the definablesubsets of Rn for arbitrary n ě 1.
Some points in case
• Cell Decomposition Theorem ñ definable subsets of Rnhave only finitely many connected components.
• Definable maps S Ñ Rn (S Ď Rm) are very regular, e.g.,• piecewise differentiable up to some fixed finite order;• the finite fibers have uniformly bounded cardinality.
• Dimension of definable sets is very well-behaved, e.g.,invariant under definable bijections: no space-filling curves.
![Page 44: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/44.jpg)
O-minimal structures
Why do we care about o-minimality?
Although the o-minimality axiom only refers to definablesubsets of R, it implies finiteness properties for the definablesubsets of Rn for arbitrary n ě 1.
Some points in case
• Cell Decomposition Theorem ñ definable subsets of Rnhave only finitely many connected components.
• Definable maps S Ñ Rn (S Ď Rm) are very regular, e.g.,• piecewise differentiable up to some fixed finite order;• the finite fibers have uniformly bounded cardinality.
• Dimension of definable sets is very well-behaved, e.g.,invariant under definable bijections: no space-filling curves.
![Page 45: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/45.jpg)
O-minimal structures
As logicians we know that geometric and topologicalconstructions of a finitary nature preserve definability.
Example
If S Ď Rn is definable, then so is its closure
clpSq “
x P Rn : @ε ą 0 Dy P S : |x´ y| ă ε(
.
(Here we assume that R expands pR;ă, 0, 1,`,ˆq.)
Each o-minimal structure R gives rise to a self-containeduniverse for a kind of “tame topology” (no pathologies) asenvisaged by GROTHENDIECK (1980s).
![Page 46: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/46.jpg)
O-minimal structures
As logicians we know that geometric and topologicalconstructions of a finitary nature preserve definability.
Example
If S Ď Rn is definable, then so is its closure
clpSq “
x P Rn : @ε ą 0 Dy P S : |x´ y| ă ε(
.
(Here we assume that R expands pR;ă, 0, 1,`,ˆq.)
Each o-minimal structure R gives rise to a self-containeduniverse for a kind of “tame topology” (no pathologies) asenvisaged by GROTHENDIECK (1980s).
![Page 47: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/47.jpg)
O-minimal structures
As logicians we know that geometric and topologicalconstructions of a finitary nature preserve definability.
Example
If S Ď Rn is definable, then so is its closure
clpSq “
x P Rn : @ε ą 0 Dy P S : |x´ y| ă ε(
.
(Here we assume that R expands pR;ă, 0, 1,`,ˆq.)
Each o-minimal structure R gives rise to a self-containeduniverse for a kind of “tame topology” (no pathologies) asenvisaged by GROTHENDIECK (1980s).
![Page 48: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/48.jpg)
O-minimal structures: examples
Ralg “ pR;ă, 0, 1,`,ˆq
TARSKI, 1940s
Ran “`
Ralg,
f : r´1, 1sn Ñ Rrestricted analytic, n P Ně1
(˘
VAN DEN DRIES, 1986VAN DEN DRIES-DENEF, 1988
Rexp “ pRalg, expq
WILKIE, 1991
Ran,exp “ pRan, expq
VAN DEN DRIES-MILLER, 1992MACINTYRE-MARKER-VAN DEN DRIES, 1994
semialgebraic sets
![Page 49: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/49.jpg)
O-minimal structures: examples
Ralg “ pR;ă, 0, 1,`,ˆq
TARSKI, 1940s
Ran “`
Ralg,
f : r´1, 1sn Ñ Rrestricted analytic, n P Ně1
(˘
VAN DEN DRIES, 1986VAN DEN DRIES-DENEF, 1988
Rexp “ pRalg, expq
WILKIE, 1991
Ran,exp “ pRan, expq
VAN DEN DRIES-MILLER, 1992MACINTYRE-MARKER-VAN DEN DRIES, 1994
semialgebraic sets
![Page 50: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/50.jpg)
O-minimal structures: examples
Ralg “ pR;ă, 0, 1,`,ˆq
TARSKI, 1940s
Ran “`
Ralg,
f : r´1, 1sn Ñ Rrestricted analytic, n P Ně1
(˘
VAN DEN DRIES, 1986VAN DEN DRIES-DENEF, 1988
Rexp “ pRalg, expq
WILKIE, 1991
Ran,exp “ pRan, expq
VAN DEN DRIES-MILLER, 1992MACINTYRE-MARKER-VAN DEN DRIES, 1994
semialgebraic sets
![Page 51: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/51.jpg)
O-minimal structures: examples
Ralg “ pR;ă, 0, 1,`,ˆq
TARSKI, 1940s
Ran “`
Ralg,
f : r´1, 1sn Ñ Rrestricted analytic, n P Ně1
(˘
VAN DEN DRIES, 1986VAN DEN DRIES-DENEF, 1988
Rexp “ pRalg, expq
WILKIE, 1991
Ran,exp “ pRan, expq
VAN DEN DRIES-MILLER, 1992MACINTYRE-MARKER-VAN DEN DRIES, 1994
semialgebraic sets
![Page 52: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/52.jpg)
O-minimal structures: examples
Ralg “ pR;ă, 0, 1,`,ˆq
TARSKI, 1940s
Ran “`
Ralg,
f : r´1, 1sn Ñ Rrestricted analytic, n P Ně1
(˘
VAN DEN DRIES, 1986VAN DEN DRIES-DENEF, 1988
Rexp “ pRalg, expq
WILKIE, 1991
Ran,exp “ pRan, expq
VAN DEN DRIES-MILLER, 1992MACINTYRE-MARKER-VAN DEN DRIES, 1994
semialgebraic sets
![Page 53: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/53.jpg)
The citation for the Karp Prize 2014 mentions . . .
• J. PILA, O-minimality and the André-Oort conjecture for Cn
Ann. of Math. 173 (2011), 1779–1840.
• J. PILA and A. J. WILKIE, The rational points of a definable setDuke Math. J. 133 (2006), 591–616.
• Y. PETERZIL and S. STARCHENKO, Uniform definability of theWeierstrass ℘-functions and generalized tori of dimension oneSelecta Math. (N.S.) 10 (2004), 525–550.
• , Definability of restricted theta functions and families ofabelian varietiesDuke Math. J. 162 (2013), 731–765.
. . . plus four more papers!
![Page 54: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/54.jpg)
O-minimal structures: diophantine properties
Fix an o-minimal expansion R “ pR;ă, 0, 1,`,ˆ, . . . q of Ralg.
More than ten years ago, WILKIE realized that the geometry ofdefinable sets influences the distribution of integer points (=points with integer coordinates) on 1-dimensional definablesets.
Around the same time, and developing earlier ideas ofBOMBIERI-PILA (1989), PILA studied rational points on curvesand surfaces definable in Ran.
These developments culminated in the theorem ofPILA-WILKIE (2006):
Definable sets which are sufficiently “transcendental” containfew rational points.
![Page 55: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/55.jpg)
O-minimal structures: diophantine properties
Fix an o-minimal expansion R “ pR;ă, 0, 1,`,ˆ, . . . q of Ralg.
More than ten years ago, WILKIE realized that the geometry ofdefinable sets influences the distribution of integer points (=points with integer coordinates) on 1-dimensional definablesets.
Around the same time, and developing earlier ideas ofBOMBIERI-PILA (1989), PILA studied rational points on curvesand surfaces definable in Ran.
These developments culminated in the theorem ofPILA-WILKIE (2006):
Definable sets which are sufficiently “transcendental” containfew rational points.
![Page 56: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/56.jpg)
O-minimal structures: diophantine properties
Fix an o-minimal expansion R “ pR;ă, 0, 1,`,ˆ, . . . q of Ralg.
More than ten years ago, WILKIE realized that the geometry ofdefinable sets influences the distribution of integer points (=points with integer coordinates) on 1-dimensional definablesets.
Around the same time, and developing earlier ideas ofBOMBIERI-PILA (1989), PILA studied rational points on curvesand surfaces definable in Ran.
These developments culminated in the theorem ofPILA-WILKIE (2006):
Definable sets which are sufficiently “transcendental” containfew rational points.
![Page 57: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/57.jpg)
O-minimal structures: diophantine properties
Fix an o-minimal expansion R “ pR;ă, 0, 1,`,ˆ, . . . q of Ralg.
More than ten years ago, WILKIE realized that the geometry ofdefinable sets influences the distribution of integer points (=points with integer coordinates) on 1-dimensional definablesets.
Around the same time, and developing earlier ideas ofBOMBIERI-PILA (1989), PILA studied rational points on curvesand surfaces definable in Ran.
These developments culminated in the theorem ofPILA-WILKIE (2006):
Definable sets which are sufficiently “transcendental” containfew rational points.
![Page 58: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/58.jpg)
O-minimal structures: diophantine properties
NotationGiven non-zero coprime a, b P Z define the height of x “ a
b byHpxq :“ maxt|a|, |b|u, and set Hp0q :“ 0.
We also define a height function Qn Ñ N, still denoted by H:
Hpx1, . . . , xnq :“ max
Hpx1q, . . . ,Hpxnq(
.
Given X Ď Rn and t P R, put
XpQ, tq :“
x P X XQn : Hpxq ď t(
(a finite set).
We’d like to understand the asymptotic behavior of |XpQ, tq|.
![Page 59: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/59.jpg)
O-minimal structures: diophantine properties
NotationGiven non-zero coprime a, b P Z define the height of x “ a
b byHpxq :“ maxt|a|, |b|u, and set Hp0q :“ 0.
We also define a height function Qn Ñ N, still denoted by H:
Hpx1, . . . , xnq :“ max
Hpx1q, . . . ,Hpxnq(
.
Given X Ď Rn and t P R, put
XpQ, tq :“
x P X XQn : Hpxq ď t(
(a finite set).
We’d like to understand the asymptotic behavior of |XpQ, tq|.
![Page 60: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/60.jpg)
O-minimal structures: diophantine properties
NotationGiven non-zero coprime a, b P Z define the height of x “ a
b byHpxq :“ maxt|a|, |b|u, and set Hp0q :“ 0.
We also define a height function Qn Ñ N, still denoted by H:
Hpx1, . . . , xnq :“ max
Hpx1q, . . . ,Hpxnq(
.
Given X Ď Rn and t P R, put
XpQ, tq :“
x P X XQn : Hpxq ď t(
(a finite set).
We’d like to understand the asymptotic behavior of |XpQ, tq|.
![Page 61: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/61.jpg)
O-minimal structures: diophantine properties
NotationGiven non-zero coprime a, b P Z define the height of x “ a
b byHpxq :“ maxt|a|, |b|u, and set Hp0q :“ 0.
We also define a height function Qn Ñ N, still denoted by H:
Hpx1, . . . , xnq :“ max
Hpx1q, . . . ,Hpxnq(
.
Given X Ď Rn and t P R, put
XpQ, tq :“
x P X XQn : Hpxq ď t(
(a finite set).
We’d like to understand the asymptotic behavior of |XpQ, tq|.
![Page 62: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/62.jpg)
O-minimal structures: diophantine properties
Example#
X “ ΓpP q where P : Rn´1 Ñ Ris a polynomial function with in-teger coefficients of degree d
+
ñ |XpQ, tq| „ Ct2pn´1q{d
QuestionWhen does |XpQ, tq| grow sub-polynomially as tÑ8?
Given X Ď Rn we let
Xalg :“! union of all infinite connected semial-
gebraic subsets of X
)
algebraicpart of X
Xtrans :“ XzXalg transcendentalpart of X.
(A caveat: even if X is definable, then Xalg in general is not.)
![Page 63: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/63.jpg)
O-minimal structures: diophantine properties
Example#
X “ ΓpP q where P : Rn´1 Ñ Ris a polynomial function with in-teger coefficients of degree d
+
ñ |XpQ, tq| „ Ct2pn´1q{d
QuestionWhen does |XpQ, tq| grow sub-polynomially as tÑ8?
Given X Ď Rn we let
Xalg :“! union of all infinite connected semial-
gebraic subsets of X
)
algebraicpart of X
Xtrans :“ XzXalg transcendentalpart of X.
(A caveat: even if X is definable, then Xalg in general is not.)
![Page 64: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/64.jpg)
O-minimal structures: diophantine properties
Example#
X “ ΓpP q where P : Rn´1 Ñ Ris a polynomial function with in-teger coefficients of degree d
+
ñ |XpQ, tq| „ Ct2pn´1q{d
QuestionWhen does |XpQ, tq| grow sub-polynomially as tÑ8?
Given X Ď Rn we let
Xalg :“! union of all infinite connected semial-
gebraic subsets of X
)
algebraicpart of X
Xtrans :“ XzXalg transcendentalpart of X.
(A caveat: even if X is definable, then Xalg in general is not.)
![Page 65: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/65.jpg)
O-minimal structures: diophantine properties
Theorem (PILA-WILKIE, 2006)
Let X Ď Rn be definable. Then for each ε ą 0 there is somet0 “ t0pεq such that
|XtranspQ, tq| ď tε for all t ě t0.
Remark
• The theorem continues to hold if given d ě 1, we replace
Q set of algebraic numbers of degree ď d
H a suitable height function on Qalg.
(PILA, 2009)
![Page 66: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/66.jpg)
O-minimal structures: diophantine properties
Theorem (PILA-WILKIE, 2006)
Let X Ď Rn be definable. Then for each ε ą 0 there is somet0 “ t0pεq such that
|XtranspQ, tq| ď tε for all t ě t0.
Remark
• The theorem continues to hold if given d ě 1, we replace
Q set of algebraic numbers of degree ď d
H a suitable height function on Qalg.
(PILA, 2009)
![Page 67: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/67.jpg)
O-minimal structures: diophantine properties
Two crucial ingredients in the proof:• a parametrization theorem for bounded definable sets by
maps with bounded derivatives(generalizing YOMDIN and GROMOV);
• a result about covering the rational points of suchparametrized sets by few algebraic hypersurfaces(no definability assumptions here).
![Page 68: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/68.jpg)
O-minimal structures: diophantine properties
Theorem I (parametrization)
Let X Ď p0, 1qn be definable, non-empty, and p P N. There is afinite set Φ of definable maps φ : p0, 1qdimX Ñ p0, 1qn such that
• the union of the images of the φ P Φ equals X;• each φ P Φ is Cp with ||φpαq|| ď 1 for |α| ď p.
Theorem II (covering by hypersurfaces)
Let m ă n and d be given. Then there are p P N and ε, C ą 0with the following properties:
• if φ : p0, 1qm Ñ Rn is Cp with image X such that ||φpαq|| ď 1for |α| ď p, then for each t, XpQ, tq is contained in theunion of Ctε hypersurfaces of degree d;
• ε “ εpm,n, dq Ñ 0 as dÑ8.
![Page 69: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/69.jpg)
O-minimal structures: diophantine properties
Theorem I (parametrization)
Let X Ď p0, 1qn be definable, non-empty, and p P N. There is afinite set Φ of definable maps φ : p0, 1qdimX Ñ p0, 1qn such that
• the union of the images of the φ P Φ equals X;• each φ P Φ is Cp with ||φpαq|| ď 1 for |α| ď p.
Theorem II (covering by hypersurfaces)
Let m ă n and d be given. Then there are p P N and ε, C ą 0with the following properties:
• if φ : p0, 1qm Ñ Rn is Cp with image X such that ||φpαq|| ď 1for |α| ď p, then for each t, XpQ, tq is contained in theunion of Ctε hypersurfaces of degree d;
• ε “ εpm,n, dq Ñ 0 as dÑ8.
![Page 70: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/70.jpg)
O-minimal structures: diophantine properties
Toy case of the PILA-WILKIE Theorem:
X “ Γpfq where f : p0, 1q Ñ p0, 1q is definable.
Let ε ą 0 be given. Choose d so that εp1, 2, dq ď ε, and thenchoose C, p as in Theorem II. Let Φ be a parametrization of Xas in Theorem I. Then XpQ, tq is contained in the union of C1t
ε
hypersurfaces of degree d, where C1 :“ C ¨ |Φ|.
The hypersurfaces H of degree d come in a definable (in fact,semialgebraic) family. So for each such H, either X XH isinfinite (and then X XH Ď Xalg) or finite of uniformly boundedsize (by o-minimality). This allows us to count XtranspQ, tq.
![Page 71: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/71.jpg)
O-minimal structures: diophantine properties
Toy case of the PILA-WILKIE Theorem:
X “ Γpfq where f : p0, 1q Ñ p0, 1q is definable.
Let ε ą 0 be given. Choose d so that εp1, 2, dq ď ε, and thenchoose C, p as in Theorem II. Let Φ be a parametrization of Xas in Theorem I. Then XpQ, tq is contained in the union of C1t
ε
hypersurfaces of degree d, where C1 :“ C ¨ |Φ|.
The hypersurfaces H of degree d come in a definable (in fact,semialgebraic) family. So for each such H, either X XH isinfinite (and then X XH Ď Xalg) or finite of uniformly boundedsize (by o-minimality). This allows us to count XtranspQ, tq.
![Page 72: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/72.jpg)
O-minimal structures: diophantine properties
Toy case of the PILA-WILKIE Theorem:
X “ Γpfq where f : p0, 1q Ñ p0, 1q is definable.
Let ε ą 0 be given. Choose d so that εp1, 2, dq ď ε, and thenchoose C, p as in Theorem II. Let Φ be a parametrization of Xas in Theorem I. Then XpQ, tq is contained in the union of C1t
ε
hypersurfaces of degree d, where C1 :“ C ¨ |Φ|.
The hypersurfaces H of degree d come in a definable (in fact,semialgebraic) family. So for each such H, either X XH isinfinite (and then X XH Ď Xalg) or finite of uniformly boundedsize (by o-minimality). This allows us to count XtranspQ, tq.
![Page 73: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/73.jpg)
Elliptic curves: viewed algebraically
PILA’s Theorem deals with elliptic curves:
y2 “ x3 ` ax` b where a, b P C, 4a3 ‰ ´27b2.
y2 “ x3 ´ 2x
x
y
• plotted in R2
instead of C2;• “elliptic” here has
nothing to do withellipses.
It is natural to add a “point 0 at infinity”:
E “
px, yq P C2 : y2 “ x3 ` ax` b(
Y t0u
![Page 74: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/74.jpg)
Elliptic curves: viewed algebraically
PILA’s Theorem deals with elliptic curves:
y2 “ x3 ` ax` b where a, b P C, 4a3 ‰ ´27b2.
y2 “ x3 ´ 2x
x
y
• plotted in R2
instead of C2;• “elliptic” here has
nothing to do withellipses.
It is natural to add a “point 0 at infinity”:
E “
px, yq P C2 : y2 “ x3 ` ax` b(
Y t0u
![Page 75: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/75.jpg)
Elliptic curves: viewed algebraically
PILA’s Theorem deals with elliptic curves:
y2 “ x3 ` ax` b where a, b P C, 4a3 ‰ ´27b2.
y2 “ x3 ´ 2x
x
y
• plotted in R2
instead of C2;• “elliptic” here has
nothing to do withellipses.
It is natural to add a “point 0 at infinity”:
E “
px, yq P C2 : y2 “ x3 ` ax` b(
Y t0u
![Page 76: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/76.jpg)
Elliptic curves: viewed algebraically
Let E be an elliptic curve. Then E can be made into an abeliangroup with 0 as its identity element.
For p, q, r P E,
p` q ` r “ 0 ðñ p, q, r lie on a line.
x
y
p‚q ‚
‚r
x
y
p “ q‚
r “ 8
This makes E into an algebraic group (the group operations aregiven by rational functions of the coordinates).
![Page 77: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/77.jpg)
Elliptic curves: viewed algebraically
Let E be an elliptic curve. Then E can be made into an abeliangroup with 0 as its identity element. For p, q, r P E,
p` q ` r “ 0 ðñ p, q, r lie on a line.
x
y
p‚q ‚
‚r
x
y
p “ q‚
r “ 8
This makes E into an algebraic group (the group operations aregiven by rational functions of the coordinates).
![Page 78: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/78.jpg)
Elliptic curves: viewed algebraically
Let E be an elliptic curve. Then E can be made into an abeliangroup with 0 as its identity element. For p, q, r P E,
p` q ` r “ 0 ðñ p, q, r lie on a line.
x
y
p‚q ‚
‚r
x
y
p “ q‚
r “ 8
This makes E into an algebraic group (the group operations aregiven by rational functions of the coordinates).
![Page 79: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/79.jpg)
Elliptic curves: viewed analytically
Let E be an elliptic curve.
Then there is a complex analyticsurjective group morphism π : CÑ E whose kernel is a lattice,which (after a change of coordinates) we may express as
Λ “ Z` Zτ where τ P H :“
z P C : Im z ą 0(
.
Conversely, for every τ P H there is an elliptic curve E “ Eτ andan analytic group morphism C� E with kernel Λ “ Z` Zτ .
0
τ
1 C
C{Λ
E
π
–
![Page 80: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/80.jpg)
Elliptic curves: viewed analytically
Let E be an elliptic curve. Then there is a complex analyticsurjective group morphism π : CÑ E whose kernel is a lattice,which (after a change of coordinates) we may express as
Λ “ Z` Zτ where τ P H :“
z P C : Im z ą 0(
.
Conversely, for every τ P H there is an elliptic curve E “ Eτ andan analytic group morphism C� E with kernel Λ “ Z` Zτ .
0
τ
1 C
C{Λ
E
π
–
![Page 81: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/81.jpg)
Elliptic curves: viewed analytically
Let E be an elliptic curve. Then there is a complex analyticsurjective group morphism π : CÑ E whose kernel is a lattice,which (after a change of coordinates) we may express as
Λ “ Z` Zτ where τ P H :“
z P C : Im z ą 0(
.
Conversely, for every τ P H there is an elliptic curve E “ Eτ andan analytic group morphism C� E with kernel Λ “ Z` Zτ .
0
τ
1 C
C{Λ
E
π
–
![Page 82: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/82.jpg)
Elliptic curves: viewed analytically
Let E be an elliptic curve. Then there is a complex analyticsurjective group morphism π : CÑ E whose kernel is a lattice,which (after a change of coordinates) we may express as
Λ “ Z` Zτ where τ P H :“
z P C : Im z ą 0(
.
Conversely, for every τ P H there is an elliptic curve E “ Eτ andan analytic group morphism C� E with kernel Λ “ Z` Zτ .
0
τ
1 C
C{Λ
E
π
–
![Page 83: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/83.jpg)
Elliptic curves: viewed analytically
Thus H is a parameter space for elliptic curves.
But the map
τ ÞÑ (isomorphism class of Eτ )
is not one-to-one:
SLp2,Zq “"ˆ
a bc d
˙
: a, b, c, d P Z, ad´ bc “ 1
*
acts on H via`
a bc d
˘
¨ τ “ aτ`bcτ`d , and for σ, τ P H:
Eσ – Eτ ðñ σ “ Aτ for some A P SLp2,Zq.
The j-invariant (19th century: KLEIN . . . )
There exists a holomorphic surjection j : HÑ C such that
jpσq “ jpτq ðñ Eσ – Eτ .
(So the “correct” parameter space is C – SLp2,ZqzH.)
![Page 84: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/84.jpg)
Elliptic curves: viewed analytically
Thus H is a parameter space for elliptic curves. But the map
τ ÞÑ (isomorphism class of Eτ )
is not one-to-one:
SLp2,Zq “"ˆ
a bc d
˙
: a, b, c, d P Z, ad´ bc “ 1
*
acts on H via`
a bc d
˘
¨ τ “ aτ`bcτ`d , and for σ, τ P H:
Eσ – Eτ ðñ σ “ Aτ for some A P SLp2,Zq.
The j-invariant (19th century: KLEIN . . . )
There exists a holomorphic surjection j : HÑ C such that
jpσq “ jpτq ðñ Eσ – Eτ .
(So the “correct” parameter space is C – SLp2,ZqzH.)
![Page 85: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/85.jpg)
Elliptic curves: viewed analytically
Thus H is a parameter space for elliptic curves. But the map
τ ÞÑ (isomorphism class of Eτ )
is not one-to-one:
SLp2,Zq “"ˆ
a bc d
˙
: a, b, c, d P Z, ad´ bc “ 1
*
acts on H via`
a bc d
˘
¨ τ “ aτ`bcτ`d , and for σ, τ P H:
Eσ – Eτ ðñ σ “ Aτ for some A P SLp2,Zq.
The j-invariant (19th century: KLEIN . . . )
There exists a holomorphic surjection j : HÑ C such that
jpσq “ jpτq ðñ Eσ – Eτ .
(So the “correct” parameter space is C – SLp2,ZqzH.)
![Page 86: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/86.jpg)
Elliptic curves: viewed analytically
Thus H is a parameter space for elliptic curves. But the map
τ ÞÑ (isomorphism class of Eτ )
is not one-to-one:
SLp2,Zq “"ˆ
a bc d
˙
: a, b, c, d P Z, ad´ bc “ 1
*
acts on H via`
a bc d
˘
¨ τ “ aτ`bcτ`d , and for σ, τ P H:
Eσ – Eτ ðñ σ “ Aτ for some A P SLp2,Zq.
The j-invariant (19th century: KLEIN . . . )
There exists a holomorphic surjection j : HÑ C such that
jpσq “ jpτq ðñ Eσ – Eτ .
(So the “correct” parameter space is C – SLp2,ZqzH.)
![Page 87: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/87.jpg)
PILA’s Theorem
This is an analogue of LAURENT’s Theorem with
0
H Cj jpτq “
1
q`744`196884q`¨ ¨ ¨
(q “ e2πiτ )
instead of z ÞÑ e2πiz : CÑ Cˆ.
Theorem (PILA, 2011)
Let X Ď Cn be an irreducible variety. If X contains a dense setof special points, then X is special.
Of course, we need to define what “special” should mean.
![Page 88: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/88.jpg)
PILA’s Theorem
This is an analogue of LAURENT’s Theorem with
0
H Cj jpτq “
1
q`744`196884q`¨ ¨ ¨
(q “ e2πiτ )
instead of z ÞÑ e2πiz : CÑ Cˆ.
Theorem (PILA, 2011)
Let X Ď Cn be an irreducible variety. If X contains a dense setof special points, then X is special.
Of course, we need to define what “special” should mean.
![Page 89: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/89.jpg)
PILA’s Theorem
Let E “ C{Λ be an elliptic curve, where Λ “ Z` Zτ (τ P H).
EndpEq –
α P C : αΛ Ď Λ(
.
Usually EndpEq “ Z. But it may be bigger:
EndpEq ‰ Z ðñ τ satisfies a quadratic equation over Q.
The jpτq P C with τ P H quadratic over Q (“E has CM”) will beour special points. KRONECKER: special ñ algebraic integer.
Why are these points special?
One possible explanation from algebraic number theory:• A finite field extension K of Q has abelian GALOIS groupðñ K Ď Qpζq for some ζ P U. (KRONECKER-WEBER)
• If Eτ has CM, all abelian extensions of Qpτq can similarlybe constructed from U, jpτq, and torsion points of Eτ .
![Page 90: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/90.jpg)
PILA’s Theorem
Let E “ C{Λ be an elliptic curve, where Λ “ Z` Zτ (τ P H).
EndpEq –
α P C : αΛ Ď Λ(
.
Usually EndpEq “ Z. But it may be bigger:
EndpEq ‰ Z ðñ τ satisfies a quadratic equation over Q.
The jpτq P C with τ P H quadratic over Q (“E has CM”) will beour special points. KRONECKER: special ñ algebraic integer.
Why are these points special?
One possible explanation from algebraic number theory:• A finite field extension K of Q has abelian GALOIS groupðñ K Ď Qpζq for some ζ P U. (KRONECKER-WEBER)
• If Eτ has CM, all abelian extensions of Qpτq can similarlybe constructed from U, jpτq, and torsion points of Eτ .
![Page 91: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/91.jpg)
PILA’s Theorem
Let E “ C{Λ be an elliptic curve, where Λ “ Z` Zτ (τ P H).
EndpEq –
α P C : αΛ Ď Λ(
.
Usually EndpEq “ Z. But it may be bigger:
EndpEq ‰ Z ðñ τ satisfies a quadratic equation over Q.
The jpτq P C with τ P H quadratic over Q (“E has CM”) will beour special points. KRONECKER: special ñ algebraic integer.
Why are these points special?
One possible explanation from algebraic number theory:• A finite field extension K of Q has abelian GALOIS groupðñ K Ď Qpζq for some ζ P U. (KRONECKER-WEBER)
• If Eτ has CM, all abelian extensions of Qpτq can similarlybe constructed from U, jpτq, and torsion points of Eτ .
![Page 92: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/92.jpg)
PILA’s Theorem
Let E “ C{Λ be an elliptic curve, where Λ “ Z` Zτ (τ P H).
EndpEq –
α P C : αΛ Ď Λ(
.
Usually EndpEq “ Z. But it may be bigger:
EndpEq ‰ Z ðñ τ satisfies a quadratic equation over Q.
The jpτq P C with τ P H quadratic over Q (“E has CM”) will beour special points. KRONECKER: special ñ algebraic integer.
Why are these points special?
One possible explanation from algebraic number theory:• A finite field extension K of Q has abelian GALOIS groupðñ K Ď Qpζq for some ζ P U. (KRONECKER-WEBER)
• If Eτ has CM, all abelian extensions of Qpτq can similarlybe constructed from U, jpτq, and torsion points of Eτ .
![Page 93: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/93.jpg)
PILA’s Theorem
Let E “ C{Λ be an elliptic curve, where Λ “ Z` Zτ (τ P H).
EndpEq –
α P C : αΛ Ď Λ(
.
Usually EndpEq “ Z. But it may be bigger:
EndpEq ‰ Z ðñ τ satisfies a quadratic equation over Q.
The jpτq P C with τ P H quadratic over Q (“E has CM”) will beour special points. KRONECKER: special ñ algebraic integer.
Why are these points special?
One possible explanation from algebraic number theory:• A finite field extension K of Q has abelian GALOIS groupðñ K Ď Qpζq for some ζ P U. (KRONECKER-WEBER)
• If Eτ has CM, all abelian extensions of Qpτq can similarlybe constructed from U, jpτq, and torsion points of Eτ .
![Page 94: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/94.jpg)
PILA’s Theorem
What should be the special varieties V Ď Cn?
While e is a group morphism, j is not. (H is not even a group!)Still, there are algebraic relations between jpτq and jpnτq:
The nth classical modular polynomial (n P Ně1)
There is an irreducible Φn P ZrX,Y s such that for x, y P C:
Φnpx, yq “ 0 ðñ x “ jpτq, y “ jpnτq for some τ P H.
The Φn are symmetric in X and Y .
For example,
Φ2 “ X3 ´X2Y 2 ` 1488X2Y ´ 162000X2 ` 1488XY 2`
40773375XY ` 8748000000X ` Y 3 ´ 162000Y 2`
8748000000Y ´ 157464000000000
![Page 95: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/95.jpg)
PILA’s Theorem
What should be the special varieties V Ď Cn?
While e is a group morphism, j is not. (H is not even a group!)Still, there are algebraic relations between jpτq and jpnτq:
The nth classical modular polynomial (n P Ně1)
There is an irreducible Φn P ZrX,Y s such that for x, y P C:
Φnpx, yq “ 0 ðñ x “ jpτq, y “ jpnτq for some τ P H.
The Φn are symmetric in X and Y .
For example,
Φ2 “ X3 ´X2Y 2 ` 1488X2Y ´ 162000X2 ` 1488XY 2`
40773375XY ` 8748000000X ` Y 3 ´ 162000Y 2`
8748000000Y ´ 157464000000000
![Page 96: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/96.jpg)
PILA’s Theorem
What should be the special varieties V Ď Cn?
While e is a group morphism, j is not. (H is not even a group!)Still, there are algebraic relations between jpτq and jpnτq:
The nth classical modular polynomial (n P Ně1)
There is an irreducible Φn P ZrX,Y s such that for x, y P C:
Φnpx, yq “ 0 ðñ x “ jpτq, y “ jpnτq for some τ P H.
The Φn are symmetric in X and Y .
For example,
Φ2 “ X3 ´X2Y 2 ` 1488X2Y ´ 162000X2 ` 1488XY 2`
40773375XY ` 8748000000X ` Y 3 ´ 162000Y 2`
8748000000Y ´ 157464000000000
![Page 97: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/97.jpg)
PILA’s Theorem
If τ P H is quadratic, then so is nτ P H, so tΦn “ 0u contains adense set of special points.
DefinitionA variety V Ď Cn is special if it is an irreducible component of avariety defined by equations
Φnpxi, xjq “ 0 and xi “ a where a P C is special.
The proof of PILA’s Theorem follows the earlier pattern:
1 The upper bound: j has a natural (semialgebraic)fundamental domain D, and j æ D is definable in Ran,exp.
2 The lower bound: C. L. SIEGEL’s theorem (1935) on thegrowth of the class number of quadratic number fields.
3 Analysis of the algebraic part: A LINDEMANN-WEIERSTRASS Theorem for j.
![Page 98: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/98.jpg)
PILA’s Theorem
If τ P H is quadratic, then so is nτ P H, so tΦn “ 0u contains adense set of special points.
DefinitionA variety V Ď Cn is special if it is an irreducible component of avariety defined by equations
Φnpxi, xjq “ 0 and xi “ a where a P C is special.
The proof of PILA’s Theorem follows the earlier pattern:
1 The upper bound: j has a natural (semialgebraic)fundamental domain D, and j æ D is definable in Ran,exp.
2 The lower bound: C. L. SIEGEL’s theorem (1935) on thegrowth of the class number of quadratic number fields.
3 Analysis of the algebraic part: A LINDEMANN-WEIERSTRASS Theorem for j.
![Page 99: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/99.jpg)
PILA’s Theorem
If τ P H is quadratic, then so is nτ P H, so tΦn “ 0u contains adense set of special points.
DefinitionA variety V Ď Cn is special if it is an irreducible component of avariety defined by equations
Φnpxi, xjq “ 0 and xi “ a where a P C is special.
The proof of PILA’s Theorem follows the earlier pattern:
1 The upper bound: j has a natural (semialgebraic)fundamental domain D, and j æ D is definable in Ran,exp.
2 The lower bound: C. L. SIEGEL’s theorem (1935) on thegrowth of the class number of quadratic number fields.
3 Analysis of the algebraic part: A LINDEMANN-WEIERSTRASS Theorem for j.
![Page 100: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/100.jpg)
PILA’s Theorem
If τ P H is quadratic, then so is nτ P H, so tΦn “ 0u contains adense set of special points.
DefinitionA variety V Ď Cn is special if it is an irreducible component of avariety defined by equations
Φnpxi, xjq “ 0 and xi “ a where a P C is special.
The proof of PILA’s Theorem follows the earlier pattern:
1 The upper bound: j has a natural (semialgebraic)fundamental domain D, and j æ D is definable in Ran,exp.
2 The lower bound: C. L. SIEGEL’s theorem (1935) on thegrowth of the class number of quadratic number fields.
3 Analysis of the algebraic part: A LINDEMANN-WEIERSTRASS Theorem for j.
![Page 101: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/101.jpg)
PILA’s Theorem
If τ P H is quadratic, then so is nτ P H, so tΦn “ 0u contains adense set of special points.
DefinitionA variety V Ď Cn is special if it is an irreducible component of avariety defined by equations
Φnpxi, xjq “ 0 and xi “ a where a P C is special.
The proof of PILA’s Theorem follows the earlier pattern:
1 The upper bound: j has a natural (semialgebraic)fundamental domain D, and j æ D is definable in Ran,exp.
2 The lower bound: C. L. SIEGEL’s theorem (1935) on thegrowth of the class number of quadratic number fields.
3 Analysis of the algebraic part: A LINDEMANN-WEIERSTRASS Theorem for j.
![Page 102: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/102.jpg)
PILA’s Theorem
If τ P H is quadratic, then so is nτ P H, so tΦn “ 0u contains adense set of special points.
DefinitionA variety V Ď Cn is special if it is an irreducible component of avariety defined by equations
Φnpxi, xjq “ 0 and xi “ a where a P C is special.
The proof of PILA’s Theorem follows the earlier pattern:
1 The upper bound: j has a natural (semialgebraic)fundamental domain D, and j æ D is definable in Ran,exp.
2 The lower bound: C. L. SIEGEL’s theorem (1935) on thegrowth of the class number of quadratic number fields.
3 Analysis of the algebraic part: A LINDEMANN-WEIERSTRASS Theorem for j.
![Page 103: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/103.jpg)
Further developments
Many recent results employ and further develop ingredientsfrom this circle of ideas:
• in number theory: ULLMOV-YAFAEV, KLINGER-ULLMO-YAFAEV,PILA-TSIMERMAN, MASSER-ZANNIER, HABEGGER-PILA . . .
• in logic: PETERZIL-STARCHENKO, FREITAG-SCANLON,BIANCONI, THOMAS, JONES-THOMAS, JONES-THOMAS-WILKIE,BAYS-KIRBY-WILKIE, CLUCKERS-COMTE-LOESER, . . .
“It is now widely accepted that a new method has emerged inthis subject.”
![Page 104: Logic meets number theory in o-minimalitymatthias/pdf/Wien.pdf · Logic meets number theory in o-minimality The work of PETERZIL, PILA, STARCHENKO, and WILKIE Logic Colloquium, Vienna](https://reader033.vdocuments.net/reader033/viewer/2022060313/5f0b46c87e708231d42fb7b4/html5/thumbnails/104.jpg)
If you want to know more
SCANLON’s excellent surveys:
• O-minimality as an approach to the André-Oort conjecture,Panor. Synth., to appear.
• Counting special points: logic, Diophantine geometry, andtranscendence theory, Bull. Amer. Math. Soc. 49 (2012), 51–71.
• A proof of the André-Oort conjecture via mathematical logic[after Pila, Wilkie and Zannier ], Séminaire Bourbaki: Vol.2010/2011, Astérisque 348 (2012), 299–315.