on some mathematical connections between fermat’s last theorem, modular functions, modular...

8/7/2019 On some mathematical connections between Fermats Last Theorem, Modular Functions, Modular Elliptic Curves a

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On some mathematical connections between Fermats Last Theorem, Modular Functions,Modular Elliptic Curves and some sector of String Theory

Michele Nardelli 2,1

1 Dipartimento di Scienze della Terra Universit degli Studi di Napoli Federico IILargo S. Marcellino, 10 80138 Napoli (Italy)

2 Dipartimento di Matematica ed Applicazioni R. CaccioppoliUniversit degli Studi di Napoli Federico II Polo delle Scienze e delle Tecnologie

Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli (Italy)

Abstract

This paper is fundamentally a review, a thesis, of principal results obtained in some sectors of Number Theory and String Theory of various authoritative theoretical physicists andmathematicians.Precisely, we have described some mathematical results regarding the Fermats Last Theorem, theMellin transform, the Riemann zeta function, the Ramanujans modular equations, how primes andadeles are related to the Riemann zeta functions and the p-adic and adelic string theory.Furthermore, we show that also the fundamental relationship concerning the Palumbo-Nardelli

model (a general relationship that links bosonic string action and superstring action, i.e. bosonic andfermionic strings in all natural systems), can be related with some equations regarding the p-adic(adelic) string sector.Thence, in conclusion, we have described some new interesting connections that are been obtainedbetween String Theory and Number Theory, with regard the arguments above mentioned.In the Chapters 1 and 2, we have described the mathematics concerning the Fermats LastTheorem, precisely the Wiles approach in the Chapter 1 and further mathematical aspectsconcerning the Fermats Last Theorem, precisely the modular forms, the Euler products, theShimura map and the automorphic L-functions in the Chapter 2 . Furthermore. In this chapter, wehave described also some mathematical applications of the Mellin transform, only mentioned in theChapter 1, the zeta-function quantum field theory and the quantum L-functions.

In the Chapter 3 , we have described how primes and adeles are related to the Riemann zetafunction, precisely the Connes approach. In the Chapter 4 , we have described the p-adic and adelicstrings, precisely the open and closed p-adic strings, the adelic strings, the solitonic q-branes of p-adic string theory and the open and closed scalar zeta strings.In the Chapter 5 , we have described some correlations obtained between some solutions in stringtheory, Riemann zeta function and Palumbo-Nardelli model. Precisely, we have showed thecosmological solutions from the D3/D7 system, the classification and stability of cosmologicalsolutions, the solution applied to ten dimensional IIB supergravity, the connections with someequations concerning the Riemann zeta function, the Palumbo-Nardelli model and the Ramanujansidentities. Furthermore, we have described the interactions between intersecting D-branes and thegeneral action and equations of motion for a probe D3-brane moving through a type IIBsupergravity background. Finally, in the Chapter 6 , we have showed the connections between theequations of the various chapters.


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Introduzione e riassunto

Lultimo teorema di Fermat una generalizzazione dellequazione diofantea 222 cba =+ . Gi gliantichi Greci ed i Babilonesi sapevano che questa equazione ha delle soluzioni intere, come (3, 4, 5)

)543( 222 =+ o (5, 12, 13) )13125( 222 =+ . Queste soluzioni sono conosciute come ternepitagoriche e ne esistono infinite, anche escludendo le soluzioni banali per cui a, b e c hanno undivisore in comune e quelle ancor pi banali in cui almeno uno dei numeri uguale a zero.Secondo lultimo teorema di Fermat, non esistono soluzioni intere positive quando lesponente 2 sostituito da un numero intero maggiore. Il teorema particolarmente noto per la sua correlazionecon molti argomenti matematici che apparentemente non hanno nulla a che vedere con la Teoria deiNumeri. Inoltre, la ricerca di una dimostrazione stata allorigine dello sviluppo di importanti areedella matematica, anche legate a moderni settori della fisica teorica, quali ad esempio la Teoriadelle Stringhe.Lultimo teorema di Fermat pu essere dimostrato per n = 4 e nel caso in cui n un numero primo:

se infatti si trova una soluzionekpkpkp

cba =+ , si ottiene immediatamente una soluzione( ) ( ) ( )pk pk pk cba =+ . Nel corso degli anni il teorema venne dimostrato per un numero sempremaggiore di esponenti speciali n, ma il caso generale rimaneva evasivo. Il caso n = 5 statodimostrato da Dirichlet e Legendre nel 1825 ed il caso n = 7 da Gabriel Lam nel 1839. Nel 1983G. Faltings dimostr la congettura di Mordell, che implica che per ogni n > 2, c al massimo unnumero finito di interi co-primi a, b e c con nnn cba =+ . (In matematica, gli interi a e b sidicono co-primi o primi tra loro se e solo se essi non hanno nessun divisore comune eccetto 1 e-1, o, equivalentemente, se il loro massimo comune divisore 1).Utilizzando i sofisticati strumenti della geometria algebrica (in particolare curve ellittiche e formemodulari), della teoria di Galois e dellalgebra di Hecke, il matematico di Cambridge Andrew John

Wiles, dellUniversit di Princeton, con laiuto del suo primo studente, Richard Taylor, diede unadimostrazione dellultimo teorema di Fermat, pubblicata nel 1995 nella rivista specialistica Annalsof Mathematics.Nel 1986, Ken Ribet aveva dimostrato la Congettura Epsilon di Gerhard Frey secondo la qualeogni contro-esempio nnn cba =+ allultimo teorema di Fermat avrebbe prodotto una curva ellitticadefinita come: ( ) ( )nn bxaxxy +=2 , che fornirebbe un contro-esempio alla Congettura diTaniyama-Shimura. Questultima congettura propone un collegamento profondo fra le curveellittiche e le forme modulari. Wiles e Taylor hanno stabilito un caso speciale della Congettura diTaniyama-Shimura sufficiente per escludere tali contro-esempi in seguito allultimo teorema diFermat. In pratica, la dimostrazione che le curve ellittiche semistabili sui razionali sono modulari,rappresenta una forma ridotta della Congettura di Taniyama-Shimura che tuttavia sufficiente perprovare lultimo teorema di Fermat.Le curve ellittiche sono molto importanti nella Teoria dei Numeri e ne costituiscono il maggiorcampo di ricerca attuale. Nel campo delle curve ellittiche, i numeri p-adici sono conosciuti comenumeri l-adici, a causa dei lavori di Jean-Pierre Serre. Il numero primo p spesso riservato perlaritmetica modulare di queste curve.Il sistema dei numeri p-adici stato descritto per la prima volta da Kurt Hensel nel 1897. Per ogninumero primo p, il sistema dei numeri p-adici estende laritmetica dei numeri razionali in mododifferente rispetto lestensione verso i numeri reali e complessi. Luso principale di questostrumento viene fatto nella Teoria dei Numeri. Lestensione ottenuta da uninterpretazionealternativa del concetto di valore assoluto. Il motivo della creazione dei numeri p-adici era il

tentativo di introdurre il concetto e le tecniche delle serie di potenze nel campo della Teoria deiNumeri. Pi concretamente per un dato numero primo p, il campo pQ dei numeri p-adici

unestensione dei numeri razionali. Se tutti i campi pQ vengono considerati collettivamente, si


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arriva al principio locale-globale di Helmut Hasse, il quale, a grandi linee, afferma che certeequazioni possono essere risolte nellinsieme dei numeri razionali se e solo se possono essere risoltenegli insiemi dei numeri reali e dei numeri p-adici per ogni p. Il campo pQ possiede una topologia

derivata da una metrica, che , a sua volta, derivata da una stima alternativa dei numeri razionali.Questa metrica completa, nel senso che ogni serie di Cauchy converge.Scopo del presente lavoro quello di evidenziare le connessioni ottenute tra la matematica inerentela dimostrazione dellultimo teorema di Fermat ed alcuni settori della Teoria di Stringa,precisamente la supersimmetria p-adica e adelica in teoria di stringa.I settori inerenti la dimostrazione dellultimo teorema di Fermat, riguardano quelle funzionichiamate L p-adiche connesse alla funzione zeta di Riemann, quale estensione analitica al pianocomplesso della serie di Dirichlet. Tali funzioni sono strettamente correlate sia ai numeri primi, siaalla funzione zeta, i cui teoremi sono gi stati connessi matematicamente con la teoria di stringa inalcuni precedenti lavori.Quindi, per concludere, anche dalla matematica che riguarda lultimo teorema di Fermat possibileottenere, come vedremo nel corso del lavoro, ulteriori connessioni tra Teoria di Stringa (p-adicstring theory), Numeri Primi, Funzione zeta di Riemann (numeri p-adici, funzioni L p-adiche) eSerie di Fibonacci (quindi identit e funzioni di Ramanujan), che, a loro volta, verranno correlateanche al modello Palumbo-Nardelli.

Chapter 1 .

The mathematics concerning the Fermats Last Theorem

1.1 The Wiles approach.[1]

An elliptic curve over Q is said to be modular if it has a finite covering by a modular curve of the

form )(0 N X . Any such elliptic curve has the property that its Hasse-Weil zeta function has ananalytic continuation and satisfies a functional equation of the standard type. If an elliptic curveover Q with a given j-invariant is modular then it is easy to see that all elliptic curves with the samej-invariant are modular. A well-known conjecture which grew out of the work of Shimura andTaniyama in the 1950s and 1960s asserts that every elliptic curve over Q is modular.In 1985 Frey made the remarkable observation that this conjecture should imply Fermats LastTheorem. The Wiles approach to the study of elliptic curves is via their associated Galoisrepresentations. Suppose that p is the representation of ( )QQGal / on the p-division points of anelliptic curve over Q, and suppose that 3 is irreducible. The choice of 3 is critical because acrucial theorem of Langlands and Tunnell shows that if 3 is irreducible then it is also modular.Thence, under the hypothesis that 3 is semistable at 3, together with some milder restrictions onthe ramification of 3 at the other primes, every suitable lifting of 3 is modular. Furthermore,Wiles has obtained that E is modular if and only if the associated 3-adic representation is modular .The key development in the proof is a new and surprising link between two strong but distincttraditions in number theory, the relationship between Galois representations and modular forms onthe one hand and the interpretation of special values of L-functions on the other.The restriction that 3 be irreducible at 3 is bypassed by means of an intriguing argument withfamilies of elliptic curves which share a common 5 . Using this, we complete the proof that allsemistable elliptic curves are modular. In particular, this yields to the proof of Fermats Last

Theorem.Now we present the methods and results in more detail.


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Let f be an eigenform associated to the congruence subgroup ( )N 1 of ( )Z SL2 of weight 2k andcharacter . Thus if nT is the Hecke operator associated to an integer n there is an algebraic integer

( )f nc , such that ( )f f ncf T n ,= for each n. We let f K be the number field generated over Q bythe ( ){ }f nc , together with the values of and let f be its ring of integers. For any prime of

f let ,f be the completion of f at . The following theorem is due to Eichler and Shimura(for k > 2).

THEOREM 1.

For each prime Z p and each prime p of f there is a continuous representation

( ) ( ) ,2, / : f f GLQQGal (1)

which is unramified outside the primes dividing Np and such that for all primes q | Np,

trace ,f (Frob q) = ( )f qc , , det ,f (Frob q) = ( ) 1k qq . (2)

We will be concerned with trying to prove results in the opposite direction, that is to say, withestablishing criteria under which a -adic representation arises in this way from a modular form .Assume

( ) ( )pF GLQQGal 20 / : (3)

is a continuous representation with values in the algebraic closure of a finite field of characteristic p

and that 0det is odd. We say that 0 is modular if 0 and mod,f are isomorphic over pF for some f and and some embedding of / f in pF . Serre has conjectured that everyirreducible 0 of odd determinant is modular.If is the ring of integers of a local field (containing pQ ) we will say that

( ) ( ) 2/ : GLQQGal (4)

is a lifting of 0 if, for a specified embedding of the residue field of in pF , and 0 are

isomorphic over pF . We will restrict our attention to two cases:

(I) 0 is ordinary (at p) by which we mean that there is a one-dimensional subspace of 2

pF , stable

under a decomposition group at p and such that the action on the quotient space is unramifiedand distinct from the action on the subspace.

(II) 0 is flat (at p), meaning that as a representation of a decomposition group at p, 0 isequivalent to one that arises from a finite flat group scheme over pZ , and 0det restricted to aninertia group at p is the cyclotomic character.

CONJECTURE.


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Suppose that ( ) ( ) 2/ : GLQQGal is an irreducible lifting of 0 and that is unramified outside of a finite set of primes. There are two cases :

(i)

Assume that 0 is ordinary. Then if is ordinary and 1det = k for some integer

2k and some of finite order, comes from a modular form.(ii)

Assume that 0 is flat and that p is odd. Then if restricted to a decomposition groupat p is equivalent to a representation on a p-divisible group, again comes from amodular form.

Now we will assume that p is an odd prime, we have the following theorem:

THEOREM 2.

Suppose that 0 is irreducible and satisfies either (I) or (II) above. Suppose also that

(i) 0 is absolutely irreducible when restricted to ( )

pQp2

1

1 .

(ii) If 1q pmod is ramified in 0 then either qD0 is reducible over the algebraic closure

where qD is a decomposition group at q or qI 0 is absolutely irreducible where qI is aninertia group at q.

Then any representation as in the conjecture does indeed come from a modular form .

The only condition which really seems essential to our method is the requirement that 0 ismodular. The most interesting case at the moment is when p = 3 and 0 can be defined over 3F .Then since ( ) 432 SF PGL every such representation is modular by the theorem of Langlands andTunnell. In particular, every representation into ( )32 Z GL whose reduction satisfies the givenconditions is modular. We deduce:

THEOREM 3.

Suppose that E is an elliptic curve defined over Q and that 0 is the Galois action on the 3-division

points. Suppose that E has the following properties:

(i)

E has good or multiplicative reduction at 3.(ii)

0 is absolutely irreducible when restricted to ( )3Q .

(iii)

For any 1q 3mod either qD0 is reducible over the algebraic closure or qI 0 isabsolutely irreducible.

Then E should be modular.

The important class of semistable curves, i.e., those with square-free conductor, satisfies (i) and (iii)but not necessarily (ii).

THEOREM 4.


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Suppose that E is a semistable elliptic curve defined over Q. Then E is modular .

In 1986, Serre conjectured and Ribet proved a property of the Galois representation associated tomodular forms which enabled Ribet to show that Theorem 4 implies Fermats Last Theorem.

Furthermore, we have the following theorems:

THEOREM 5.

Suppose that 0=++ ppp wvu with Qwvu ,, and 3p then 0=uvw . (Equivalently there are

no non-zero integers a,b,c,n with n > 2 such that nnn cba =+ .)

THEOREM 6.

Suppose that 0 is irreducible and satisfies the hypothesis of the conjecture, including (I) above.

Suppose further that

(i) 00 QLInd = for a character 0 of an imaginary quadratic extension L of Q which is unramified

at p.(ii) =pI 0det .

Then a representation as in the conjecture does indeed come from a modular form.

Wiles has worked on the Iwasawa conjecture for totally real fields and some applications of it, withthe assumption that the reduction of a given l -adic representation was reducible and tried to proveunder this hypothesis that the representation itself would have to be modular. Thence, we write p forl because of the connections with Iwasawa theory .In the solution to the Iwasawa conjecture for totally real fields, Wiles has introduced a newtechnique in order to deal with the trivial zeroes.It involved replacing the standard Iwasawa theory method of considering the fields in thecyclotomic pZ -extension by a similar analysis based on a choice of infinitely many distinct primes

1iq inpmod with in as i . Wiles has developed further the idea of using auxiliary

primes to replace the change of field that is used in Iwasawa theory .

Let p be an odd prime. Let be a finite set of primes including p and let Q be the maximal

extension of Q unramified outside this set and . Throughout we fix an embedding of Q , and soalso of Q , in C. We will also fix a choice of decomposition group qD for all primes q in Z.

Suppose that k is a finite field characteristic p and that

( ) ( )k GLQQGal 20 / : (5)

is an irreducible representation. We will assume that 0 comes with its field of definition k and that

0det is odd.We will restrict our choice of

0 further by assuming that either:

(i) 0 is ordinary. The restriction of 0 to the decomposition group pD has (for a suitable choice of


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basis) the form

2

10 0

pD (6)

where 1 and 2 are homomorphisms from pD tok with 2 unramified. Moreover we require

that 21 .

(ii) 0 is flat at p but not ordinary. Then pD0 is the representation associated to a finite flat group

scheme over pZ but is not ordinary in the sense of (i). We will assume also that =pI 0det

where pI is an inertia group at p and is the Teichmuller character giving the action onthp

roots of unity.

Furthermore, we have the following restrictions on the deformations:

(i) (a) Selmer deformations . In this case we assume that 0 is ordinary, with notion as above, andthat the deformation has a representative ( ) )(/ : 2 AGLQQGal with the property that(for a suitable choice of basis)

2

1

~0

~

pD

with 2~ unramified, 2~ mmod , and 21

1det =pI where is the cyclotomic

character, ( ) pZ QQGal / : , giving the action on all p-power roots of unity, is of order prime to p satisfying pmod , and 1 and 2 are the characters of (i) viewed astaking values in Ak a .

(i) (b) Ordinary deformations . The same as in (i) (a) but with no condition on the determinant.

(i) (c) Strict deformations . This is a variant on (i) (a) which we only use when pD0 is not

semisimple and not flat. We also assume that =121 in this case. Then a strict

deformation is an in (i) (a) except that we assume in addition that ( ) =pD21~

/ ~

.

(ii) Flat (at p) deformations . We assume that each deformations to ( )AGL2 has the propertythat for any quotient A / a of finite order pD amod is the Galois representation associated

to the pQ -points of a finite flat group scheme over pZ .

In each of these four cases, as well as in the unrestricted case one can verify that Mazurs use of Schlessingers criteria proves the existence of a universal deformation

( ) ( )RGLQQGal 2/ : (7)

With regard the primes pq which are ramified in 0 , we distinguish three special cases :


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(A)

=

2

10

qD for a suitable choice of basis, with 1 and 2 unramified, =

121 and

the fixed space of qI of dimension 1,

(B) ,10

00

= qqI

1q , for a suitable choice of basis,

(C) ( ) 0,1 = W QH q where ( ){ } ( ) 012 det0:, == Symtracef U U Homf W k .

Then in each case we can define a suitable deformation theory by imposing additional restrictionson those we have already considered, namely:

(A)

=

2

1

qD for a suitable choice of basis of 2A with 1 and 2 unramified and

=121 ;

(B)

=10

0qqI

for a suitable choice of basis ( q of order prime to p, so the same character as

above);

(C) qq I I 0detdet = , i.e., of order prime to p.

Thus if is a set of primes in distinct from p and each satisfying one of (A), (B) or (C) for 0 ,

we will impose the corresponding restriction at each prime in .Thus to each set of data { }= ,,,D where . is Se, str, ord, flat or unrestricted, we can associatea deformation theory to 0 provided

( ) ( )k GLQQGal 20 / : (8)

is itself of type D and is the ring of integers of a totally ramified extension of ( )k W ; 0 isordinary if . is Se or ord, strict if . is strict and flat if . is flat; 0 is of type , i.e., of type (A), (B)or (C) at each ramified primes pq , q .

Suppose that q is a prime not dividing N. Let ( ) ( ) ( )qN qN 011 , = I and let( ) ( )QqN X qN X / 11 ,, = be the corresponding curve. The two natural maps ( ) ( )N X qN X 11 ,

induced by the maps zz and qzz on the upper half plane permit us to define a map( ) ( ) ( )qN J N J N J ,111 . Using a theorem of Ihara, Ribet shows that this map is injective. Thus

we can define by

( ) ( ) ( )qN J N J N J ,0 111

. (9)

Dualizing, we define B by

( ) ( ) ( ) 0,0 11

1 N J N J qN J B .


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Let ( )qN T ,1 be the ring of endomorphism of ( )qN J ,1 generated by the standard Hecke operators.One can check that pU preserves B either by an explicit calculation or by noting that B is the

maximal abelian subvariety of ( )qN J ,1 with multiplicative reduction at q. We set( )N J N J J 112 )( = . More generally, one can consider ( )N J H and ( )qN J H , in place of ( )N J 1 and

( )qN J ,1 (where ( )qN J H , corresponds to ( ) H qN X / ,1 ) and we write ( )N T H and ( )qN T H , for theassociated Hecke rings.In the following lemma if m is a maximal ideal of 11

r Nq or r Nq1 we use( )qm to denote the

maximal ideal of ( )( )11 , + r r q qNq compatible with m , the ring ( )( ) ( )1111 ,, ++ r r r r q qNqqNq beingthe sub-ring obtained by omitting qU from the list of generators.

LEMMA 1.

If pq is a prime and 1r then the sequence of abelian varieties

( ) ( ) ( ) ( )111111 ,021

+ r r r r r qNqJ NqJ NqJ NqJ

(10)

where ( ) ( )) = oo r r ,2,11 , and = r r ,3,42 , induces a corresponding sequence of p-divisible groups which becomes exact when localized at any ( )qm for which m is irreducible .

Now, we have the following theorem:

THEOREM 7.

Assume that 0 is modular and absolutely irreducible when restricted to ( )

pQp

21

1 . Assume

also that 0 is of type (A), (B) or (C) at each pq in . Then the map DDD R : (remember that D is an isomorphism) is an isomorphism for all D associated to 0 , i.e., where

( )= ,,,D with = Se, str, fl or ord. In particular if = Se, str or fl and f is any newform for which ,f is a deformation of 0 of type D then

( ) ( )


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There is an isomorphism

( ) ( )( ) ( )( )( ) ( )( )LLGalunr K LM GalHomY QQH / 1 / ,/ ,/

(13)

where 1unr H denotes the subgroup of classes which are Selmer at p and unramified everywhere else .

Now we write ( ) nstr Y QQH ,/ 1 (where = nY Y n and similarly for nY ) for the subgroup of ( ) ( ) ( )( ){ }0111 / ,0:,/ ,/ == nnppnunr nunr Y Y QinH Y QQH Y QQH where ( )0nY is the first step in the

filtration under pD , thus equal to ( )0/ nn Y Y or equivalently to ( )

0nY

where ( )0Y is the divisiblesubmodule of Y on which the action of pI is via

2 . It follows from an examination of the action

pI on Y that

( ) ( )nunr nstr Y QQH Y QQH ,/ ,/ 11 = . (14)

In the case of Y we will use the inequality

Y QQH Y QQH unr str ,/ #,/ #11 . (15)

Furthermore, for n sufficiently large the map

( ) ( ) Y QQH Y QQH str nstr ,/ ,/ 11 (16)

is injective.The above map is then injective whenever the connecting homomorphism

( )( ) ( )( )nK LH K LH pp / ,/ ,

10

is injective, which holds for sufficiently large n. Furthermore, we have

( )

( )( )( ) ( )

( )

=n

nnp

nstr

nstr

Y QH

Y QH Y QH

Y QQH

Y QQH

,#

,#,#

,/ #

,/ #0

000

1

1

. (17)

Thence, setting ( )( )( )qt q = 1/ #inf if 1mod = or 1=t if 1mod (17b), we get

( ) ( )( ) ( )( )( ) ( )( )

LLGal

qSe K LM GalHomt Y QQH / 1 / ,/ #

1,/ # l (18)

where ( )= Y QH qq ,# 0l for pq , ( )( )= 00 ,#lim npnp Y QH l . This follows from Proposition 1, (14)-(17) and the elementary estimate

( ) ( )( ){ }

pq

qunr Se Y QQH Y QQH l,/ / ,/ #11 , (19)


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which follows from the fact that ( ) ( ) qQQGalunr

qq

unr qY QH l=

/ 1 ,# . ( Remember that l is the l -adic

representation ).Let f w denote the number of roots of unity of L such that f mod1 ( f an integral ideal of

L ). We choose an f prime to p such that 1=f w . Then there is a grossencharacter of L satisfying ( )( ) = for f mod1 . According to Weil, after fixing an embedding pQQ a wecan associate a p-adic character p to . We choose an embedding corresponding to a prime abovep and then we find =p for some of finite order and conductor prime to p.

The grossencharacter (or more precisely LF N / o ) is associated to a (unique) elliptic curve Edefined over ( )f LF = , the ray class field of conductor f , with complex multiplication by L andisomorphic over C to LC / . We may even fix a Weierstrass model of E over F which has good

reduction at all primes above p . For each prime of F above p we have a formal group E ,

and this is a relative Lubin-Tate group with respect to F over pL . We let = E be thelogarithm of this formal group.Let U be the product of the principal local units at the primes above p of ( )fpL ; i.e.,

=p

U U , where = ,lim, nU U .

To an element = U uu nlim we can associate a power series ( ) [ ]

,uf where is the

ring of integers of F . For we will choose the prime above p corresponding to our chosen

embedding pQQ a . This power series satisfies ( )( )nun f u = ,, for all ( )d nn 0,0 > wherepLF d := and { }n is chosen as an inverse system of n division points of E . We define a

homomorphism U k : by

( ) ( )( )

( ) 0,

, log'1

: =

==

u

k

E

k k f d d

uu

. (20)

Then

( ) ( ) ( )uu k k k = (21) for ( )F F Gal /

where denotes the action on [ ]pE . Now p = on ( )F F Gal / . We want a homomorphismon u with a transformation property corresponding to on all of ( )LLGal / . We observe that

2p = on ( )F F Gal / .

Let S be a set of coset representatives for ( ) ( )F LGalLLGal / / / and define

( ) ( ) [ ]

=

S

uu

21

2 . (22)

Each term is independent of the choice of coset representative by (17b) and it is easily checked that


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( ) ( )uu 22 = .

It takes integral values in [ ] . Let ( ) U denote the product of the groups of local principal unitsat the primes above p of the field ( ) L . Then 2 factors through ( ) U and thus defines a

continuous homomorphism ( ) pC U :2 .

Let C be the group of projective limits of elliptic units in ( ) L . Then we have a crucial theorem of Rubin:

THEOREM 8.

There is an equality of characteristic ideals as ( )( )[ ][ ]LLGalZ p / = -modules :

( )( )( ) ( )( ) = C U char LM Galchar / / .

Let mod0 = . For any ( )( )[ ]LLGalZ p / 0 -module X we write ( )0 X for the maximal quotientof

pZ X on which the action of ( )( )LLGal / 0 is via the Teichmuller lift of 0 . Since

( )( )LLGal / decomposes into a direct product of a pro-p group and a group of order prime to p,

( )( ) ( ) ( )( ) ( )( )LLGalLLGalLLGal / / / 00 ,

we can also consider any ( )( )[ ][ ]LLGalZ p / -module also as a ( )( )[ ]LLGalZ p / 0 -module. Inparticular ( )0 X is a module over ( )( )[ ]( ) 0/ 0

LLGalZ p . Also( ) [ ][ ] 0 .

Now according to results of Iwasawa, ( )( )0 U is a free ( )0 -module of rank one. We extend 2 -linearly to ( ) pZ U and it then factors through ( )

( )0 U . Suppose that u is a generator of

( )( )0 U and an element of ( )0 C . Then ( ) = uf 1 for some ( ) [ ][ ]f and atopological generator of ( ) ( )( )0/ LLGal . Computing 2 on both u and gives

( )( ) ( ) ( )uf 22 / 1 = . (23)

We have that can be interpreted as the grossencharacter whose associated p-adic character , viathe chosen embedding pQQ a , is , and is the complex conjugate of .Furthermore, we can compute ( )u2 by choosing a special local unit and showing that ( )u2 is ap-adic unit .Now, if we have that

( ) ( )

qqf Se LY QQH l ,2/ #,/ # 0

21 ,

and ( ){ }

pq

qLh l/ # , (24)


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where ( )( )( ) )= / / ,# 0 K K QH qq l and Lh is the class number of L , combining these weobtain the following relation:

( ) ( ) ( )

qqLf Se hLV QQH l/ #,2/ #,/ # 0

21 , (25)

where ( )= V QH qq ,# 0l (for pq ), ( )( )= 00 ,# Y QH ppl . (Also here, we remember that l is p-adic ).Let 0 be an irreducible representation as in (5). Suppose that f is a newform of weight 2 andlevel N, a prime of f above p and ,f a deformation of 0 . Let m be the kernel of thehomomorphism ( ) / 1 f N arising from f .We now give an explicit formula for developed by Hida by interpreting , in terms of the cupproduct pairing on the cohomology of ( )N X 1 , and then in terms of the Petersson inner product of f with itself. Let

( ) ( )( ) ( )( ) f f f N X H N X H ,,:, 1111 (26)

be the cup product pairing with f as coefficients. Let f p be the minimal prime of ( ) f ON 1 associated to f , and let

( )( )[ ]f f f pN X H L = ,11 .

If = nnqaf let = nnqaf . Then f is again a newform and we define f L by replacingf by

f in the definition of f L . Then the pairing ( ), induces another by restriction

( ) f f f LL :, . (27)

Replacing by the localization of f at p (if necessary) we can assume that f L and f L are free

of rank 2 and direct summands as f -modules of the respective cohomology groups. Let 21 , be

a basis of f L . Then also 21 , is a basis of f f LL = . Here complex conjugation acts on

( )( )f N X H ,11 via its action on f . We can then verify that

( ) ( )ji ,det:, =

is an element of f whose image in ,f is given by ( )2 (unit).

To give a more useful expression for ( ) , we observe that f and f can be viewed as elementsof ( )( ) ( )( )C N X H C N X H DR ,, 1111 via ( ) ,dzzf f a zd f f a . Then { } f f , form a basis for

C Lf f

. Similarly { } f f , form a basis for C Lf f

. Define the vectors ( ),,1 f f = f f ,2 = and write C =1 and C =2 with ( )C M C 2 . Then writing f f f f == 21 ,

we set( ) ( )( ) ( ) ( )C C f f ji det,,det:, == .


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Now ( ) , is given explicitly in terms of the (non-normalized) Petersson inner product , :( ) 2,4, f f = where

( ) = N dxdyf f f f 1/ , . Hence, we have the following equation :

( )( )

2

/ 14,

= N dxdyf f . (28)

To compute ( )C det we consider integrals over classes in ( ) )f N X H ,11 . By Poincar duality thereexist classes 21,cc in ( ) )f N X H ,11 such that

jc

i det is a unit in f . Hence C det generates

the same f -module as is generated by

jcif det for all such choices of classes ( 21,cc ) and

with { } { }

f f f f ,, 21=

. Letting f u be a generator of the f

-module

jc if det we have thefollowing formula of Hida:

( ) = f f uuf f / , 22 (unit in ,f ).

Now, we choose a (primitive) grossencharacter on L together with an embedding pQQ a corresponding to the prime p above p such that the induced p-adic character p has the properties :

(i) 0mod =pp ( =p maximal ideal of pQ ).(ii) p factors through an abelian extension isomorphic to T Z p with T of finite order prime to

p.(iii) ( )( ) = for ( )f 1 for some integral ideal f prime to p.

Let ( ) f f N p = 10 :ker and let ( ) ( )N J pN J Af 101 / = be the abelian variety associated to f byShimura. Over +F there is an isogeny ( )d

F F f E A ++ / / where = :f d .

We have that the p-adic Galois representation associated to the Tate modules on each side are

equivalent to ( pf F F K Ind p ,0 +

where pf pf QK =, and where ( ) pp F F Gal / : is the p-adic character associated to and restricted to F .We now give an expression for

f f , in terms

of the L-function of . We note that ( ) ( ) ( ) ,2,2,22

N N N LLL == and remember that is the p-adic character, and is the complex conjugate of , we have that:

( ) ( )

,1,21

116

1, 223 N N

N q

LLq

N f f

Sq

=

, (29)

where is the character of f and its restriction to L ; is the quadratic character

associated to L ; ( )N L denotes that the Euler factors for primes dividing N have been removed; S is the set of primes q N such that q = 'qq with q | cond and ',qq primes of L , not

necessarily distinct .

THEOREM 9.


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Suppose that 0 as in (5) is an irreducible representation of odd determinant such that

00 QLInd = for a character 0 of an imaginary quadratic extension L of Q which is unramified

at p. Assume also that:

(i) =pI 0det ;(ii) 0 is ordinary.Then for every ( ) ,,, =D such that 0 is of type D with = Se or ord,

DDR

and D is a complete intersection .

COROLLARY.

For any 0 as in the theorem suppose that

( ) ( ) 2/ : GLQQGal

is a continuous representation with values in the ring of integers of a local field, unramified outsidea finite set of primes, satisfying 0 when viewed as representations to ( )pF GL2 . Supposefurther that:

(i) pD

is ordinary;

(ii) 1

det

=k

I p with of finite order, 2k .Then is associated to a modular form of weight k .

THEOREM 10. (Langlands-Tunnell)

Suppose that ( ) ( )C GLQQGal 2/ : is a continuous irreducible representation whose image isfinite and solvable. Suppose further that det is odd. Then there exists a weight one newform f such that ( ) ( ) ,, sLf sL = up to finitely many Euler factors .

Suppose then that( ) ( )320 / : F GLQQGal

is an irreducible representation of odd determinant. This representation is modular in the sense thatover 3F , mod,0 g for some pair ( ) ,g with g some newform of weight 2. There exists arepresentation

( ) ( ) ( ).2: 2232 C GLGLF GLi a

By composing i with an automorphism of ( )32 F GL if necessary we can assume that i induces theidentity on reduction ( )21mod + . So if we consider ( ) ( )C GLQQGali 20 / : o we obtain anirreducible representation which is easily seen to be odd and whose image is solvable.Now pick a modular form E of weight one such that ( )31E . For example, we can take ,16E E = where ,1E is the Eisenstein series with Mellin transform given by ( ) ( ) ,ss for the quadratic


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character associated to ( )3Q . Then 3modf fE and using the Deligne-Serre lemma we canfind an eigenform 'g of weight 2 with the same eigenvalues as f modulo a prime ' above( )21 + . There is a newform g of weight 2 which has the same eigenvalues as ' g for almost all

lT s, and we replace ( )',' g by ( ) ,g for some prime above ( )21 + . Then the pair ( ) ,g satisfies our requirements for a suitable choice of (compatible with ' ).We can apply this to an elliptic curve E defined over Q , and we have the following fundamentaltheorems:

THEOREM 11.

All semistable elliptic curves over Q are modular .

THEOREM 12.

Suppose that E is an elliptic curve defined over Q with the following properties:(i) E has good or multiplicative reduction at 3, 5,(ii) For p = 3, 5 and for any prime pq mod1 either qpE D, is reducible over pF or qpE I , is

irreducible over pF .

Then E is modular .

Chapter 2 .

Further mathematical aspects concerning the Fermats Last Theorem

2.1 On the modular forms, Euler products, Shimura map and automorphic L-functions.

A. Modular forms [2]

We know that there is a direct relation with elliptic curves, via the concept of modularity of ellipticcurves over Q .Let E be an elliptic curve over Q , given by some Weierstrass equation. Such a Weierstrassequation can be chosen to have its coefficients in Z . A Weierstrass equation for E withcoefficients in Z is called minimal if its discriminant is minimal among all Weierstrass equationsfor E with coefficients in Z ; this discriminant then only depends on E and will be denoteddiscr( E ). Thence, E has a Weierstrass minimal model over Z , that will be denoted by Z E .For each prime number p, we let

pF E denote the curve over pF given by reducing a minimal

Weierstrass equation modulo p; it is the fibre of Z E over pF . The curve pF E is smooth if and only if

p does not divide discr( E ).The possible singular fibres have exactly one singular point: an ordinary double point with rationaltangents, or with conjugate tangents, or an ordinary cusp. The three types of reduction are called


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split multiplicative, non-split multiplicative and additive, respectively, after the type of group lawthat one gets on the complement of the singular point. For each p we then get an integer pa by

requiring the following identity:pp F E ap #1 =+ . (1)

This means that for all p, pa is the trace of pF on the degree one tale cohomology of pF E , with

coefficients in lF , or in Z lZ n/ or in the l -adic numbers lZ . For p not dividing discr( E ) we know

that 2/ 12 pa p . If pF E is multiplicative, then 1=pa or 1 in the split and non-split case. If pF E

is additive, then 0=pa . We also define, for each p an element ( )p in { }1,0 by setting ( ) 1=p forp not dividing discr( E ). The Hasse-Weil L-function of E is then defined as:

( ) ( )sLsLp

pE E = , , ( ) ( )( )12

, 1 += ssppE ppppasL , (2)

for s in C with ( ) 2/ 3>sR . We note that for all p and for all pl we have the identity:

( ) ( )( )let F pp QE H tF t pt a ,,1det1 ,12 =+ . (3)

We use tale cohomology with coefficients in lQ , the field of l -adic numbers, and not in lF .

The function E L was conjectured to have a holomorphic continuation over all of C , and to satisfy acertain precisely given functional equation relating the values at s and s2 . In that functionalequation appears a certain positive integer E N called the conductor of E , composed of the primes p

dividing discr( E ) with exponents that depend on the behaviour of E at p, i.e., on pZ E . Thisconjecture on continuation and functional equation was proved for semistable E (i.e., E such thatthere is no p where E has additive reduction) by Wiles and Taylor-Wiles, and in the general caseby Breuil, Conrad, Diamond and Taylor. In fact, the continuation and functional equation are directconsequences of the modularity of E that was proved by Wiles, Taylor-Wiles, etc.The weak Birch and Swinnerton-Dyer conjecture says that the dimension of the Q -vector space

( )QE Q is equal to the order of vanishing of E L at 1. Anyway, the function E L gives us integersna for all 1n as follows:

( )

=1n

snE nasL , for ( ) 2/ 3>sR . (4)

From these na one can then consider the following function:

( ){ } C C H f E >= 0: , 1

2

n

innea

a . (5)

Equivalently, we have:

=1n

nnE qaf , with C H q : ,

ie 2a . (6)

A more conceptual way to state the relation between E L and E f is to say that E L is obtained, up toelementary factors, as the Mellin transform of E f :


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( ) ( ) ( ) ( ) =

02 sLs

t dt

t it f E ss

E , for ( ) 2/ 3>sR . (7)

Hence, we can finally state what the modularity of E means:E f is a modular form of weight two for the congruence subgroup ( )E N 0 of ( )Z SL2 .

The last statement means that E f has an enormous amount of symmetry.A typical example of a modular form of weight higher than two is the discriminant modular form,usually denoted . One way to view is as the holomorphic function on the upper half plane H given by:

( )

=1

241

n

nqq , (8)

where q is the function from H to C given by ( )izz 2expa . The coefficients in the power seriesexpansion:

( )

=1n

nqn (9)

define the famous Ramanujan -function .To say that is a modular form of weight 12 for the group ( )Z SL2 means that for all elements

d c

baof ( )Z SL2 the following identity holds for all z in H :

( ) ( )zd czd czbaz +=

++

12 , (10)

which is equivalent to saying that the multi-differential form ( )( ) 6 dzz is invariant under the

action of ( )Z SL2 . As ( )Z SL2 is generated by the elements

10

11and

01

10, it suffices to

check the identity in (10) for these two elements. The fact that is q times a power series in q means that is a cusp form : it vanishes at 0=q . It is a fact that is the first example of a non-zero cusp form for ( )Z SL2 : there is no non-zero cusp form for ( )Z SL2 of weight smaller than 12,i.e., there are no non-zero holomorphic functions on H satisfying (10) with the exponent 12replaced by a smaller integer, whose Laurent series expansion in q is q times a power series.

Moreover, the C -vector space of such functions of weight 12 is one-dimensional, and hence is abasis of it.The one-dimensionality of this space has as a consequence that is an eigenform for certainoperators on this space, called Hecke operators , that arise from the action on H of ( )+QGL2 , thesubgroup of ( )QGL2 of elements whose determinant is positive. This fact explains that thecoefficients ( )n satisfy certain relations which are summarised by the following identity of Dirichlet series:

( ) ( ) ( )( )

+==

1

12111:n p

sss ppppnnsL . (11)

These relations:( ) ( ) ( )nmmn = if m and n are relatively prime;


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( ) 2111 = nnn ppppp if p is prime and 2n

were conjectured by Ramanujan, and proved by Mordell. Using these identities, ( )n can beexpressed in terms of the ( )p for p dividing n . As L is the Mellin transform of , L is

holomorphic on C , and satisfies the functional equation (Hecke):

( ) ( ) ( ) ( ) ( ) ( ) ( )sLssLs ss

= 212122 12 . (12)

The famous Ramanujan conjecture states that for all primes p one has the inequality:

( ) 2/ 112 pp sR and gives rise by analytic continuation to a meromorphic function ( )s inC . For ( ) 1>sR ( )s admits the absolutely convergent infinite product expansion

p sp11

, (15)

taken over the set of primes . This Euler product may be regarded as an analytic formulation of theprinciple of unique factorization in the ring Z of integers. It is, as well, the product taken over allthe non-Archimedean completions of the rational field Q (which completions pQ are indexed by

the set of primes ) of the Mellin transform in pQ

( ) sp ps = 11

, (16)

(where the Mellin transform is, more or less, Fourier transform on the multiplicative group.

Classically, the Mellin transform of f is given formally by ( ) ( ) ( )

=0

/ xdxxxf s s . (17))

of the canonical Gaussian density ( )= xp 1 if x closure of Z in pQ ; 0 otherwise, whichGaussian density is equal to its own Fourier transform. For the Archimedean completion RQ = of the rational field Q one forms the classical Mellin transform

( )( )

( )2/ 2/

sss

=

(18)

of the classical Gaussian density


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Observing that the formula

2

222

ydydx

ds+

= for H iyx += , (27)

gives a (the hyperbolic) ( )RSL2

-invariant metric in H with associated invariant measure

2ydxdy

d = , (28)

one introduces the Petersson (Hermitian) inner product in the space of cuspform of weight w for with the definition:

( ) ( ) ( ) ( ) = / , H w d gf gf . (29) (see also page 13 eq. (28))

(Integration over the quotient / H makes sense since the integrand ( ) ( ) wygf (30) is -invariant).For the modular group ( )1 the thn Hecke operator ( ) ( )nT nT w= is the linear endomorphism of thespace of cuspforms of weight w arising from the following considerations. Let nS be the set of

22 matrices in Z with determinant n . For

nSd c

baM

= (31)

and for a function f in H one defines

( )( ) ( ) ( ) ( ) f d cM f M www += 1det , (32)

and then, observing that ( )1 under w acts trivially on the modular forms of weight w , one maydefine the Hecke operator ( )nT w by

( )( ) ( )( )( )

=1/ nSM

ww f M f nT , (33)

where the quotient ( )1/ nS refers to the action of ( )1 by left multiplication on the set nS . Onefinds for nm, coprime that

( ) ( ) ( )nT mT mnT = , (34)

and furthermore one has( ) ( )( ) ( )111 + = ewee pT ppT pT pT . (35)

Consequently, the operators ( )nT commute with each other, and, therefore, generate a commutativealgebra of endomorphisms of the space of cusp forms of weight w for ( )1 . It is not difficult to seethat the Hecke operators are self-adjoint for the Petersson inner product on the space of cuspforms .Consequently, the space of cuspforms of weight w admits a basis of simultaneous eigenforms for


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the Hecke algebra. A Hecke eigencuspform is said to be normalized if its Fourier coefficient11 =c . If f is a normalized Hecke eigencuspform, then:

(i)

The Fourier coefficient mc of f is the eigenvalue of f for ( )mT .(ii)

The Fourier coefficients ( )

mcmc = of f satisfy

( ) ( ) ( )ncmcmnc = for nm, coprime , and( ) ( )( ) ( )111 + = ewee pcppcpcpc for p prime .

Consequently, the Dirichlet series associated with a simultaneous Hecke eigencuspform of level 1and weight w admits an Euler product

( ) += p swsp ppcs 211

1 . (36)

For example, when f is the unique normalized cuspform of level 1 and weight 12, one has

( )( ) += p ss ppp

s 21111

, (37) (in fact, if 12=w , then ssw 21121 = )

where ( )pc p = is the function of Ramanujan .

C. Shimura map [3]

Shimura showed for a given N W -compatible Hecke eigencuspform f of weight 2 for the group

( )N 0 with rational Fourier coefficients how to construct an elliptic curve f E defined over Q suchthat the Dirichlet series ( )s associated with f is the same as the L -function )sE L f , .Let be a congruence subgroup of ( )Z SL2 , and let ( ) X denote the compact Riemann surface

/ H . The inclusion of in ( )1 induces a branched covering

( ) ( ) 11 PX X . (38)

One may use the elementary Riemann-Hurwitz formula from combinatorial topology to determinethe Euler number, and consequently the genus, of ( ) X . The genus is the dimension of the space of cuspforms of weight 2. Even when the genus is zero one obtains embeddings of ( ) X in projectivespaces r P through holomorphic maps

( ) ( ) ( )( ) r

f f f ,...,,10

a , (39)

where r f f f ,...,, 10 is a basis of the space of modular forms of weight w with w sufficiently large.


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Using the corresponding projective embedding one finds a model for ( ) ( )( )N X N X 00 = over Q ,i.e., an algebraic curve defined over Q in projective space that is isomorphic as a compact Riemannsurface to ( )N X 0 .Associated with any complete non-singular algebraic curve X of genus g is a complex torus, the

Jacobian ( )X J of X , that is the quotient of g -dimensional complex vector spaceg

C by thelattice generated by the period matrix, which is the gg 2 matrix in C obtained byintegrating each of the g members i of a basis of the space of holomorphic differentials over eachof the g2 loops in X representing the members of a homology basis in dimension 1. Furthermore,if one picks a base point 0z in X , then for any z in X , the path integral from 0z to z of each of the g holomorphic differentials is well-defined modulo the periods of the differential. One obtainsa holomorphic map ( )X J X from the formula

mod,...,0 0

1

z

z

z

zgz a . (40)

This map is universal for pointed holomorphic maps from X to complex tori. Furthermore, theJacobian ( )X J is an algebraic variety that admits definition over any field of definition for X and

0z , and the universal map also admits definition over any such field. The complex tori that admitembeddings in projective space are the abelian group objects in the category of projective varieties.They are called abelian varieties . Every abelian variety is isogenous to the product of simpleabelian varieties: abelian varieties having no abelian subvarieties. Shimura showed that one of thesimple isogeny factors of ( )( )N X J 0 is an elliptic curve f E defined over Q characterized by thefact that its one-dimensional space of holomorphic differentials induces on ( )N X 0 , via thecomposition of the universal map with projection on f E , the one-dimensional space of differentials

on ( )N X 0 determined by the cuspform f .He showed further that )sE L f , is the Dirichlet series ( )s with Euler product given by f . Anelliptic curve E defined over Q is said to be modular if it is isogenous to f E for some N W -

compatible Hecke eigencuspform of weight 2 for ( )N 0 . Equivalently E is modular if and only if ( )sE L , is the Dirichlet series given by such a cuspform. The Shimura-Taniyama-Weil Conjecture

states that every elliptic curve defined over Q is modular . Shimura showed that this conjecture istrue in the special case where the Z -module rank of the ring of endomorphisms of E is grater than

one. In this case the point of the upper-half plane corresponding to ( )C E is a quadratic imaginarynumber, and ( )sE L , is a number-theoretic L -function associated with the corresponding imaginaryquadratic number field.

D. Automorphic L -functions [4]

Talking about zeta functions in general one inevitably is led to start with the Riemann zeta function( )s . It is defined as a Dirichlet series :

( )

=

=1n

sns , (41)


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which converges for each complex number s of real part greater than one. In the same region itpossesses a representation as a Mellin integral :

( )( ) t

dt t

ess st

=

0 111 . (42)

Let f be a cusp form of weight k 2 for some natural number k , i.e., the function f is holomorphicon the upper half plane H in C , and has a certain invariance property under the action of themodular group ( )Z SL2 on H . Then f admits a Fourier expansion

( )

=

=1

2

n

iznneazf

. (43)

Define its L -function for ( ) 1Re >s by

( )

=

=1

,n

sn

nasf L . (44)

The easily established integral representation

( ) ( ) ( ) ( ) ( ) = =

0,2,

t dt

t it f sf Lssf L ss , (45)

implies that ( )sf L , extends to an entire function satisfying the functional equation( ) ( ) ( )sk f Lsf L k = 2,1, . With ( ) ( )ksf Lsf 2,, = this becomes

( ) ( ) ( )sf sf k = 1,1, . (46)

This construction can be extended to cusp forms for suitable subgroups of the modular group. TheseL -functions look like purely analytical objects. Thus it was particularly daring of A. Weil, G.Shimura, and Y. Taniyama in 1955 to propose the conjecture that the zeta function of any ellipticcurve over Q coincides with a ( )sf , for a suitable cusp form f . This conjecture was proved inpart by A. Wiles and R. Taylor providing a proof of Fermats Last Theorem as a consequence.The upper half plane is a homogeneous space of the group ( )RSL2 , and so cusp forms may beviewed as functions on this group, in particular, they are vectors in the natural unitaryrepresentation of ( )RSL2 on the space

( ) ( )( )RSLZ SLL 222 \ . (47)

Going even further one can extend this quotient space to the quotient of the adele group ( )AGL2 modulo its discrete subgroup ( )QGL2 , so cusp forms become vectors in

( ) ( ))1222 \ AGLQGLL , (48)


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where ( )12 AGL denotes the set of all matrices in ( )AGL2 whose determinant has absolute value one.Now 2GL can be replaced by nGL for N n and one can imitate the methods of Tates thesis (thecase n = 1) to arrive at a much more general definition of an automorphic L -function: this is anEuler product ( )sL , attached to an automorphic representation of ( )1AGLn , i.e., an irreduciblesubrepresentation of ( ) ( ))12 \ AGLQGLL nn . As in the 1GL -case it has an integral representationas a Mellin transform and it extends to a meromorphic representation, which is entire if iscuspidal and 1>n . Furthermore it satisfies a functional equation

( ) ( ) ( )sLssL = 1,~,, , (49)

where ~ is the contragredient representation and ( )s, is a constant multiplied by an exponential.We conclude remember that extending the Weil-Shimura-Taniyama conjecture, R.P. Langlandsconjectured in the 1960s that any motivic L -function coincides with ( )sL , for some cuspidal .

2.2 On some mathematical applications of the Mellin transform.[5]

Harmonic sums are sums of the form

( ) ( )=k

k k xgxG , (50)

where the k are the amplitudes , the k are the frequencies and ( )xg is the base function . Weconsider harmonic sums because we wish to evaluate ( )xG at a set of particular points ,..., 10 xx or atall Rx .

Definition of the harmonic sum and computation of the appropriate Mellin transform .

Now, let k k / 1= , k k / 1= and ( ) ( ) ( )xxxxg / 11/ 11/ +=+= ; and we consider the harmonic sum

( ) ( )

+=

+==

xk k k xk k x

k xk

xgxh11

/ 1/ 1 . (51)

This sum is of interest because

( ) += =

===

+=

1 1

11111

nk

n

k nH k k k k nk

nh , (52)

the n th harmonic number.The principal operation in the evaluation of harmonic sums is the computation of the Mellintransform of the base function ( )yg and the computation of the Dirichlet generating function ( )s .We first compute the transform of the base function. We have ( )[ ] ( )ssx sin/ ;1/ 1 =+ andhence

( )ss

x

x

sin;

1=

+

. (53)

Now we compute the Dirichlet generating function ( )s . We have


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( ) ( ) === sk k k s ss 1

111 . (54)

We conclude that the Mellin transform of ( )xh is

( )( )s

s 1

sin

. (55)

Inversion of the map .

Now, by Mellin inversion we obtain:

( )( ) ( )xhxs

s= ;1

sin1

. (56)

This is equivalent to the inversion integral

( )( ) ( )

+

=

ic

ic

s xhdsxss

1sin

. (57)

This integral representation permits the computation of ( )xh , because the integral can evaluated bythe Cauchy Residue theorem, i.e., it is a sum of residues of ( ) sxsh .

Computation of the poles of the transform function and the corresponding terms in the asymptoticexpansion .

We use the fact that( ) ( )

( )( )

=H xshSing

s

s

sxshsxhI

;Re , (58)

where H is the right half-plane, chosen for an expansion at infinity. We must compute the set of poles ( )( ) H xshSing s I and map them back to the terms of the expansion of ( )xh . The poles of

( )sh in the right half-plane are at 0=s , where we have a double pole and

( ) ...12 +=

sssh

(59)

and at += Z k k s , , where we have

( ) ( ) ( ) ...11 +

=k sk

sh k

. (60)

These poles map back to xlog (61) and x21

for 1=k ,( )

k k

k

xk B 11

for 2k . (62)

We conclude that Harmonic numbers satisfy the asymptotic expansion


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( )

+++

2

1121

logk

k k

k

n nk B

nnH . (63)

This expansion is exact; it converges for 1 n .The Mellin transform maps the space of functions that are integrable along the positive real line tothat of complex functions that are analytic on a vertical strip of the complex plane. This strip may inmany cases be extended to a larger domain. The map is given by the fundamental formula:

( )[ ] ( ) ( ) + ==

0

1; dxxxf sf sxf s . (64)

The Mellin-Perron formula is a specific instance of generalized Mellin summation. The traditionalproof uses the discontinuous factor described by the following lemma:

( )( ) ( )

+

=

++= ic

ic

ms

ymds

msssy

iy 11

!1

...121

if y1 ;

( )( ) ( )

+

=

++=

ic

ic

s

dsmsss

yi

y 0...12

1

if 10 < y , (65)

where ++ Z mRy , and 1c .The above equality for the discontinuous factor ( )y is easily verified with the Cauchy residuetheorem.

Hence, there are two cases.

Case 1 . y1 .

The term( ) ( )msss

y s

++ ...1(66) is meromorphic with residues

( )( ) ( )( ) ( )( )( )!!

1...11...1 k mk

ymk k k k k k k

y k k k

=+++++

(67)

where mk 0 . Therefore the sum of these residues is

( )( ) = =

=

=

m

k

m

k

m

k m

k k k

ymyk

m

mk mk y

0 0

11

!1

11

!1

!!1

. (68)

Now consider the left contour. The integral along the vertical segment at c in the right-half planeapproaches

( ) ( ) +

++

ic

ic

s

dsmsss

y...1

(69)


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28

as T goes to infinity. Along the two horizontal segments from iT T to iT c , the integrand is

bounded by mT y

and because the term mT y

+

+

1

11

with T = , c= vanishes as T goes to

infinity (recall that y1 ), the contribution from these two segments is zero. The integrand is

bounded by ( ) ( )mT T T yT

...1on the vertical segment in the left half-plane; hence the integral is

bounded by( ) ( )mT T

y T

...12

and its contribution is zero also.

Case 2 . 10 < y .

Consider the contour in the right half-plane. Along the horizontal segments we may use the samebound as in the first case, with c= and T = ; hence these integrals vanish ( )10 < y . The

integrand is bounded by ( ) ( )mT T T yT

++ ...1 on the vertical segment in the right half-plane; its

contribution is zero because 10 < y .The principal feature of the discontinuous factor is that it can be used to evaluate finite sums.Suppose we have a finite sum over the indices k from 1 to 1n . Evidently ( )y is non-zero if y/ 1lies in ( )1,0 and zero otherwise. We need only find a map such that the set { }1,...1 n maps to asubrange of ( )1,0 and { }...1, +nn to a subrange of [ ),1 . Clearly nk y / / 1 = is such a map. Weobtain

( ) ( ) +

=

++

ic

ic

ms

s nk

mds

msssn

k i1

!1

...11

21

if nk <

( ) ( ) +

=

++

ic

ic

s

s dsmsssn

k i0

...11

21

if k n (70)

By a formal argument we finally have

( ) ( )

=

+

++

=

1

1 ...121

1!

1 n

k

ic

ic

s

sk

m

k dsmsssn

k ink

m

. (71)

This is the Mellin-Perron formula .

The Mellin-transform view adds two additional perspectives. One, that the Mellin-Perron formula isa specific instance of harmonic sum formulas, and hence, two, that its evaluation corresponds toMellin inversion.

We wish to evaluate the harmonic sum


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29

( )( ) ( )msss

s++

...11

. (75)

By Mellin inversion we thus have

( ) ( )( ) ( )

+

++=

ic

ic

s

dsmsss

xs

ixG

...121

(76)

and in particular

( )( ) ( )


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30

The Mellin transform ( )xH m of ( )xH m , where N m , exists in +,0 and is given by

( )

( ) ( )msss

mxH m ++

=

...1

!. (82)

We have ( ) ( )10 +xm xH and ( ) ( )bxm xxH + for any 0>b and for N m , hence ( )xH m exists in +,0 . Note that

( ) [ ] === 101

01

011s

xs

dxxxH ss . (83)

We also have

( ) ( ) ( ) ( )

===

1

0

1

0

1

0 1

1

1

1 dxxxH dxxxH dxxxH sH sm

s

m

s

mm

( ) ( ) ( ) ( ) == 10 1111

xH ms

xH dxsxm

xxH mm

sm

m . (84)

This gives ( ) ( )xH ms

mxH mm

+= 1 (85)

Now, we will be concerned with the linearity and the rescaling property of the Mellin transform.

Theorem 2.2.1

Let Z be a finite set of integers; let k ,+

Rk . Let the fundamental strip of ( )[ ]sxf ; be , . We have

( ) ( )[ ]sxf sxf k

sk

k

k k k ;;

=

, (86)

where ,s .Let xy k = and dxdy k = . Note that

( ) ( ) ( ) ( )

===

0 0 0

111

k k k sk

k sk

sk

sk k

s

k k k sf

dyyyf dxxxf dxxxf

. (87)

We were able to exchange the integral with the summation because is finite. It can be shown thatthis operation extends to infinite as long as k sk k / converges absolutely. The extendedproperty holds in the intersection of the half-plane of convergence of k sk k / and thefundamental strip , of ( )xf .

Definition 2.2.4

1.

(Lebesgue integration)


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31

Let ( )xf be integrable with fundamental strip , . If ( ) ,c and ( )it cf + is integrable, then

( ) ( ) +

=ic

ic

s xf dsxsf

i 2

1(88)

almost everywhere. If ( )xf is continuous, the equality holds everywhere on ( )+,0 .

2.

(Riemann integration.)

Let ( )xf be locally integrable with fundamental strip , and be of bounded variation in aneighbourhood of 0x . Then

( )( ) ( )

+

+

+=

iT c

iT c x

s

T

xf xf dsxsf

i 22

1lim 00

0 (89)

for ( ) ,c . Of course if ( ) ( )xf xf xxxx +

=00

limlim then

( ) ( ) ( )000 2 xf xf xf =

+ +. (90)

Theorem 2.2.2 (Mellin-Perron formula)

Let +Rc lie in the half-plane of absolute convergence of

k s

k k / . Then we have

( ) ( )


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32

( ) ( ) ( ) +

+

=

+ iT ciT c

s

k sk

k

k T

k k k dsxsf i

xf xf

21

lim2

. (94)

Let ( ) ( )xH xf m= , N m and let k k = . Recall that the fundamental strip of ( )xH m is ,0 ; letnx / 1= . This gives

( ) ( ) =

+

=+ ++

k k

mm

k k k

k nk

H nk

H xf xf

22

( ) ( ) ( ) ( ) . We need to evaluate ( )xF . ( )xH 0 vanishes outside of [ )1,0 , hence we require( ) 1/ 0


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33

Theorem 2.2.3

Let 1>c .

( )( ) ( ) ( ) +

+=

+

ic

ic

s

sar nN k dsss

nasr in

r ak 1,

121

1/ . (101 )

This theorem has several useful corollaries. The first of these is obtained by setting 1 =r . Let( )0,1 .

Corollary 2.2.3

Let N n .

( )( )

+

=

+

i

i

s

dsssn

asi

0

1,

21

. (102)

Let 1=c . The set of poles of ( ) ( )( )1/ , +ssnas s in c, is { }0,1 . We apply the shifting lemmawith ( ) sns = and jT j = . Because nn s = we can take cnM = .

( )( )

( )( )

+

+

+=

+

i

i

ic

ic

ss

ssn

asi

dsssn

asi

1,

21

1,

21

( )( )

( )( )

=

+

=

+ 0;

1,Re1;

1,Re s

ssn

asssssn

assss

= ( ) ( ) ( ) ( ) ( )


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Suppose 14 mmn += where { }3,2,1,01m . We have [ ] amnan += 4/ 4/ 4/ 1 . If am (107)

and there is an Euler adelic representation

( ) ( ) =p

sps1

1 . (108)

Now, we have the Riemann -function

( ) ( ) ( )ssssss

=

22

1 2 (109)

which is an entire function. The zeros of the -function are the same as the nontrivial zeros of the -function. There is the functional equation

( ) ( )ss = 1 (110)

and the Hadamard representation for the -function

( ) =

/

121 sas

es

es . (111)


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Here are nontrivial zeros of the zeta-function and

4log21

121 +=a (112 )

where is Eulers constant.If ( ) F is a function of a real variable then we define a pseudo-differential operator F ( ) byusing the Fourier transform

F ( ) ( ) ( ) ( ) = dk k k F ex ixk ~2 . (113)

Here is the dAlambertian operator

2 1

2

21

2

20

2

...

++

+

=

d xxx, (114)

( )x is a function from d Rx , ( )k ~ is the Fourier transform and 2 121202 ... = d k k k k .We assume that the integral (113) converges.

One can introduce a natural field theory related with the real valued function ( )

+= iF 21

defined by means of the zeta-function. We consider the following Lagrangian

( iL += 2/ 1 ) , (115)

the integral( i+2/ 1 ) ( ) ( iex ixk += 2/ 1 ) ( )dk k ~ (116)

converges if ( )x is a decreasing function since

+ i21

is bounded.

The operator ( i+2/ 1 ) (or ( i+2/ 1 )) is the first quantization the Riemann zeta-function.From the Hadamard representation (111) we get

=

=

+

14

2

1221

n nmC

i

. (117)

It is possible to write the formula (117) in the form

+=

+n nm

C i

,2122

1

(118)

where 1= and a regularization is assumed.To quantize the zeta-function classical field ( )x which satisfies the equation in the Minkowskispace

F ( ) ( ) 0=x (119)


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36

where F ( ) ( i+= 2/ 1 ) we can try to interpret ( )x as an operator valued distribution in aHilbert space which satisfies the equation (119). We suppose that there is a representation of thePoincare group and an invariant vacuum vector 0 in . Then the Wightman function

( ) ( ) ( )00 yxyxW =

is a solution of the equationF ( ) ( ) 0=xW . (120)

By using (118) we can write the formal Kallen-Lehmann representation

( ) ( ) ( ) +=n

nnixk dk mk k f exW

22 . (121)

One introduces also another useful function

( ) ( )

+=

+

+

+ =

ieii

i

Z ii21

21

241

241

2/ . (122)

Here ( )z is the gamma function. The function ( ) Z is called the Riemann-Siegel (or Hardy)function. It is known that ( ) Z is real for real and there is a bound

( ) OZ = , 0> . (123)

One can introduce a natural field theory related with the real valued functions ( ) Z defined bymeans of the zeta-function by considering the following Lagrangian

Z L = ( ) .

The integral (113) converges if ( )x is a decreasing function since there is the bound (123).Thence, we have the following connection:

( ) ( )=

+=

+

+

+

Oieii

i

ii

21

21

241

241

2/

F ( ) ( ) ( ) ( ) = dk k k F ex ixk ~2 , 0> . (124)

For any character to modulus q one defines the corresponding Dirichlet L-function by setting

( ) ( )

=

=1

,n

snn

sL

, ( )1> . (125)


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37

If is primitive then ( ) ,sL has an analytic continuation to the whole complex plane. The zeroslie in the critical strip and symmetrically distributed about the critical line 2 / 1= .If we quantize the L-function by considering the pseudo-differential operator

( iL + , ) (126)

then we can try to avoid the appearance of tachyons and/or ghosts by choosing an appropriatecharacter .The Taniyama-Weil conjecture relates elliptic curves and modular forms. It asserts that if E is anelliptic curve over Q , then there exists a weight-two cusp form f which can be expressed as theFourier series

( ) = nzneazf 2 (127)

with the coefficientsn

a depending on the curve E. Such a series is a modular form if and only if itsMellin transformation, i.e. the Dirichlet L-series

( ) = snnaf sL , (128)

has a holomorphic extension to the full s-plane and satisfies a functional equation. For the ellipticcurve E we obtain the L-series ( )E sL , . The Taniyama-Weil conjecture was proved by Wiles andTaylor for semistable elliptic curves and it implies Fermats Last Theorem.Quantization of the L-functions can be performed similarly to the quantization of the Riemann zeta-function discussed above by considering the corresponding pseudo-differential operator ( iL + ).

Chapter 3 .

How primes and adeles are related to the Riemann zeta function [7]

A. Connes has reduced the Riemann hypothesis for L-function on a global field k to the validity of atrace formula for the action of the idele class group on the noncommutative space quotient of theadeles of k by the multiplicative group of k.Connes has devised a Hermitian operator whose eigenvalues are the Riemann zeros on the criticalline. Connes gets a discrete spectrum by making the operator act on an abstract space where theprimes appearing in the Euler product for the Riemann zeta function are built in; the space isconstructed from collections of p-adic numbers (adeles) and the associated units (ideles).


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Hence, the geometric framework involves the space X of Adele classes, where two adeles whichbelong to the same orbit of the action of ( )k GL1 ( k a global field), are considered equivalent. Thegroup ( ) ( )k GLAGLC k 11 / = of Idele classes (which is the class field theory counterpart of the Galoisgroup) acts by multiplication on X.

We have a trace formula (Theorem 3) for the action of the multiplicative groupK

of a local fieldK on the Hilbert space ( )K L2 , and (Theorem 4) a trace formula for the action of the multiplicativegroup sC of Idele classes associated to a finite set S of places of a global field k , on the Hilbert

space of square integrable functions ( )SX L2 , where SX is the quotient of Sv vk by the action of the group SO of S-units of k . The validity of the trace formula for any finite set of places followsfrom Theorem 4, but in the global case is left open and shown (Theorem 5) to be equivalent to thevalidity of the Riemann Hypothesis for all L functions with Grossencharakter.

H. Montgomery has proved (assuming RH) a weakening of the following conjecture (with0, > ),

( ) [ ]{ } ( )

duu

uM xxM jijiCard ji

2sin

1,;,...,1,;, (1)

This law, i.e. the equation (1), is precisely the same as the correlation between eigenvalues of hermitian matrices of the gaussian unitary ensemble. Moreover, numerical tests due to A. Odlyzkohave confirmed with great precision the behaviour (1) as well as the analogous behaviour for morethan two zeros. N. Katz and P. Sarnak has proved an analogue of the Montgomery-Odlyzko law forzeta and L-functions of function fields over curves.

It is thus an excellent motivation to try and find a natural pair ( )D, where naturality should meanfor instance that one should not even have to define the zeta function, let alone its analyticcontinuation, in order to obtain the pair (in order for instance to avoid the joke of defining as the

2l space built on the zeros of zeta).

Theorem 1 .

Let K be a local field with basic character . Let K Sh have compact support. Then ( )hU R is a trace class operator and when , one has

( )( ) ( ) ( ) ( ) ++=

'

1

11

log'12 oud u

uhhhU RTrace (2)

where [ ]

=

,, 1log'2

K d , and the principal value ' is uniquely determined by the pairing

with the unique distribution on K which agrees withu

du1

for 1u and whose Fourier transform

vanishes at 1 .

Proof .

We normalize the additive Haar measure to be the selfdual one on K . Let the constant 0> bedetermined by the equality,


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39

1log

d when , (3)

so that

d d 1 = . Let L be the unique distribution, extension of u

du

1

1 whose Fourier

transform vanishes at 1, ( ) 01 =L . One then has by definition,

( ) ( )

=

' 11

,1 u

uhLud

uuh

, (4)

where( )

01

=

uuh

for 1u outside the support of h . Let ( )hU T = . We can write the Schwartz

kernel of T as,

( ) ( ) ( ) = d xyhyxk 1, . (5)

Given any such kernel k we introduce its symbol,

( ) ( ) ( )duuuxxk x += ,, (6)

as its partial Fourier transform. The Schwartz kernel ( )yxr t , of the transpose t R is given by,

( ) ( )( )( )yxxyxr t = , . (7)

Thus, the symbol of t R is simply,

( ) ( ) ( ) = xx, . (8)

The operator R is of trace class and one has,

Trace ( ) ( ) ( ) = dxdyyxr yxk T R t ,, . (9)

Using the Parseval formula we thus get,

Trace ( ) ( ) = , ,x dxd xT R . (10)

Now the symbol of T is given by,

( ) ( ) ( ) ( ) += d duuxuxhx 1, . (11)

One has,

( ) ( ) ( )( ) =+ xduuxux 1 , (12)thus (11) gives,


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40

( ) ( ) ( ) = K d xgx 1, (13)

where,( ) ( )( ) 11 11 ++= hg . (14)

Since h is smooth with compact support on K the function g belongs to ( )K C c . Thus

( ) ( ) xgx , 1= andTrace ( ) ( ) = ,1 x dxd xgT R . (15)

With xu = one hasx

dxdudxd = and, for 2u ,

=xu uxdx

loglog'21

. (16)

Thus we can rewrite (15) as,

Trace ( ) ( )( ) = 2 loglog'2u duuugT R . (17)

Since ( )K C g c

one has,

( ) ( ) =2 u N Oduug N (18)

and similarly for ( ) uug log . Thus

Trace ( ) ( ) ( ) ( ) += 1loglog'02 oduuuggT R . (19)

Now for any local field K and basic character , if we take for the Haar measure da the selfdualone, the Fourier transform of the distribution ( ) uu log= is given outside 0 by

( ) aa1

1

= , (20)

with determined by (3). To see this one lets P be the distribution on K given by,

( )( )

( ) ( ) +=

x

K Mod

f xd xf f P log0lim0

. (21)

One has ( ) ( ) ( )0log f af Pf P a = which is enough to show that the function ( )xP is equal to+ xlog cst, and differs from P by a multiple of 0 . Thus the Parseval formula gives, with the

convention of Theorem 3,


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41

( ) ( ) = '1log ada

agduuug

. (22)

Replacing a by 1 and applying (14) gives the desired result.

Now, let k be a global field and S a finite set of places of k containing all infinite places. Thegroup SO of S -units is defined as the subgroup of

k , Svqk qO vS == ,1, . It is co-compact

in 1SJ where,

=Sv

vS k J and, { }1,1 == jJ jJ SS . Thus the quotient group = SSS OJ C / plays

the same role as k C , and acts on the quotient SX of = Sv vS k A by

SO .

Theorem 2 .

Let SA be as above, with basic character = v . Let ( )SC Sh have compact support. Thenwhen , one has

Trace ( )( ) ( ) ( ) ( )

++=

Svk v

oud u

uhhhU R

' 1

11

log'12 (23)

where [ ]

=

,, 1log'2

SC d , each vk is embedded in SC by the map ( )1,...,,...,1,1 uu and

the principal value ' is uniquely determined by the pairing with the unique distribution on vk

which agrees withu

du

1for 1u and whose Fourier transform relative to

v vanishes at 1 .

Proof .

We normalize the additive Haar measure dx to be the selfdual one on the abelian group SA . Let theconstant 0> be determined by the equality,

1,log

D

d when ,

so that d d 1 = . We let f be a smooth compactly supported function on SJ such that

( ) ( )

=SOq

ghqgf SC g . (24)

The existence of such an f follows from the discreteness of SO in SJ . We then have the equality

( ) ( )hU f U = , where( ) ( ) ( )

= d U f f U . (25)


42/86


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43

Since f is smooth with compact support on SA the function qg belongs to ( )Sc AC .

Thus ( ) ( ) xgx q, 1= and, using the Parseval formula we get,

( ) = ,, ,xDxq dxd xI . (37)

This gives,( )

dxd xgI

xDx qq =

,,

1 . (38)

With xu = one hasx

dxdudxd = and, for 2u ,

=xuDx uxdx

,

1

loglog'2 . (39)

Thus we can rewrite (38) as,

Trace ( ) ( )( )

=

SOqu q

duuugT R2 loglog'2 . (40)

Now

=Sv v

uu loglog , and we shall first prove that,

( ) ( ) =SOq q hduug 1 , (41)

while for any Sv ,

( )( ) ( )

=

Sv

Oqk vq

ud u

uhduuug

' 1

1log . (42)

In fact all the sums in q will have only finitely many non zero terms. It will then remain to controlthe error term, namely to show that,

( )( ) ( )

+ =SOq

N q duuug 0log'2log , (43)

for any N , where we used the notation 0=+x if 0x and xx =+ if 0>x .

Now recall that for (36), ( ) ( )( ) 11 11 ++= uuqf ug q , so that ( ) ( ) ( )qf gduug qq == 0 . Since f has compact support in SA , the intersection of

SO with the support of f is finite and by (24) we

get the equality (41). To prove (42), we consider the natural projection vpr from

Sl lk to

vl lk . The image

Sv Opr is still a discrete subgroup of

vl lk , thus there are only finitelymany SOq such that

vk meets the support of qf , where ( ) ( )qaf af q = for all a .


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For each SOq one has,

( )( ) ( )

=

'1

1log

vk

qvq ud u

uf duuug , (44)

and this vanishes except for finitely many sq ' , so that by (24) we get the equality (42).

Theorem 3 .

Let k be a global field of positive characteristic and Q be the orthogonal projection on the

subspace of ( )X L2 spanned by the ( )ASf such that ( )xf and ( )xf vanish for >x . Let ( )k C Sh have compact support. Then the following conditions are equivalent ,

a)

When , one has

Trace ( )( ) ( ) ( ) ( ) ++=

v

k voud

uuh

hhU Q' 1

11

log'12 . (45)

b)

All L functions with Grossencharakter on k satisfy the Riemann Hypothesis .

To prove that (a) implies (b), we shall prove (assuming (a)) the positivity of the Weil distribution,

+= v

vDDd 11log . (46)

We have that for 0= , the map E ,

( )( ) ( )

=k q

qgf ggf E 2/ 1

k C g , (47)

defines a surjective isometry from ( )02 X L to ( )k C L2 such that,

( ) ( )E aV aaEU 2/ 1= , (48)

where the left regular representation V of k C on ( )k C L2 is given by,

( )( )( ) ( )gagaV 1= k C ag , . (49)

Let S be the subspace of ( )k C L2 given by,

( ) ( ) == ,,,0; 12 gggC LS k . (50)

We shall denote by the same letter the corresponding orthogonal projection.


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Let 0,B be the subspace of ( )02 X L spanned by the ( )0ASf such that ( )xf and ( )xf vanish for>x and 0,Q be the corresponding orthogonal projection. Let ( )0ASf be such that ( )xf and

( )xf vanish for >x , then ( ) ( )gf E vanishes for >g , and the equality

( ) ( ) ( )

=g

f E gf E 1 ( )0ASf , (51)

shows that ( ) ( )gf E vanishes for 1


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( ) ( ) ( )

=

+

+=iRz

L

zd zf N f

N B

,~21

,~

/

021

,~

(57)

where B is the open strip ( ) ,21

,21

Re;

= C B

+ 21

,~N is the multiplicity of the

zero, ( )zd is the harmonic measure of with respect to the line C iR , and the Fourier transform f of f is defined by

( ) ( ) ( ) =k C

ud uuuf f ~,~ . (58)

Let us first recall the Weil explicit formulas. One lets k be a global field. One identifies thequotient 1,/ k k C C with the range of the module,

{ } += RC ggN k ; . (59)

One endows N with its normalized Haar measure xd . Given a function F on N such that, for

some21>b ,

( ) bF 0= 0 , ( ) bF = 0 , , (60)

one lets,( ) ( ) = N s d F s 2/ 1 . (61)

Given a Grossencharakter , i.e. a character of k C and any in the strip ( ) 1Re0

on some mathematical connections between fermat’s last theorem, modular functions, modular...

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