experiments on siegel modular forms of genus 2 …...i for siegel modular forms, the rst weight at...

Experiments on Siegel modular forms of genus 2(Not only on the Paramodular Conjecture)

Nathan Ryan

Modular Forms and Curves of Low Genus: Computational AspectsICERM

October 1st, 2015

Nathan Ryan Experiments on Siegel modular forms of genus 2 (Not only on the Paramodular Conjecture)

Experiments with L-functions of Siegel modular forms

1. Compute a basis for the space of Siegel modular forms ofgenus 2 and identify the Hecke eigenforms.

2. Compute (a lot of) coefficients of the Hecke eigenforms.

3. Compute the Hecke eigenvalues of the Hecke eigenforms.

4. Compute the Euler factors of the L-function and therefore theDirichlet series.

5. Evaluate the L-function at a point s.


Why compute Siegel modular forms and their L-functions?

I Verify conjectures. . .

I Formulate conjectures. . .

I Discovering unexpected phenomena. . .

I To understand abstract things concretely. . .

I Because we can. . .


Harder’s Conjecture

I Generalizes Ramanujan’s congruence

τ(p) ≡ p11 + 1 (mod 691).

I Let f ∈ S(1)r be a Hecke eigenform with coefficient field Qf

and let ` be an ordinary prime in Qf (i.e. such that the `-thHecke eigenvalue of f is not divisible by `). Suppose s ∈ N issuch that `s divides the algebraic critical value Λ(f , t). Then

there exists a Hecke eigenform F ∈ S(2)k,j , where k = r − t + 2,

j = 2t − r − 2, such that

µpδ (F ) ≡ µpδ (f ) + pδ(k+j−1) + pδ(k−2) (mod `s)

for all prime powers pδ.


Harder’s Conjecture

I In joint work with Ghitza and Sulon we verified the conjecturecomputationally for r ≤ 60 and for

pδ ∈ {2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 125}.

I A variant of Harder’s conjecture due to Bergstrom, Faber, vander Geer, and Harder involves critical values of the symmetricsquare L-function. We verified this conjecture for r ≤ 32 androughly the same list of prime powers.

Our computations were in weight (2, j).


Maeda’s ConjectureI Let [Tp] be the matrix of the Hecke operator on the space of

modular forms of weight k and level 1. It has been conjecturedthat the characteristic polynomial of this matrix is irreducible.

I For Siegel modular forms, the first weight at which the spacebecomes two-dimensional, the characteristic polynomialfactors into linear factors. In weights 24 and 26 we have these“terrifying example[s] due to Skoruppa”.

I As we verified Harder’s conjecture, we found terrifyingexamples of vector valued Siegel modular forms in weights(k , 2).


Rankin convolution

I A Siegel modular form F has a Fourier expansion indexed bypositive semidefinite binary quadratic forms. If we gather thecoefficients in a certain way, we can write

F (z , τ, z ′) =∑n≥0

φF ,n(z , τ)q′n

where each φF ,n is a Jacobi form of the same weight and ofindex n.

I For two modular forms F and G define the convolutionDirichlet series:

DF ,G (s) = ζ(2s − 2k + 4)∑n≥1

〈φG ,n, φF ,n〉n−s ,

where 〈·, ·〉 is the Petersson inner product of two Jacobi forms.


Rankin convolution

I A Theorem due to Skoruppa and Zagier: if F is a Siegelmodular form and G is a Saito-Kurokawa lift, then

DF ,G (s) = 〈φF ,1, φG ,1〉L(F , s)

where L(F , s) is the spin L-function of F .

I In joint work with Skoruppa and Stromberg, we asked what ifG is not a lift?

I We identified all the eigenforms in weights between 20 and 30and used those to compute the Jacobi forms used in thecomputations of DF ,G (s).


Rankin convolution

I We implemented a method to compute the Petersson innerproduct.

I We computed DF ,G (s) for all Hecke eigenforms F ,G of thesame weight k for 20 ≤ k ≤ 30.

I We showed that the Dirichlet series DF ,G (s) was not anL-function: its coefficients weren’t even multiplicative!


Formulating Bocherer’s Conjecture in the paramodularsetting

For a fundamental discriminant D < 0 coprime to the level,Bocherer’s Conjecture states:

L(F , 1/2, χD) = CF |D|1−kA(D)2

where F is a Siegel modular form of weight k , CF > 0 is aconstant that only depends on F , and A(D) is an average of thecoefficients of F of discriminant D.


Putting the Conjecture in context:

I It’s a generalization of Waldspurger’s formula relating centralvalues of elliptic curve L-functions to sums of coefficients ofhalf-integer weight modular forms.

I In general, computing coefficients of Siegel modular forms ismuch easier than computing their Hecke eigenvalues (andtherefore their L-functions). So this formula would provide acomputationally feasible way to compute lots of central values.

I A theorem of Saha states that a weak version of theconjecture implies multiplicity one for Siegel modular forms oflevel 1.


The state of the art:

I Bocherer originally proved it for Siegel modular forms that areSaito-Kurokawa lifts.

I Kohnen and Kuss verified the conjecture numerically for thefirst few rational Siegel modular eigenforms that are not lifts(these are in weight 20-26) for only a few fundamentaldiscriminants.

I Raum (very) recently verified the conjecture numerically fornonrational Siegel modular eigenforms that are not lifts for afew more fundamental discriminants.

I Bocherer and Schulze-Pillot formulated a conjecture for Siegelmodular forms with level > 1 and proved it when the form is aYoshida lift.


Suppose we are given a paramodular form F ∈ Sk(Γpara[p]) so thatfor all n ∈ Z, F |T (n) = λF ,nF = λnF where T (n) is the nth Heckeoperator. Then we can define the spin L-series by the Euler product

L(F , s) :=∏

q prime

Lq

(q−s−k+3/2)−1,

where the local Euler factors are given by

Lq(X ) := 1−λqX + (λ2q−λq2−q2k−4)X 2−λqq

2k−3X 3 +q4k−6X 4

for q 6= p, and Lp(X ) has a similar formula.


We define

AF (D) :=∑

{T>0 : disc T=D}/Γ0(p)

a(T ; F )

ε(T )

where ε(T ) := #{U ∈ Γ0(p) : T [U] = T}.

Conjecture (Paramodular Bocherer’s Conjecture, I)

Suppose F ∈ Sk(Γpara[p])+. Then, for fundamental discriminantsD < 0 we have

L(F , 1/2, χD) = ?CF |D|1−kA(D)2

where CF is a positive constant that depends only on F , and ? = 1when p - D, and ? = 2 when p | D.


Theorem (R., Tornarıa)

Let F = Grit(f ) ∈ Sk(Γpara[p])+ where p is prime and f is a Heckeeigenform of degree 1, level p and weight 2k − 2. Then thereexists a constant CF > 0 so that

L(F , 1/2, χD) = ?CF |D|1−kA(D)2

for D < 0 a fundamental discriminant, and ? = 1 when p - D, and? = 2 when p | D.


The idea of the proof is to combine four ingredients:

I the factorization of the L-function of the Gritsenko lift asgiven by Ralf Schmidt,

I Dirichlet’s class number formula,

I the explicit description of the Fourier coefficients of theGritsenko lift and

I Waldspurger’s theorem.


Theorem (R., Tornarıa)

Let F ∈ S2(Γpara[p])+ where p < 600 is prime. Then, numerically,there exists a constant CF > 0 so that

L(F , 1/2, χD) = ?CF |D|1−kA(D)2

for −200 ≤ D < 0 a fundamental discriminant, and ? = 1 whenp - D, and ? = 2 when p | D.


Results of Cris Poor and Dave Yuen:

I Determine what levels of weight 2 paramodular cuspformshave Hecke eigenforms that are not Gritsenko lifts.

I Provide Fourier coefficients (up to discriminant 2500) for allparamodular forms of prime level up to 600 that are notGritsenko – not enough to compute central values of twists.


Brumer and Kramer formulated the following conjecture:

Conjecture (Paramodular Conjecture)

Let p be a prime. There is a bijection between lines of Heckeeigenforms F ∈ S2(Γpara[p]) that have rational eigenvalues and arenot Gritsenko lifts and isogeny classes of rational abelian surfacesA of conductor p. In this correspondence we have that

L(A, s,Hasse-Weil) = L(F , s).

We remark that it is merely expected that the two L-seriesmentioned above have an analytic continuation and satisfy afunctional equation.


In our computations we assume the Paramodular conjecture forthese curves:

p ε C

277 + y2 + y = x5 − 2x3 + 2x2 − x

349 + y2 + y = −x5 − 2x4 − x3 + x2 + x

389 + y2 + xy = −x5 − 3x4 − 4x3 − 3x2 − x

461 + y2 + y = −2x6 + 3x5 − 3x3 + x

523 + y2 + xy = −x5 + 4x4 − 5x3 + x2 + x

587 + y2 = −3x6 + 18x4 + 6x3 + 9x2 − 54x + 57

587 - y2 +(x3 + x + 1

)y = −x3 − x2


The Selberg data we use are:

I L∗(F , s) =(√

p

4π2

)sΓ(s + 1/2)Γ(s + 1/2)L(F , s).

I conjecturally L∗(F , s) = ε L∗(F , 1− s) when F ∈ S2(Γpara[p])ε.

I we use Mike Rubinstein’s lcalc to compute the centralvalues using this Selberg data and Sage code we wrote tocompute the coefficients of the Hasse-Weil L-function


D A(D; F277)L(F277,1/2,χD )

C277|D| D A(D; F277)

L(F277,1/2,χD )C277

|D|

-3 -1 1.000000 -83 6 36.000000-4 -1 1.000000 -84 1 1.000000-7 -1 1.000000 -87 -3 9.000000

-19 -2 4.000000 -88 -2 4.000000-23 0 -0.000000 -91 -1 1.000000-39 1 1.000000 -116 3 9.000000-40 -6 36.000000 -120 -2 4.000000-47 0 0.000000 -123 -1 1.000000-52 5 25.000000 -131 -10 100.000000-55 -2 4.000000 -136 -6 36.000000-59 3 9.000000 -155 -10 100.000000-67 -8 64.000000 -164 -5 25.000000-71 2 4.000000 -187 8 64.000001-79 0 0.000000 -191 2 3.999999


Two surprises

Suppose F ∈ Sk(Γpara[p])−, and let D < 0 be a fundamentaldiscriminant.

I When(

Dp

)= +1, the Conjecture holds trivially. Indeed, note

that for such F the sign of the functional equation is −1 andso the central critical value L(F , s, χD) is zero. On the otherhand, A(D) can be shown to be zero using the Twin mapdefined by Poor and Yuen.

I On the other hand, the formula of Conjecture 1 fails to hold

in case(

Dp

)= −1. Since A(D) is an empty sum for this type

of discriminant, the right hand side of the formula vanishestrivially. However, the left hand side is still an interestingcentral value, not necessarily vanishing.


Two surprises

Let LD := L(F−587, 1/2, χD) |D|. This table shows fundamentaldiscriminants for which

(D

587

)= −1. The obvious thing to notice is

that the numbers in the table appear to be squares and so thenatural question to ask is: squares of what?

D -4 -7 -31 -40 -43 -47

LD/L−3 1.0 1.0 4.0 9.0 144.0 1.0


Two surprises

Up to now we have only considered twists by imaginary quadraticcharacters; namely χD =

( ·D

)for D < 0. What if we consider

positive D?

I Since

A(D) = AF (D) :=1

2

∑{T>0:discT=D}/Γ0(p)

a(T ; F )

ε(T )

we see that for D > 0 the sum is empty. And so Bocherer’sConjecture shouldn’t make sense.


Two surprises

Let LD := L(F277, 1/2, χD) |D| and(

D277

)= +1. Again, these

seem to be squares, but squares of what?

D 12 13 21 28 29 40

LD/L1 225.0 225.0 225.0 225.0 2025.0 900.0


A new conjecture

Conjecture

Let N be squarefree. Suppose F ∈ Snewk (Γpara[N]) is a Hecke

eigenform and not a Gritsenko lift. Let ` and d be fundamentaldiscriminants such that `d < 0 and such that `d is a squaremodulo 4N. Then

B`,F (`d)2 = kF ·{

2νN(`) L(F , 1/2, χ`) |`|k−1}

·{

2νN(d) L(F , 1/2, χd) |d |k−1}

for some positive constant kF independent of ` and d .


A new conjecture

Fix ρ such that ρ2 ≡ `d (mod 4N). Then

B`,F (`d) =

∣∣∣∣2νN(gcd(`,d)) ·∑

ψ`(T )a(T ; F )

ε(T )

∣∣∣∣where the sum is over {T = [Nm, r , n] > 0 : discT = `d , r ≡ ρ(mod 2N)}/Γ0(N) and where ψ`(T ) is the genus charactercorresponding to ` | disc T . This is independent of the choice of ρ.

I Essentially, B`,F (`d) is the same sum as AF (`d), butappropriately twisted by the genus character ψ`.


Proposition

Let d > 1 and assume Bocherer’s Conjecture. Then, the ratio ofspecial values 2ν277(d) L(F277, 1/2, χd) |d |/L(F277, 1/2) is divisibleby 152.

I We note that the torsion of C277 is 15.

I We conjecture that this result generalizes to the other formswe considered (the data back this up) and even moregenerally.


Computational challenges

1. Computing enough Fourier coefficients:I to get Hecke eigenvaluesI to use Bocherer’s conjecture to do statistics on Siegel modular

form L-functions

2. Computing enough Hecke eigenvalues:I to get Euler factorsI to check cogruencesI to use Faltings-Serre


Workaround I: New way to compute Hecke eigenvalues

I Joint with Ghitza, based on an idea of Voight.

I The action of Hecke is defined as follows: take a double cosetΓMΓ and decompose it as ΓMΓ = ∪Γα. Then

(F |kΓMΓ)(Z ) =∑

(F |kα)(Z )

=∑

F ((AZ + B)(CZ + D)−1) where α =(

A BC D

).

I Compute eigenvalues this way! Fix a Z in the upperhalf-space. If F is an eigenform, compute F (Z ) and theF ((AZ + B)(CZ + D)−1) above. The quotient should be theeigenvalue.

I Based on work of Broker and Lauter in which they explain howto evaluate Siegel modular forms with rigorous error bounds.


Workaround II: New way to compute a lot of Fouriercoefficients

I Joint with Rupert, Sirolli and Tornarıa.

I Identify as many of the first Fourier Jacobi coefficients of theform we want to compute as possible using existing data.Identify the Jacobi forms using the modular symbols methodto compute Jacobi forms. Use existing techniques to computea large of coefficients of those Jacobi forms.

I Bootstrap from here by using relations between the Fouriercoefficients of Siegel forms and relations between the FourierJacboi coefficients of Siegel forms.


Workaround III: Evaluating L-functions with fewcoefficients

I Joint with Farmer.

I Using the approximate functional equation we can vary thetest function that We use to evaluate L(s) for a fixed s.

I we find an optimal test function by finding the least squaresfit to minimize The error based on assuming the Ramanujanconjecture.

I using our method we were able to evaluate the degree 10L-function associated to a Siegel modular form of weight 20at s = 1

2 + 5i to an error of ±0.00016.

I This is only using 79 Euler factors!

I Computing things naively, we got the error was bigger thanthe value.


experiments on siegel modular forms of genus 2 …...i for siegel modular forms, the rst weight at...

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