Tables of Paramodular Forms · Cris and David Tables of Paramodular Forms. OutlinePart One Part...

Outline Part One Part Two Part Three

Tables of Paramodular Forms

Cris Poor David S. YuenFordham University Lake Forest College

including work in progess with:Jeff Breeding, Fordham University

Modular Forms and Curves of Low Genus: ComputationalAspects

ICERM, September 2015

Cris and David Tables of Paramodular Forms

Computational Aspects of Modularity

1. Part I . The Paramodular Conjecture.

2. Part II . Using Fourier-Jacobi expansions to make rigoroustables of paramodular forms. (Joint work with J. Breedingand D. Yuen.)

3. Part III . Using Fourier-Jacobi expansions to make heuristictables of paramodular forms. (Joint work with D. Yuen.)

4. Our paramodular website exists:math.lfc.edu/∼yuen/paramodular

All elliptic curves E/Q are modular

Theorem (Wiles; Wiles and Taylor; Breuil, Conrad, Diamond andTaylor)

Let N ∈ N. There is a bijection between

1. isogeny classes of elliptic curves E/Q with conductor N

2. normalized Hecke eigenforms f ∈ S2(Γ0(N))new with rationaleigenvalues.

In this correspondence we have L(E , s,Hasse) = L(f , s,Hecke).

Eichler (1954) proved the first examplesL(X0(11), s,Hasse) = L(η(τ)2η(11τ)2, s,Hecke).

Shimura proved 2 implies 1.

Weil added N = N.

All abelian surfaces A/Q are paramodular

Paramodular Conjecture (Brumer and Kramer 2009)

1. isogeny classes of abelian surfaces A/Q with conductor N andendomorphisms EndQ(A) = Z,

2. lines of Hecke eigenforms f ∈ S2(K (N))new that have rationaleigenvalues and are not Gritsenko lifts from Jcusp2,N .

In this correspondence we have

L(A, s,Hasse-Weil) = L(f , s, spin).

Remarks

The paramodular group of level N,

K (N) =

∗ N∗ ∗ ∗∗ ∗ ∗ ∗/N∗ N∗ ∗ ∗N∗ N∗ N∗ ∗

∩ Sp2(Q), ∗ ∈ Z,

K (N)\H2 is a moduli space for complex abelian surfaces withpolarization type (1,N).

K (N) is the stabilizer in Sp2(Q) of Z⊕ Z⊕ Z⊕ NZ.

New form theory for paramodular groups:Ibukiyama 1984; Roberts and Schmidt 2004, (LNM 1918).

Grit : Jcuspk,N → Sk (K (N)), the Gritsenko lift from Jacobi cuspforms of index N to paramodular cusp forms of level N is anadvanced version of the Maass lift.

Remarks

K (N) =

∗ N∗ ∗ ∗∗ ∗ ∗ ∗/N∗ N∗ ∗ ∗N∗ N∗ N∗ ∗

∩ Sp2(Q), ∗ ∈ Z,

Remarks

K (N) =

∗ N∗ ∗ ∗∗ ∗ ∗ ∗/N∗ N∗ ∗ ∗N∗ N∗ N∗ ∗

∩ Sp2(Q), ∗ ∈ Z,

Remarks

K (N) =

∗ N∗ ∗ ∗∗ ∗ ∗ ∗/N∗ N∗ ∗ ∗N∗ N∗ N∗ ∗

∩ Sp2(Q), ∗ ∈ Z,

Remarks

K (N) =

∗ N∗ ∗ ∗∗ ∗ ∗ ∗/N∗ N∗ ∗ ∗N∗ N∗ N∗ ∗

∩ Sp2(Q), ∗ ∈ Z,

All abelian surfaces A/Q are paramodularThe Paramodular Conjecture again after the remarks

Paramodular Conjecture

1. isogeny classes of abelian surfaces A/Q with conductor N andendomorphisms EndQ(A) = Z,

2. lines of Hecke eigenforms f ∈ S2(K (N))new that have rationaleigenvalues and are not Gritsenko lifts from Jcusp2,N .

In this correspondence we have

L(A, s,Hasse-Weil) = L(f , s, spin).

A glimpse of higher modularity?A sequence of discrete subgroups

The L-groups of symplectic groups are orthogonal groups.

Consider the following special, stable, integral orthogonal groups ofspinor norm one.

Γ0(N) ∼= SO+Z

1−2N

K (N) ∼= SO+Z

1−2N

(Gritsenko, Nikulin 1998)

A glimpse of higher modularity?A sequence of discrete subgroups

The L-groups of symplectic groups are orthogonal groups.

Consider the following special, stable, integral orthogonal groups ofspinor norm one.

Γ0(N) ∼= SO+Z

1−2N

K (N) ∼= SO+Z

1−2N

(Gritsenko, Nikulin 1998)

A glimpse of higher modularity?

Q = Q2n+1(N) =

−2N..

Q2n+1(N) = M ′M; L = MZ2n+1; U is hyperbolic plane(Q2n+1(N),Z2n+1

) ∼= (·,L) ∼= Un ⊕ (−2N)

SO+Z (Q) =

{g ∈ SL2n+1(Z) : g ′Qg = Q,∀x ∈ L∗, gx − x ∈ L, sp. nm.(g) = 1}

A glimpse of higher modularity?References

1. Pei-Yu Tsai, On Newforms for Split Special Odd OrthogonalGroups. (Harvard thesis: 2013)

2. Benedict Gross, On the Langlands correspondence forsymplectic motives. (2015)

Definition of Siegel Modular Form

Siegel Upper Half Space: Hn = {Z ∈ Msymn×n(C) : ImZ > 0}.

Symplectic group: σ =(A BC D

)∈ Spn(R) acts on Z ∈ Hn by

σ · Z = (AZ + B)(CZ + D)−1.

Γ ⊆ Spn(R) such that Γ ∩ Spn(Z) has finite index in Γ andSpn(Z)

Slash action: For f : Hn → C and σ ∈ Spn(R),(f |kσ) (Z ) = det(CZ + D)−k f (σ · Z ).

Siegel Modular Forms: Mk(Γ) is the C-vector space ofholomorphic f : Hn → C that are “bounded at the cusps” andthat satisfy f |kσ = f for all σ ∈ Γ.

Definition of Siegel Modular Form

Cusp Forms:Sk(Γ) = {f ∈ Mk(Γ) that “vanish at the cusps”}Fourier Expansion: f (Z ) =

∑T>0 a(T ; f )e(tr(ZT ))

For paramodular groups: n = 2; Γ = K (N);

symmetric T ∈(

12Z NZ

Plus and Minus SpacesParamodular Fricke involution

There is a paramodular involution µN that splits spaces ofparamodular forms into plus and minus spaces.

Sk (K (N)) = Sk (K (N))+ ⊕ Sk (K (N))−

Sk (K (N))ε = {f ∈ Sk (K (N)) : f |kµN = ε f }

µN =1√N

0 N−1 0

0 1−N 0

∈ Sp2(R)pr

Fourier-Jacobi expansions

Fourier expansion of elliptic modular form:f (τ) =

∑n≥0 a(n; f )e (nτ).

Fourier expansion of Siegel modular form:f (Z ) =

∑T≥0 a(T ; f )e(tr(ZT ))

Fourier expansion of paramodular form f ∈ Mk(K (N)) incoordinates: f ( τ z

z ω ) =∑n,r ,m∈Z, n,m≥0, 4Nnm≥r2 a(

(n r/2r/2 Nm

); f )e(nτ + rz + Nmω)

Fourier-Jacobi expansion of paramodular form f ∈ Mk(K (N)):f ( τ z

z ω ) =∑

m∈Z,m≥0 φNm(τ, z)e(Nmω)The FJE of paramodular form can be a more suggestiveanalogy to the elliptic case than the full Fourier expansion.

Coefficients φNm ∈ Jk,Nm are Jacobi forms.

φNm(τ, z) =∑

n,r∈Z: n≥0, 4Nnm≥r2 a((

n r/2r/2 Nm

); f )e(nτ + rz)

∑n≥0 a(n; f )e (nτ).

∑T≥0 a(T ; f )e(tr(ZT ))

z ω ) =∑n,r ,m∈Z, n,m≥0, 4Nnm≥r2 a(

(n r/2r/2 Nm

z ω ) =∑

φNm(τ, z) =∑

n,r∈Z: n≥0, 4Nnm≥r2 a((

n r/2r/2 Nm

); f )e(nτ + rz)

∑n≥0 a(n; f )e (nτ).

∑T≥0 a(T ; f )e(tr(ZT ))

z ω ) =∑n,r ,m∈Z, n,m≥0, 4Nnm≥r2 a(

(n r/2r/2 Nm

z ω ) =∑

φNm(τ, z) =∑

n,r∈Z: n≥0, 4Nnm≥r2 a((

n r/2r/2 Nm

); f )e(nτ + rz)

∑n≥0 a(n; f )e (nτ).

∑T≥0 a(T ; f )e(tr(ZT ))

z ω ) =∑n,r ,m∈Z, n,m≥0, 4Nnm≥r2 a(

(n r/2r/2 Nm

z ω ) =∑

φNm(τ, z) =∑

n,r∈Z: n≥0, 4Nnm≥r2 a((

n r/2r/2 Nm

); f )e(nτ + rz)

∑n≥0 a(n; f )e (nτ).

∑T≥0 a(T ; f )e(tr(ZT ))

z ω ) =∑n,r ,m∈Z, n,m≥0, 4Nnm≥r2 a(

(n r/2r/2 Nm

z ω ) =∑

φNm(τ, z) =∑

n,r∈Z: n≥0, 4Nnm≥r2 a((

n r/2r/2 Nm

); f )e(nτ + rz)

Fourier-Jacobi expansion (FJE)

FJE: f ( τ zz ω ) =

∑m∈Z,m≥0

φNm(τ, z)e(Nmω)

The Fourier-Jacobi expansion of a paramodular form is fixed termby term by the following subgroup of the paramodular group K (N):

P2,1(Z) =

∗ 0 ∗ ∗∗ ∗ ∗ ∗∗ 0 ∗ ∗0 0 0 ∗

∩ Sp2(Z), ∗ ∈ Z,

P2,1(Z)/{±I} ∼= SL2(Z) n Heisenberg(Z)

Thus the coefficients φNm are automorphic forms in their own rightand easier to compute than Siegel modular forms. This is onemotivation for the introduction of Jacobi forms.

Fourier-Jacobi expansionTwo Natural Questions

∑m∈Z,m≥0

φNm(τ, z)e(Nmω)

We ask two natural questions:

1 Which Jacobi forms can be a Fourier-Jacobi coefficient of aparamodular form?

2 Can we find the consistency conditions among a sequence ofJacobi forms that characterize the Fourier-Jacobi expansionsof paramodular forms?

Fourier-Jacobi expansionNatural Question Number One

∑m∈Z,m≥0

φNm(τ, z)e(Nmω)

The existence of the Gritsenko lift answers the first question. AnyJacobi cusp form can be the first Fourier-Jacobi coefficient of aparamodular form.

Theorem (Gritsenko)

For φ ∈ Jcuspk,m the series Grit(φ) converges and defines a map

Grit : Jcuspk,m → Sk (K (m))ε , ε = (−1)k .

Grit(φ)( τ zz ω ) =

∑`∈N

`2−k(φ|V`)(τ, z)e(`mω).

Fourier-Jacobi expansionsNatural Question Number Two

Fix N ∈ N. Consider a formal series of Jacobi forms φNm ofweight k and index Nm: ∑

m∈NφNmξ

If this sequence of Jacobi forms is the Fourier-Jacobi expansion ofa paramodular form f in Sk (K (N))ε, then the Fourier coefficientsof the Jacobi forms satisfy simple relations that follow from theµN -eigenvalue condition on f :

c(n, r ;φNm) = ε(−1)kc(m, r ;φNn)

Experimentally, these conditions seem to be sufficient as well, butthis has only been proven for N ≤ 4.

Natural Question TwoReferences

The sufficiency of the conditionsc(n, r ;φNm) = ε(−1)kc(m, r ;φNn) is not hard to believe becauseK (N)+ = 〈K (N), µN〉 = 〈P2,1(Z), µN〉. The only obstruction isconvergence.

1. Aoki, Estimating Siegel modular forms of genus 2 using Jacobiforms (2000) (N = 1)

2. Ibukiyama, Poor, Yuen, Jacobi forms that characterizeparamodular forms. (2013) (N ≤ 4)

3. Brunier, Raum, Kudla’s Modularity Conjecture and FormalFourier-Jacobi Series (2015)

Notation: K (N)∗ = 〈K (N),All paramodular AL-involutions〉

Strategy for the rigorous computation of paramodularforms

1. Find a sufficient number of Fourier-Jacobi coefficients, say L,that determine a paramodular form in Sk(K (N))ε, withoutknowing the dimension of Sk(K (N))ε.

2. Use the theory of theta blocks due to Gritsenko, Skoruppa,and Zagier, to span spaces of Jacobi forms.

3. Use the conditions c(n, r ;φNm) = ε(−1)kc(m, r ;φNn) todefine a vector space V containing all possible initialFourier-Jacobi expansions of length L from Sk(K (N))ε.

4. If you can construct dimV linearly independent paramodularforms in Sk(K (N))ε then you have provendimSk(K (N))ε = dimV .

How many Fourier-Jacobi coefficients determine aparamodular cusp form?

Definition

The Jacobsthal function j(N) is defined to be the smallest positiveinteger m such that every sequence of m consecutive positiveintegers contains an integer coprime to N.

Examples: j(2) = j(3) = j(4) = j(5) = 2,j(6) = 4, j(10) = 4, j(15) = 3.

j(N) ∈ O((lnN)2); H. Iwaniec, On the problem of Jacobsthal(1978)

How many FJCs determine a paramodular form?

Theorem (Breeding, Poor, Yuen)

Let k,N ∈ N. Let χ : K (N)∗ → {±1} be a character trivial onK (N). Let f ∈ Sk(K (N)∗, χ) be a common eigenfunction of theparamodular Atkin-Lehner involutions and have Fourier-Jacobiexpansion

f =∞∑j=1

φjNξjN .

Let N = pα11 · · · p

α`` be the prime factorization of N and set

N = p1 · · · p`. Choose µ ∈ N such that 2µ+ 1 ≥ j(N/pi ) for all i .Let κ be 1 when N is prime, 2 when N is a composite prime powerand 1 + µ+ µ2 otherwise. If φjN = 0 for

j ≤ κ k

10N∏pr ||N

pr + pr−2

pr + 1, then f = 0.

Rigorous dimensions of weight two paramodular forms

Table 1. Dimension of S2(K (N)) and number of FJ-coefficientsneeded in proof for N ≤ 60. Omitted levels N indicate thatdim S2(K (N)) = 0.

N 37 43 53 57 58

dim 1 1 1 1++ 1++

FJCs 7 8 10 9 8

These eigenforms are all Gritsenko lifts.Paramodular Conjecture (vacuously) true for (odd) levels N ≤ 60.

Rigorous dimensions of weight three paramodular forms

Table 2. Dimension of S3(K (N)) and number of FJ-coefficientsneeded in proof for N ≤ 40. Omitted levels N indicate thatdim S3(K (N)) = 0.

N 13 17 19 21 22 23 25 26 27 28

dim 1 1 1 1+− 1+− 1 1 1+− 1 1−+

FJCs 3 4 5 4 3 6 8 5 7 4

N 29 31 32 33 34 35 37 38 39 40

dim 2 2 1 2+−−+ 2+−2 1+− 4 2+−2 2+−2 1−+

FJC 7 9 10 6 7 3 10 6 9 7

Strategy for heuristic computation of paramodular forms

Run everything in Fq, for an auxillary prime q, to save memory.

1. Use enough Fourier-Jacobi coefficients to achieve insight butnot rigor for Sk(K (N))ε.

2. Use the theory of theta blocks due to Gritsenko, Skoruppa,and Zagier, to span spaces of Jacobi forms.

3. Use the conditions c(n, r ;φNm) = ε(−1)kc(m, r ;φNn) todefine a vector space V containing all possible initialFourier-Jacobi expansions from Sk(K (N))ε for a fixed shortlength.

4. Hope that dim Sk(K (N))ε = dimV .

Heuristic tables: k = 2 paramodular newforms: N ≤ 600.

+new = dim(

(S2(K (N))new)+ /Grit(Jcusp2,N

))−new = dim (S2(K (N))new)− .

N +new −new various comments

249 ≥ 1 ss-Jac

277 = 1 ss-Jac

295 ≥ 1 ss-Jac

349 = 1 ss-Jac

353 = 1 ss-Jac

388 ≥ 1 ss-Jac

389 = 1 ss-Jac

394 1 ss-Jac

427 1 ss-Jac

461 ≤ 1 ss-Jac

464 1 ss-Jac

472 1 ss-Jac

511 2 unknown 4 dim A/Q, quad pair,√

523 ≤ 1 ss-Jac

550 1 surface unknown

555 = 1 ss-Jac

561 1 ss-Prym

574 1 ss-Jac

587 ≤ 1 = 1 both ss-Jac

597 1 ss-Jac

· · ·

657 1 Weil Res., E/Q(√−3)

775 1 Weil Res., E/Q(√

954 2 twists by√−3

Thanks to Armand Brumer for all the abelian surfaces in this table.

(And thank you to Andrew Sutherland for correcting some of myerrors.)

Thank you!

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