paramodularitysiegelmodularforms.org/pages/degree2/paramodular-wt2-all/... · 2018. 6. 13. · 2....

Paramodularity

Cris PoorFordham University

TORA IXTexas-Oklahoma Representations and Automorphic Forms

Norman, April 2018

Cris Paramodularity TORA 1 / 58

A Survey of the Paramodular Conjecture

1. Part I . All elliptic curves defined over Q are modular.

2. Part II . All abelian surfaces defined over Q with minimalendomorphisms are paramodular.

3. Part III . Systematic Evidence for the Paramodular Conjecture

4. Part IV . Paramodular Conjecture 2.0

5. Part V . Heuristic tables of paramodular forms.


All elliptic curves E/Q are modular

Theorem (Wiles; Wiles & Taylor; Breuil, Conrad, Diamond & Taylor)

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 gave a construction from 2 to 1.

Weil added N = N.


L-functions of elliptic curves over Q

1. The local Hasse p-Euler factor is the characteristic polynomial ofFrobenius on the Tate module T`(E ) of the elliptic curve E .

Qp(E , t) = det(I − t Frobp |T`(E )Ip

)2. The local Hasse Zeta function can be computed by counting points

on the elliptic curve E over finite fields.

Zp(E , t) = exp

( ∞∑n=1

#{Points on E/Fpn}tn

n

)=

Qp(E , t)

(1− t)(1− pt)

3. Global Hasse L-function

L(E , s,Hasse) =∏

primes p

Qp(E , p−s)−1


An example: LMFDB 11.a3

1. E [F] = {(x , y , z) ∈ P2(F) : y2z + yz2 = x3 − x2z}

n 1 2 3 4 5 6 7

#E [F2n ] 5 5 5 25 25 65 145

2. The Hasse Zeta function at p = 2:

Z2(E , t) = exp

(5t + 5

t2

2+ 5

t3

3+ 25

t4

4+ · · ·

)=

1 + 2t + 2t2

(1− t)(1− 2t)

3. ∴ Q2(E , t) = 1 + 2t + 2t2


An example: LMFDB 11.a3

1. Global Hasse L-function

L(E , s,Hasse) =∏

primes p

Qp(E , p−s)−1 = 1− 2

2s− 1

3s+

2

4s+

1

5s+ · · ·

2. There is an elliptic modular newform f11 ∈ S2 (Γ0(11))

f11(τ) = q − 2q2 − q3 + 2q4 + q5 + · · ·

= η(τ)2η(11τ)2 = q∞∏n=1

(1− qn)2(1− q11n)2

=1

2θ

2 1 1 11 2 0 11 0 8 41 1 4 8

(τ)− 1

2θ

2 0 1 00 2 0 11 0 6 00 1 0 6

(τ)


All elliptic curves E/Q are modular (again)

Theorem (Wiles; Wiles & Taylor; Breuil, Conrad, Diamond & Taylor)


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).

Credit Taniyama (1956) and Shimura (∼ 1963) for importantmodularity conjectures.

In its final form, this is a classification theorem.

Cremona has led the classification of E/Q up toconductor N ≤ 400 000. (johncremona.github.io/ecdtata)


Part TwoThe Paramodular Conjecture

1. What are abelian surfaces A/Q?

2. What are paramodular forms f ∈ Sk(K (N))?

3. How does the Paramodular Conjecture relate the two?


Abelian Varieties

Let K ⊆ C be an algebraic number field.

Definition

An abelian variety A/K is a projective variety defined over K with analgebraic group law also defined over K .

g=1 Elliptic curves: Ahol∼= C1/ (Z + τZ) for τ ∈ H.

g=2 Abelian surfaces: Ahol∼= C2/

(DZ2 + ZZ2

)for Z ∈ H2.

D = diag(1, d) for d ∈ N gives the type of “polarization” of A.

H2 = {Z ∈ Msym2×2(C) : ImZ > 0}, the Siegel upper half space.

But for g > 1 the equations defining A inside a projective space arecomplicated!


How can we construct some abelian surfaces?

1. If C is a curve of genus 2 defined over Q, then

A = Jac(C)

is an abelian surface defined over Q with principal polarization, i.e.,type D = diag(1, 1).

2. If a genus 3 curve C3 is a ramified double cover of a genus 1 curve C1,then the abelian surface

A = Prym(C3/C1) = Jac(C3)/ Jac(C1)

has a natural polarization of type D = diag(1, 2).


Definition of Siegel Modular Forms

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 and Spn(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 of holomorphicf : Hn → C that are “bounded at the cusps” and that satisfyf |kσ = f for all σ ∈ Γ.

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


Definition of paramodular form

A paramodular form is a Siegel modular form for a paramodulargroup. In degree 2, the paramodular group of level N, is

Γ = K (N) =

∗ N∗ ∗ ∗∗ ∗ ∗ ∗/N∗ N∗ ∗ ∗N∗ N∗ N∗ ∗

∩ Sp2(Q), ∗ ∈ Z,

K (N) is the stabilizer in Sp2(Q) of Z⊕ Z⊕ Z⊕ NZ.TK (N)\H2 is a moduli space for complex abelian surfaces withpolarization type (1,N). (T is “transpose” here.)

The paramodular Fricke involution splits paramodular forms into plusand minus spaces.

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


All abelian surfaces A/Q with a minimal endomorphismgroup over 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).

The direction 1→ 2 is still believed, but 2→ 1 will be amended later.


Remarks

The endomorphisms defined over Q should be minimal:EndQ(A) = Z. (Where is this condition in the elliptic case?)The Paramodular conjecture addresses the typical case. WhenEndQ(A) > Z, modularity is known.

Newform theory for paramodular groups:Ibukiyama 1984; Roberts and Schmidt 2007 (LNM 1918).

Grit : Jcuspk,N → Sk (K (N)), the Gritsenko lift from Jacobi cusp forms ofindex N to paramodular cusp forms of level N is an advanced versionof the Maass lift.


Do the arithmetic and automorphic data match up?Looks like it.

By 1997, Brumer has a long list of N that could possibly be the conductorof an abelian surface A/Q and begins campaigning for the computation ofcorresponding automorphic forms.

Theorem (PY 2009)

Let p < 600 be prime. If p 6∈ {277, 349, 353, 389, 461, 523, 587} thenS2(K (p)) consists entirely of Gritsenko lifts.

This exactly matches Brumer’s “Yes” list for prime levels p < 600.


L-functions of abelian surfaces over Q

1. The local Hasse-Weil p-Euler factor is the characteristic polynomial ofFrobenius on the Tate module T`(A) of the abelian surface A.

Qp(A, t) = det(I − t Frobp |T`(A)Ip

)2. When the isogeny class of A/Q contains a Jacobian or a Prym then

computation of the Hasse-Weil p-Euler factors will be easy. However,it may be difficult to find a representative abelian surface.

3. Global Hasse-Weil L-function

L(A, s,H-W) =∏

primes p

Qp(A, p−s)−1


L-functions of abelian surfaces over Q

1. For N < 1000, we have seen mainly Jacobians, Pryms, and Weilrestrictions, but the algorithmic challenges searching for A/Q willgrow with the conductor N.

2. In the special case when A = Jac(C ) is the Jacobian of a genus twocurve C , we have

L(A, s,H-W) = L(C , s,H-W)

3. The local Hasse-Weil p-Euler factors for A = Jac(C ) are accessible bycounting points

Zp(C , t) = exp

( ∞∑n=1

#{Points on C/Fpn}tn

n

)=

Qp(C , t)

(1− t)(1− pt)


An example: LMFDB: Genus 2 Curve 277.a.277.1

1. A277 = Jac(C277) for C277 given by the hyperelliptic curvey2 + (x3 + x2 + x + 1)y = −x2 − x

2. Magma will compute Hasse-Weil Euler factors of a curve:> G:=x^ 3 + x^ 2 + x + 1; F:=-x^ 2 - x;

(y2 + G (x)y = F (x) is the equation)> C:=HyperellipticCurve(F,G);

> J:=Jacobian(C);

> h2:=EulerFactor(J,GF(2)); h2;

4*x^ 4 + 4*x^ 3 + 4*x^ 2 + 2*x + 1


9*x^ 4 + 3*x^ 3 + x^ 2 + x + 1


25*x^ 4 + 5*x^ 3 - 2*x^ 2 + x + 1


An example: a paramodular newform of level 277

How can we make a newform in S2 (K (277))?

Both Gritsenko lifts and Borcherds products can help.Both are built from Jacobi forms.

Dimension formulas. No— not in weight two.

Theta series. No— the trace from Γ0(p)→ K (p) gives zero.

Modular symbols. — no such theory yet.


Fourier-Jacobi expansions

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)) in coordinates:

f ( τ zz 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 zz ω ) =

∑m∈Z:m≥0

φm(τ, z)e(Nmω)


Fourier-Jacobi expansion (FJE)

FJE: f ( τ zz ω ) =

∑m∈Z:m≥0

φm(τ, z)e(Nmω)

The Fourier-Jacobi expansion of a paramodular form is fixed term-by-termby 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 φm are automorphic forms in their own right andeasier to compute than Siegel modular forms. This is one motivation forthe introduction of Jacobi forms.


Definition of Jacobi Forms: AutomorphicityLevel one

Assume φ : H× C→ C is holomorphic.

φ : H2 → C( τ zz ω ) 7→ φ(τ, z)e(mω)

Assume that φ transforms by χ det(CZ + D)k for

P2,1(Z) =

∗ 0 ∗ ∗∗ ∗ ∗ ∗∗ 0 ∗ ∗0 0 0 ∗

∩ Sp2(Z), ∗ ∈ Z,


Definition of Jacobi Forms: Support

Jacobi forms are tagged with additional adjectives to reflect thesupport supp(φ) = {(n, r) ∈ Q2 : c(n, r ;φ) 6= 0} of the Fourierexpansion

φ(τ, z) =∑n,r∈Q

c(n, r ;φ)qnζr , q = e(τ), ζ = e(z).

φ ∈ Jcuspk,m : automorphic and c(n, r ;φ) 6= 0 =⇒ 4mn − r2 > 0

φ ∈ Jk,m: automorphic and c(n, r ;φ) 6= 0 =⇒ 4mn − r2 ≥ 0

φ ∈ Jweakk,m : automorphic and c(n, r ;φ) 6= 0 =⇒ n ≥ 0

φ ∈ Jwhk,m: automorphic and c(n, r ;φ) 6= 0 =⇒ n >> −∞

(“wh” stands for weakly holomorphic)


Index Raising Operators V (`) : Jk ,m → Jk ,m`from Eichler-Zagier

The Jacobi V (`) are images of the elliptic T (`).

Elliptic Hecke Algebra −→ Jacobi Hecke Algebra

∑SL2(Z)

(a bc d

)7→∑

P2,1(Z)

a 0 b 00 ad − bc 0 0c 0 d 00 0 0 1

∑ad=`

b mod d

SL2(Z)

(a b0 d

)7→

∑ad=`

b mod d

P2,1(Z)

a 0 b 00 ` 0 00 0 d 00 0 0 1

T (`) 7→ V (`)


Fourier-Jacobi expansion of the Gritsenko lift

Any Jacobi cusp form can be the leading 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

(φ|V`)(τ, z)e(`mω).


Borcherds Product Summary

Theorem (Borcherds, Gritsenko, Nikulin)

Given ψ ∈ Jwh0,N(Z), a weakly holomorphic weight zero, index N Jacobi

form with integral coefficients

ψ(τ, z) =∑

n,r∈Z: n≥−No

c (n, r) qnζr

there is a weight k ′ ∈ Z, a character χ, and a meromorphic paramodularform Borch(ψ) ∈ Mmero

k ′ (K (N))(χ)

Borch(ψ)(Z )=qAζBξC∏

n,m,r∈Z

(1− qnζrξNm

)c(nm,r)converging in a nghd of infinity and defined by analytic continuation.


Theta Blocks: a great way to make Jacobi formsdue to Gritsenko, Skoruppa, and Zagier

Dedekind Eta function η ∈ Jcusp1/2,0(ε)

η(τ) = q1/24∏n∈N

(1− qn)

Odd Jacobi Theta function ϑ ∈ Jcusp1/2,1/2(ε3vH)

ϑ(τ, z) = q1/8(ζ1/2 − ζ−1/2

)∏n∈N

(1− qn)(1− qnζ)(1− qnζ−1)

TBk [d1, d2, . . . , d`](τ, z) = η(τ)2k−`∏`

j=1 ϑ(τ, djz) ∈ Jmerok,m (ε2k+2`)

where 2m = d21 + d2

2 + · · ·+ d2` and di ∈ N.


There are 10 dimensions of Gritsenko lifts in S2 (K (277))

We have dimS2(K (277)) = 11 whereas the dimension of Gritsenko lifts inS2(K (277)) is dim Jcusp2,277 = 10.

Let Gi = Grit (TB2(Σi )) ∈ S2 (K (277)) for 1 ≤ i ≤ 10 be the lifts of the10 theta blocks given by:

Σi ∈ { [2, 4, 4, 4, 5, 6, 8, 9, 10, 14], [2, 3, 4, 5, 5, 7, 7, 9, 10, 14],[2, 3, 4, 4, 5, 7, 8, 9, 11, 13], [2, 3, 3, 5, 6, 6, 8, 9, 11, 13],[2, 3, 3, 5, 5, 8, 8, 8, 11, 13], [2, 3, 3, 5, 5, 7, 8, 10, 10, 13],[2, 3, 3, 4, 5, 6, 7, 9, 10, 15], [2, 2, 4, 5, 6, 7, 7, 9, 11, 13],[2, 2, 4, 4, 6, 7, 8, 10, 11, 12], [ 2, 2, 3, 5, 6, 7, 9, 9, 11, 12] }.

Remark: The Gritsenko lifts of these ten theta blocks are all Borcherdsproducts as well.


The nonlift paramodular eigenform f277 ∈ S2 (K (277))

f277 =Q

L(proven holomorphic)

Q = −14G 21 − 20G8G2 + 11G9G2 + 6G 2

2 − 30G7G10 + 15G9G10 + 15G10G1

− 30G10G2 − 30G10G3 + 5G4G5 + 6G4G6 + 17G4G7 − 3G4G8 − 5G4G9

− 5G5G6 + 20G5G7 − 5G5G8 − 10G5G9 − 3G 26 + 13G6G7 + 3G6G8

− 10G6G9 − 22G 27 + G7G8 + 15G7G9 + 6G 2

8 − 4G8G9 − 2G 29 + 20G1G2

− 28G3G2 + 23G4G2 + 7G6G2 − 31G7G2 + 15G5G2 + 45G1G3 − 10G1G5

− 2G1G4 − 13G1G6 − 7G1G8 + 39G1G7 − 16G1G9 − 34G 23 + 8G3G4

+ 20G3G5 + 22G3G6 + 10G3G8 + 21G3G9 − 56G3G7 − 3G 24 ,

L = −G4 + G6 + 2G7 + G8 − G9 + 2G3 − 3G2 − G1.


A newform f −587 ∈ S2(K (587))−.Gritsenko, P–, Yuen (2016)

Construct theta blocks φ ∈ Jcusp2,587 and Ξ ∈ Jcusp2,2·587:

φ = TB2[1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14]

Ξ = TB2[1, 10, 2, 2, 18, 3, 3, 4, 4, 15, 5, 6, 6, 7, 8, 16, 9, 10, 22, 12, 13, 14]

ψ =φ |V (2)− Ξ

φ∈ Jwh

0,587(Z)

ψ(τ, z) =∑n,r

c(n, r)qnζr = 4 +1

q+ ζ−14 + · · ·+ q134ζ561 + · · ·

f −587( τ zz ω ) = q2ζ68ξ587

∏(n,m,r)≥0

(1− qnζrξNm

)c(nm,r)


A nonlift newform f249 ∈ S2 (K (249)).P–, Shurman, Yuen (2017)

ψ249(τ, z) =ϑ(τ, 8z)

ϑ(τ, z)

ϑ(τ, 18z)

ϑ(τ, 6z)

ϑ(τ, 14z)

ϑ(τ, 7z)∈ Jw.h.

0,249(Z)

f249( τ zz ω ) = 14 q2ζ63ξ498

∏n,m,r∈Z: m≥0

if m = 0 then n ≥ 0if m = n = 0 then r < 0

(1− qnζrξmN)c(nm,r ;ψ249)

− 6 Grit(TB2(2, 3, 3, 4, 5, 6, 7, 9, 10, 13))

− 3 Grit(TB2(2, 2, 3, 5, 5, 6, 7, 9, 11, 12))

+ 3 Grit(TB2(1, 3, 3, 5, 6, 6, 6, 9, 11, 12))

+ 2 Grit(TB2(1, 1, 2, 3, 4, 5, 6, 9, 10, 15))

+ 7 Grit(TB2(1, 2, 3, 3, 4, 5, 6, 9, 11, 14)).


siegelmodularforms.org

Our paramodular website: www.siegelmodularforms.org

(Joint with J. Shurman, D. Yuen.)


Pulling back: G a reductive algebraic group

Modularity

Arithmetic Automorphic

Motives −→ Galois representations ←− Automorphic reps

Motives −→ ρ : Gal(Q/Q)→ G (C) ←− Auto reps of G (A)

↑ ↑

Etale cohomology of varieties Automorphic forms

Abelian surfaces → Gal(Q/Q)→ GSp(4,C) ← paramodular, G = SO(5)

Abelian surfaces → L-functions ← paramodular f , G = SO(5)

Elliptic curves → Gal(Q/Q)→ GL(2,C) ← elliptic G = GL(2)


Proven paramodularity!

Q: How can you prove the L-functions of A and f agree?

A: You associate Galois representations to each of A and f and provethat their Galois representations are equivalent.

Q: How can you prove two Galois representations are equivalent?

A: Use a generalization of Faltings-Serre to GSp(4) to show that theGalois representations agree if enough traces of Frobenius agree.

The Hasse-Weil and spin L-functions of some A and f agree.N = 277: need primes up to p = 43N = 353: need primes as large as p = 137N = 587−: need primes up to p = 41(On arXiv— On the paramodularity of typical abelian surfaces,Brumer, Pacetti, Poor, Tornarıa, Voight, Yuen)


Modularity of A731, a composite level: 731 = 17 · 43

1. Berger and Klosin have proven the modularity of

A731 = Jac(y2 = x6 − 6x4 + 4x3 + 9x2 − 16x − 4

),

the first modularity proof of a typical abelian surface with compositeconductor.

2. A731 has rational 5-torsion.

The candidate eigenform f731 ∈ S2 (K (731)) is congruent to aGritsenko lift modulo p = 5.

3. The Galois representations σA and σf associated to A731 and f731 arethe unique characteristic zero deformation of σA.

Tobias Berger and Krzysztof Klosin: Deformations of Saito-Kurokawa typeand the Paramodular Conjecture.

arXiv:1710.10228v2 (appendix by Poor, Shurman, Yuen)


Part Three

Systematic Evidence for the Paramodular Conjecture


Twisting

Let χ be the nontrivial quadratic Dirichlet character modulo p.

Twist L-functions:∑

n

anns7→∑

n χ(n)anns

Twist abelian surfaces: A 7→ Aχ

I only indicate the special case of Jacobians:Jac(y2 = f (x)

)7→ Jac

(py2 = f (x)

).

According to the Paramodular Conjecture, there should be a way totwist paramodular forms. (Well, weight two newforms . . . )

Atwist //

PC��

Aχ

PC��

f?? // f χ


Twisting

Johnson-Leung and Roberts have a theory of twisting paramodular forms.

Jennifer Johnson-Leung and Brooks Roberts: Twisting of SiegelParamodular Forms, Int. J. Number Theory 13 (2017).

Theorem. Let χ be the nontrivial quadratic Dirichlet charactermodulo an odd prime p - N. There exists a linear twisting map

Tχ : Sk (K (N))→ Sk(K (Np4)

)such that if f is a new eigenform and Tχ(f ) 6= 0 then

L (Tχ(f ), s, spin) = Lχ (f , s, spin) .

Therefore if f shows the modularity of A then Tχ(f ) shows themodularity of Aχ.

For example, for all p 6= 277, the surface Aχ277 is modular!


Weil Restriction

Let E/K be an elliptic curve over a quadratic field K .

There exists an abelian surface over Q, AE = ResK/Q (E ), the Weilrestriction of E , whose Q-points are in bijection with the K -points of E .

If E is not isogenous to its conjugate over K then EndQ (AE ) = Z, andthe Paramodular Conjecture applies to the Weil restriction AE .

When K is real quadratic, E/K is modular with respect to someweight (2, 2) Hilbert modular form F for SL(OK ⊕ a).

Nuno Freitas and Bao V. Le Hung and Samir Siksek: Elliptic curvesover real quadratic fields are modular. Invent. Math. 201 (2015)

According to the Paramodular Conjecture, there should be a lift fromHilbert to paramodular forms.


Weil Restriction

Johnson-Leung and Roberts have a theory of lifting Hilbert eigenforms toparamodular eigenforms

Jennifer Johnson-Leung, and Brooks Roberts: Siegel modular formsof degree two attached to Hilbert modular forms. J. Number Theory132 (2012)

For real quadratic K , the modularity of the Weil restrictionsAE = ResK/Q (E ) is shown by the J-LR paramodular lift of theHilbert eigenform that shows the modularity of E/K .

E/KWR //

Modularity��

A/Q

PC��

Hilbert formJL-R lift // Paramodular form


Weil Restriction

For imaginary quadratic K , Berger, Dembele, Pacetti, and Sengun have asimilar theory lifting Bianchi eigenforms to paramodular eigenforms.

Tobias Berger, and Lassina Dembele, and Ariel Pacetti, and MehmetHaluk Sengun: Theta lifts of Bianchi modular forms and applicationsto paramodularity. J. Lond. Math. Soc. (2) 92 (2015)

E/KWR //

modularity��

A/Q

PC��

Bianchi formBDPS lift // Paramodular form


Do the arithmetic and automorphic data match up?

Theorem (P– Yuen 2009)

Let p < 600 be prime. If p 6∈ {277, 349, 353, 389, 461, 523, 587} thenS2(K (p)) consists entirely of Gritsenko lifts.

Paramodular Conjecture verified for prime levels p < 600 not listed above.Brumer and Kramer prove the absence of abelian surfaces.

The dimension formula of Ibukiyama (2007: Hamana Lake) forS4 (K (p)) was crucial to these computations.



Theorem (Breeding, P– Yuen 2016)

For all N ≤ 60, S2(K (N)) consists entirely of Gritsenko lifts.

Paramodular Conjecture verified for many odd levels N ≤ 60. Brumer andKramer prove the absence of semistable abelian surfaces of odd conductor.

We proved an a priori bound on the number of Fourier-Jacobicoefficients needed to determine the space S2 (K (N)∗, χ).

Jeffery Breeding, Cris Poor, and David S. Yuen: Computations ofspaces of paramodular forms of general level. J. Korean Math. Soc.53 (2016)



Theorem (P– Shurman, Yuen 2017)

Let N < 300 be square-free. If N 6∈ {249, 277, 295} then S2(K (N))consists entirely of Gritsenko lifts. Furthermore, there is exactly onedimension of nonlift eigenforms for S2(K (249)),S2(K (277)),S2(K (295)).

Paramodular Conjecture verified for odd squarefree levels N < 300, exceptas noted.

Cris Poor, Jerry Shurman, David S. Yuen: Siegel paramodular forms ofweight 2 and squarefree level, Int. J. Number Theory 13 (2017)

The dimension formula of Ibukiyama and Kitayama for S4 (K (N)) andsquarefree N was crucial to these computations.


Part Four

The Paramodular Conjecture 2.0


Perspective on classification results.

Elliptic curves over Q ↔ elliptic Q-newforms in S2(Γ0(N))

E over real quad K → Hilbert Q-newforms in S2(SL(OK ⊕ a))(but (←) is not quite done— see discussion in Freitas, Siksek:arxiv.org/pdf/1307.3162.pdf)

E over imaginary quad K → Bianchi Q-newforms in S2(Γ0(n))

But there is a problem going (←) from Bianchi newforms to E/K asnoted by John Cremona.


Counterexamples

Ciaran Schembri is my source for this example.

Define a hyperelliptic curve Co/Q(i) of genus two byy2 = x6 + 4ix5− (6 + 2i)x4 + (7− i)x3− (9− 8i)x2− 10ix + (3 + 4i)

Ao = Jac(Co) is an abelian surface over Q(i) of conductor p45,1p437,2 of

norm 342252 = 1854.

O6 ↪→ EndQ(i)(Ao) where O6 is the maximal order of the rationalquaternion algebra of discriminant 6

There is a Bianchi newform fo ∈ S2(Γ0(p45,1p437,2)) with Q-rational

eigenvalues such that L(Ao , s,Hasse-Weil) = L(fo , s)2.

By Faltings, there can be no E/Q(i) with L(E , s,Hasse) = L(fo , s).(L-functions determine isogeny classes of abelian varieties)

Thus, the pairing between E/Q(i) and Bianchi newforms is not perfect.


A counterexample to the Paramodular conjecture 1.0

Frank Calegari pointed out counterexamples to the ParamodularConjecture in January, 2018.

By Weil restriction, B = WR(Ao/Q(i)) is an abelian fourfold definedover Q with EndQ(B)⊗Q an indefinite quaternion algebra.

The lift of Berger, Dembele, Pacetti, and Sengun givesf = BDPS-lift(fo) ∈ S2(K (N)). Note N = (16 · 185)2 = 8 761 600.

L(B, s,H-W) = L(f , s, spin)2 and there can be no abeliansurface A/Q with L(A, s,H-W) = L(f , s, spin) due to the differentendomorphism rings EndQ(B)⊗Q 6= EndQ(A⊕ A)⊗Q, these beingisogeny invariants.


The Paramodular conjecture 2.0 (2018)

An abelian fourfold B/Q has quaternionic multiplication (QM) if EndQ(B)is an order in a non-split quaternion algebra over Q. A cuspidal, nonliftSiegel paramodular newform f ∈ S2(K (N)) with rational Heckeeigenvalues will be called a suitable paramodular form of level N.

Paramodular Conjecture (Brumer–Kramer)

Let N ∈ N. Let AN be the set of isogeny classes of abelian surfaces A/Qof conductor N with EndQ A = Z. Let BN be the set of isogeny classes ofQM abelian fourfolds B/Q of conductor N2. Let PN be the set of suitableparamodular forms of level N, up to nonzero scaling. There is a bijectionAN ∪ BN ↔ PN such that

L(C , s,H-W) =

{L(f , s, spin), if C ∈ AN ,

L(f , s, spin)2, if C ∈ BN .

Brumer and Kramer: QM implies N = M2s with s | gcd(30,M).


Part Five

Heuristic Tables for the Paramodular Conjecture

(found by classifying initial Fourier-Jacobi expansions)

With updated dimensions from: Nonlift weight two paramodular eigenformconstructions, by Poor, Shurman, and Yuen. In progress.

And from: Antisymmetric paramodular forms of weights 2 and 3, byGritsenko, Poor, and Yuen: arXiv:1609.04146v1.


Heuristic tables: k = 2 paramodular newforms: N ≤ 800.

+new nonlift = dim(

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

))−new = dim (S2(K (N))new)− .

The “=” means “proven.”

N +new −new various comments

249 = 1 BP+Grit; Jac

277 = 1 modular! Q/L; Jac



353 = 1 modular! BP+Grit; Jac

388 1 Jac




394 1 Jac

427 1 Jac

461 = 1 Tr(BP)+Grit; Jac

464 1 Jac

472 1 Jac

511 2 quad pair,√

5; 4-dim A/Q?


550 1 no match known



555 1 Jac

561 1 Prym

574 1 Jac

587 = 1 = 1 modular ! Tr(BP)+Grit and BP-; Jacs

597 1 Jac

603 1 Jac

604 1 Jac

623 1 Jac

633 1 Jac



637 2 quad pair,√

2; 4-dim A/Q?

644 1 Jac

645 2 quad pair,√

2; 4-dim A/Q?

657 ≥ 1 modular! WR: E(9ζ6−8)/Q(√−3)

665 1 Prym

688 1 Jac

691 1 Jac


ζ6 = exp(2πi 16)



704 1 Jac

708 1 Jac

709 1 Jac

713 1 ≥ 1 BP-; Jacs

731 = 1 Berger and Klosin: modular! Jac

737 1 Prym

741 1 Jac

743 1 Jac



745 1 Jac


762 1 Jac

763 1 Jac

768 1 Jac

775 ≥ 1 modular! WR: E(5φ−2)/Q(√

5)

797 1 Jac

φ =1 +√

5

2


Thanks to Armand Brumer for all his help, in particular for providing mewith the majority of the abelian surfaces in this talk.


Thank you!


paramodularitysiegelmodularforms.org/pages/degree2/paramodular-wt2-all/... · 2018. 6. 13. · 2....

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