Explicit methods in the theory of Jacobi forms of lattice index and - - PowerPoint PPT Presentation

Explicit methods in the theory of Jacobi forms

• f lattice index and over number fields

Nils Skoruppa

Universit¨ at Siegen

September 29, 2015

ICERM Modular Forms and Curves of Low Genus: Computational Aspects Sept 28 – October 2, 2015

Nils Skoruppa Explicit methods September 29, 2015 1 / 23

Jacobi forms over Q: Basic example. I

Jacobi’s theta function

ϑ(τ, z) =

• r∈Z

−4

r

• q

r2 8 ζ r 2

= q

1 8

ζ

1 2 − ζ− 1 2

n>0

• 1 − qn

1 − qnζ

• 1 − qnζ−1

Notation

q = e2πiτ, ζ = e2πiz for τ ∈ H, z ∈ C

Theorem

ϑ ∈ J 1

2 , 1 2 (ε3). Nils Skoruppa Explicit methods September 29, 2015 2 / 23

Jacobi forms over Q: Basic example. II

Automorphic Properties of ϑ(τ, z)

For all A = a b

c d

• in SL(2, Z) and all integers λ, µ:

ϑ

• Aτ,

z cτ+d

• e

−c 1

2 z2

cτ+d

• (cτ + d)

1 2 = ζ8(A) ϑ(τ, z)

ϑ

• τ, z + λτ + µ
• e
• τ 1

2λ2 + 2λ 1 2z

• = e
• 1

2(λ + µ)2

ϑ(τ, z)

1 2 = weight, 1 2 = index

Nils Skoruppa Explicit methods September 29, 2015 3 / 23

Jacobi forms over Q I

Theorem (Zagier-S.)

For every m > 0 and (integral) k ≥ 2, there exist Hecke-equivariant isomorphisms Jk,m

≃

− → M−

2k−2(m),

where M−

2k−2(m) is a certain subspace of M2k−2

• Γ0(m)
• , containing all

newforms whose L-series have a minus sign in their functional equation.

Nils Skoruppa Explicit methods September 29, 2015 4 / 23

Jacobi forms over Q II

Let f be a newform in M−

2k−2(m), let

φ =

• 4mn−r2≥0

cφ(n, r)qnζr be its associated Jacobi form.

Theorem (Waldpurger,Gross-Kohnen-Zagier)

For every negative fundamental discriminant D ≡ mod 4m, say, D = r2 − 4mn, one has |cφ(n, r)|2 |φ|2 = const(k, m, D) L(f ⊗ χD, k − 1) |f |2 .

Nils Skoruppa Explicit methods September 29, 2015 5 / 23

Computation of Jacobi forms over Q

Generating explicit formulas for Jacobi forms is as easy as for elliptic modular forms (over congruence subgroups) — in fact, easier.

Methods for generating Jacobi forms

1 Theta blocks, 2 Taylor expansion around z = 0, 3 Modular symbols, 4 Vector valued modular forms. 5 Invariants of Weil representations and pullback. Nils Skoruppa Explicit methods September 29, 2015 6 / 23

Computation via theta blocks

The first elliptic curve over Q of positive rank is E : y2 + y = x3 − x conductor = 37. The associated newform f (so that L(E, s) = L(f , s)) is fE = q − 2q2 − 3q3 + 2q4 − 2q5 + 6q6 − q7 + 6q9 + O(q10). The associated Jacobi form is a theta block: φE = ϑa ϑb ϑc ϑd ϑa+b ϑb+c ϑc+d ϑa+b+c ϑb+c+d ϑa+b+c+d/η6, where (a, b, c, d) = (1, 1, 1, 2), and where ϑa(τ, z) = ϑ(τ, az).

Recall

ϑ(τ, z) =

• r∈Z

−4

r

• q

r2 8 ζ r 2 = q 1 8

ζ

1 2 −ζ− 1 2

n>0

• 1−qn

1−qnζ

• 1−qnζ−1

.

Nils Skoruppa Explicit methods September 29, 2015 7 / 23

Computation via periods

Example

The elliptic curve of congruent numbers: C : y2 = x(x − D)(x + D). Associated Jacobi form φC is in (spans) Jcusp,+

2,32

, cφ(n, r) = ν+(r2 − 128n, r) − ν−(r2 − 128n, r), For D > 0, D ≡ r2 mod 128: νǫ(D, r) = #

• (a, b, c) ∈ Z3 : b2 − 4ac = D, b2 < D, ǫa > 0,

a ≡ 3b + r 2 mod 32, 3c ≡ b − r 2 mod 32

• .

Remark

Slightly cheated: we used skew-holomorphic Jacobi forms.

Nils Skoruppa Explicit methods September 29, 2015 8 / 23

A first long-term project

Goal

Develop a similar theory for Jacobi forms of several z-variables (which we shall call “Jacobi forms of lattice index”).

Motivation

Seemingly complicated Jacobi forms are pullbacks of simple universal Jacobi forms of several variables (e.g. the m = 37 example and infinitely many others.) This yields a unified arithmetic theory for all kind of (elliptic) vector valued modular forms, namely:

Theorem (S. 2012)

Any given space of elliptic modular forms of vector valued elliptic modular forms of integral or half integral weight on a congruence subgroup can be naturally embedded into a space of Jacobi forms of integral weight on the full modular group.

Nils Skoruppa Explicit methods September 29, 2015 9 / 23

Definitions, notations

Definition

(Even integral positive) Lattice L = (L, β): Finite free Z-module L, symmetric, positive definite Z-bilinear map β : L × L → Z and β(x) := 1 2β(x, x) integral.

Definition

Jk,L (k integral): space of holomorphic φ(τ, z) (τ ∈ H, z ∈ C ⊗Z L) such that: φ(τ + 1, z) = φ(−1/τ, z/τ) e

• − β(z)/τ
• τ −k = φ(τ, z),

φ(τ, z + τx + y) e

• τβ(x) + β(z, x)
• = φ(τ, z) (x, y ∈ L),

φ holomorphic at infinity.

Nils Skoruppa Explicit methods September 29, 2015 10 / 23

Remarks

Fourier expansion

φ is called holomorphic at infinity if its Fourier expansion is of the form φ =

• n∈Z, r∈L♯

n≥β(r)

cφ(n, r) qn e

• β(r, z)
• .

(L♯ = {y ∈ Q ⊗Z L : β(y, L) ⊆ Z})

Proposition

For fixed D ≤ 0, the map Cφ(D, r) := cφ(β(r) − D, r) for D ≡ β(r) mod Z, and Cφ(D, r) := 0 otherwise, depends only on r + L.

Remark

Let Z(2m) := (Z, (x, y) → 2mxy). Then Jk,Z(2m) equals “classical” Jk,m.

Nils Skoruppa Explicit methods September 29, 2015 11 / 23

Examples

A simple effective construction method

Let α = (α1, . . . , αm) be an isometric embedding of L into Zm. Then, for sufficiently large (possibly negative) t ≡ −3m mod 24, the function ϑ(τ, α1(z)) · · · ϑ(τ, αm(z)) ηt defines an element of Jk,L. (If zj are coordinate functions with respect to a Z-basis of L, the αj(z) become linear forms in zj with integral coefficients.)

Examples

ϑ(τ, z1)ϑ(τ, z2)ϑ(τ, z1 + z2)η15 ∈ J9,A2, ϑ(τ, z1)ϑ(τ, z2)3ϑ(τ, z1 + z2)η9 ∈ J7,”[ 2 1

1 4]”,

ϑ(τ, z1)ϑ(τ, z2)ϑ(τ, z1 − z2)ϑ(τ, z1 + z2)ϑ(τ, z1 + 2z2)ϑ(τ, 2z1 + z2)/η(τ)4 ∈ J1,”[ 8 4

4 8]”.

Nils Skoruppa Explicit methods September 29, 2015 12 / 23

Basic features of the theory

What is known

dim Jk,L = explicit formula (for all k, including singular or critical)

• k∈Z Jk,L is finite free C[E4, E6]-module with explicit Hilber-Poincar´

e series

p(x) (1−x4)(1−x6). (p polynomial of weight < 12, coefficients give

number of generators in a given weight.) Various methods for generating explicit closed formulas for Jacobi forms:

1

⊗-products (≈ orthogonal sums of lattices),

2

pull-backs (≈ isometric maps between lattices),

3

Taylor expansion in around z = 0 yields Jk,L as finite direct sum of spaces of quasi-modular forms,

4

Forms of singular (k = n

2) and critical weight (k = n+1 2 ) are in 1–1

correspondence with invariants of Weil representations.

Nils Skoruppa Explicit methods September 29, 2015 13 / 23

Hecke operators: odd rank

Theorem (Ajouz-S. 2015)

Let L be a lattice of odd rank n, level N, discriminant ∆ = (−1)

n−1 2 2 det(L), and let φ be a Jacobi form in Jk,L. For a positive

integer ℓ, relatively prime to N, let T(ℓ)φ :=

• D≤0, r∈L♯

D≡β(r) mod Z

CT(ℓ)(D, r) e

• (β(r) − D)τ + β(r, z)
• ,

where CT(ℓ)φ(D, r) =

• a

a2k−n−1 ρ(D, a) Cφ( ℓ2

a2 D, ℓa′r).

Here a is over all a|ℓ2, a2|ℓ2ND, a′a ≡ 1 mod N, and ρ(D, a) equals

• D∆/f 2

a/f 2

• if gcd(ND, a) = f 2, and it equals 0 otherwise.

The application φ → T(ℓ)φ defines an endomorphism of Jk,L.

Nils Skoruppa Explicit methods September 29, 2015 14 / 23

Hecke operators: even rank

Theorem (Ajouz-S. 2015)

Let L be a lattice of even rank n, level N, discriminant ∆ = (−1)

n 2 det(L),

and let φ be a Jacobi form in Jk,L. For a positive integer ℓ, relatively prime to N, let T(ℓ)φ :=

• D≤0, r∈L♯

D≡β(r) mod Z

CT(ℓ)(D, r) e

• (β(r) − D)τ + β(r, z)
• ,

where CT(ℓ)φ(D, r) =

• a|ℓ2, ND

ak−n/2 ∆ a

• Cφ( ℓ2

a2 D, ℓa′r).

The application φ → T(ℓ)φ defines an endomorphism of Jk,L.

Nils Skoruppa Explicit methods September 29, 2015 15 / 23

L-series

Theorem

Jk,L possesses a basis of simultaneous Hecke eigenforms for all ℓ (with gcd(ℓ, N) = 1).

Theorem

Let φ be a simultaneous Hecke eigenform with eigenvalues λ(ℓ), and let L(φ, s) =

gcd(ℓ,N)=1 λ(ℓ)ℓ−s. Then one has, for odd rank n,

L(φ, s) =

• p|N

(1 − λ(p)p−s + p2k−n−2−2s)−1, and, for even rank n, with λ′(p) = λ(p) − pk−n/2−1

∆ p

• L(φ, s) = L(

∆

·

• , s − k + n/2 + 1)

ζ(N)(2s − 2k + n + 2)

• p|N

(1 − λ′(p)p−s + p2(k−n/2−1−2s))−1.

Nils Skoruppa Explicit methods September 29, 2015 16 / 23

Consequences: odd rank

Conjecture

For each k > n+1

2

and each L, there are Hecke equivariant injections Jk,L → M2k−n−1(N/4).

Remark

The conjecture is true if L is stably isomorphic to a rank 1 lattice. The conjecture is true for Eisenstein series. The conjecture is true for many examples.

Expectation

More (new) finite closed formulas for L(f ⊗ D

·

• , k − n−1

2 ) for Hecke

eigenforms f in M2k−n−1(N/4), in particular, for f with +1 in functional equation.

Nils Skoruppa Explicit methods September 29, 2015 17 / 23

Consequences: even rank

Observation

For even rank, the shape of the L-series of a Hecke eigenform φ is like the L-series ξ(ℓ)γ(ℓ2)ℓ−s, where γ(ℓ) are the eigenvalues of a Hecke eigenform in Mk−n/2(N, ξ ∆

·

• ) for some ξ.

Conjecture

For each k > n

2 and each L, there are maps Mk−n/2(N, ξ

∆

·

• ) → Jk,L

such that T(ℓ2) on the left corresponds to ξ(ℓ)T(ℓ) on the right. The space Jk,L is the sum of the images of all these maps

Remark

The conjecture is true if det(L) is a prime. The conjecture is true for Eisenstein series. The conjecture is true for many examples.

Nils Skoruppa Explicit methods September 29, 2015 18 / 23

A second long term project

Goal

Develop a similar theory over (totally real) number fields.

Problems to be solved (joint with Boylan, Hayashida, Str¨

• mberg)

Correct definition of Jacobi forms over number fields. Answer various natural questions concerning Hilbert modular groups: Non-trivial central twofold extensions, Weil representations, linear characters, . . . but ... H2(SL(2, o), {±1}) = 2?, [SL(2, o),

v|∞ κv] does not split?

Hecke theory. ? Kernel functions, dimension formulas, trace formulas. Generate examples of Jacobi forms over number fields. Develop algorithms to generate such examples systematically.

Nils Skoruppa Explicit methods September 29, 2015 19 / 23

The correct definition

Remarks

1 The definitions of Jacobi forms over number fields found in the

literature were incomplete.

2 In the literature there were no examples - except for some Jacobi

theta series associated to lattices (H. Stark, O. Richter, . . . ).

For the correct definition

Recall: The index of a Jacobi form is actually the Gram matrix of the lattice (Z, (x, y) → 2mxy). A lattice over a number field K is in general not free, and thus has no Gram matrix. In general one has to consider JFs over SL(g ⊕ o) =

• g−1

g

• ∩ SL(2, K), where g is an ideal and o the ring of

integers of K.

Nils Skoruppa Explicit methods September 29, 2015 20 / 23

Examples: The forms of singular weight

Theorem (H. Boylan)

Let a be a fractional o-ideal and ω a totally positive element in K such that 2m(= a2ωd) = g for a product g of primes of degree 1 over 3. Suppose 2 splits completely in K. Set ϑ[a,ω](τ, z) :=

• s∈ag−1

χ4g(s′)q

1 8 ωs2ζωs/2.

Then ϑ[a,ω] is a Jacobi form on the full modular group of weight 1/2, index [a, ω] :=

• a, (x, y) → ωxy
• (with a certain character ε[a,ω]).

Here χ4g is the totally odd Dirichlet character modulo 4g, and s → s′ is an isomorphism of o-modules ag−1/4a → o/4g. and

Theorem (H. Boylan)

There are no other Jacobi forms of singular weight then the ones above.

Nils Skoruppa Explicit methods September 29, 2015 21 / 23

A glimpse on a beautiful theory

Construction method: Taylor expansion around 0

A classical Jacobi form φ(τ, z) of index m has, for fixed τ as function

• f z, exactly 2m zeros in C/Zτ + Z.

Therefore, writing φ(τ, z) =

n≥0 αn(τ) zn, the first m many αk(τ)

determine φ. The αj(τ) are quasi-modular forms (i.e. elements of C[E2, E4, R6]). Leads to useful description of Jacobi forms in terms of quasi-modular forms.

Somehow surprising fact

This holds true for Jacobi forms over number fields too.

Consequence

For given lattice over a number field the Jacobi forms of this lattice index (and arbitrary weight) form a finitely generated module over the ring of Hilbert modular forms on the full Hilbert modular group.

Nils Skoruppa Explicit methods September 29, 2015 22 / 23

References

S., Boylan: Jacoby forms of lattice index, monograph in preparation Ali Ajouz: Hecke operators on Jacobi forms of lattice index and the relation to elliptic modular forms. http://dokumentix.ub.uni-siegen.de/opus/volltexte/2015/938/ Hatice Boylan: Jacobi Forms, Finite Quadratic Modules and Weil Representations over Number Fields. Lecture Notes in Mathematics 2130

Thank you!

Nils Skoruppa Explicit methods September 29, 2015 23 / 23