Eisenstein series for Jacobi forms of lattice index

Transcription

1 The University of Nottingham Building Bridges 4 Alfréd Rényi Institute of Mathematics 17th of July, 018

2 Structure of the talk 1 Introduction to Jacobi forms JacobiEisenstein series 3 Fourier coecients of Eisenstein series

3 1. Jacobi forms the arithmetic theory of scalar Jacobi forms was developed in 1985 by Eichler & Zagier Eisenstein series Taylor expansions Hecke operators relation with half-integral weight elliptic modular forms and with vector-valued modular forms (theta expansion) relation with Siegel modular forms Jacobi forms and Algebraic Geometry (from the work of V. Gritsenko) orthogonal modular forms can be obtained as lifts of Jacobi forms; the former determine Lorentzian KacMoody Lie (super) algebras of Borcherds type Jacobi forms are solutions to the mirror symmetry problem for K3 surfaces the elliptic genus of a CalabiYau manifold is a weak Jacobi form and much more...

4 Notation e c(x) = exp ( ) πix c and e(x) = e1(x) the weight of a Jacobi form will be k in N and the index L = (L, β): L Z is a free, nite rank Z-module β : L L Z is a symmetric, positive-denite, even Z-bilinear form set β(λ) := 1 β(λ, λ) the dual lattice of L: L # := {t L Z Q : β(λ, t) Z for all λ in L} the determinant of L is det(l) := L # /L the integral Jacobi group associated to L is J L := Γ L J L acts on Hol(H (L C) C); given g = (A, (λ, µ)) in J L, set ( φ k,l g(τ, z) := φ Aτ, z + λτ + µ ) (cτ + d) k cτ + d ( ) cβ(z + λτ + µ) e + τβ(λ) + β(λ, z) cτ + d

5 Jacobi forms of lattice index Denition A function φ in Hol(H (L C) C) is called a Jacobi form of weight k and index L if: 1 φ k,l (A, h) = φ, for all (A, h) in J L ; φ has a Fourier expansion of the form φ(τ, z) = C(D, t)e ((β(t) D)τ + β(t, z)). D Q 0,t L # β(t) D Z for xed k and L, denote the C-vector space of all such functions by J k,l Jacobi cusp forms: D < 0; denote the subspace of cusp forms of weight k and index L by S k,l

6 . JacobiEisenstein series Why the interest? they are an example of Jacobi forms ϑ L,y(τ, z) = t L # t y mod L e (β(t)τ + β(t, z)) they should be perpendicular to Jacobi cusp forms with respect to a suitably dened Petersson scalar product and hence we obtain the following decomposition: J k,l = S k,l J Eis k,l we are interested in a theory of newforms with respect to Hecke operators (dened by Ajouz in 015)

7 JacobiEisenstein series the isotropy set of L is Iso(D L) := {r L # /L : β(r) Z} dene J L := {(( 1 0 n 1 ), (0, µ)) : n Z, µ L} Denition For every r in Iso(D L), let g L,r(τ, z) := e(β(r)τ + β(r, z)) and dene the Eisenstein series of weight k and index L associated to r as E k,l,r (τ, z) := 1 g L,r k,lγ(τ, z). γ J \J L L it is absolutely and uniformly convergent on compact subsets of H (L C) for k > + their twists by primitive Dirichlet characters modulo divisors of N r (order of r in L # /L) form a basis of eigenforms of Span{E k,l,r : r Iso(D L)} E k,l,r,χ (τ, z) := χ(d)e k,l,dr (τ, z) d (Z/ZN r) with eigenvalues given by twisted divisor sums

8 Example (scalar case - Eichler & Zagier) consider L m = (Z, (x, y) mxy), with m in N the dual is 1 m Z then J k,lm is the space of scalar Jacobi forms J k,m take L m as the index, r = 0 and k even; then E k,lm,0 has Fourier coecients C ( ) t 0, m = 1 for all t in Z and m ( ) t C 4m n, t 1 ) = (π)k i k k 3 (n m (m) 1 Γ(k 1 ) t c k 4m c 1 e c(md 1 λ mrλ + nd). λ,d( mod c) (d,c)=1 ( ) when m = 1, we have C t 4 n, t = L t 4n ( k) ζ(3 k) when m is square-free, we have ( t C 4m n, t m ) = 1 ζ(3 k)σ k 1 (m) d (n,t,m) d k 1 L t 4nm d ( k)

9 Theorem (M., 017) For k > + and for every r in Iso(D L), the Eisenstein series E k,l,r is an element of J k,l and it is orthogonal to S k,l with respect to a suitably dened Petersson scalar product. It has the following Fourier expansion: E k,l,r (τ, z) = 1 ( ) ϑ L,r(τ, z) + ( 1) k ϑ L, r(τ, z) + C k,l,r (D, t)e ((β(t) D)τ + β(t, z)), where D Q <0,t L # β(t) D Z k (π) i k k 1 C k,l,r (D, t) = ( ) ( D) det(l) 1 Γ k ( ) c k H L,c(r, D, t) + ( 1) k H L,c( r, D, t) c 1 and H L,c(r, D, t) is the lattice sum ( β(λ + r)d 1 + (β(t) D)d + β(t, λ + r) ). λ L/cL,d (Z/cZ) e c

10 Sketch of proof E k,l,r (τ, z) = 1 g L,r k,lγ(τ, z) γ J \J L L (Reminder) invariance under k,l action of J L follows by construction the given Fourier expansion completes the modularity argument for orthogonality to cusp forms, use unfolding technique: φ, E k,l,r := φ(τ, z) g r k,lγ(τ, z)i(τ) k e 4πβ(I(z)) I(τ) dv L(τ, z) F J L γ J \J L L = φ(τ, z)g r(τ, z)i(τ) k e 4πβ(I(z)) I(τ) dv L(τ, z) F L J choose a convenient fundamental domain insert Fourier expansion of φ and use orthogonality relations for exponential function, which imply that the integral in R(z) vanishes

11 3. Fourier coecients of Eisenstein series E k,l,r (τ, z) = 1 g L,r k,lγ(τ, z) γ J \J L L (Reminder) choose a convenient set of coset representatives for J L \ J L write each γ as (A, (aλ, bλ)) and separate contributions coming from terms with c = 0 and c 0 c = 0 gives the singular term ( ) e (β(t)τ) e (β(t, z)) + ( 1) k e (β( t, z)) t L # t r mod L terms with c < 0 give same contribution as those with c > 0, multiplied by ( 1) k and with z replaced by z contribution coming from c > 0 is: c k ( β(λ + r)d 1 ) F (τ + dc, z 1c ) λ, c>0 λ(c),d(c) e c where F(τ, z) has period Z in τ and period L in z.

12 Theorem (M., 017) For every r in Iso(D L), the Eisenstein series E k,l,r is an element of J k,l and it is orthogonal to S k,l with respect to a suitably dened Petersson scalar product. It has the following Fourier expansion: E k,l,r (τ, z) = 1 ( ) ϑ L,r(τ, z) + ( 1) k ϑ L, r(τ, z) + C k,l,r (D, t)e ((β(t) D)τ + β(t, z)), where D Q <0,t L # β(t) D Z k (π) i k k 1 C k,l,r (D, t) = ( ) ( D) det(l) 1 Γ k ( ) c k H L,c(r, D, t) + ( 1) k H L,c( r, D, t) c 1 and H L,c(r, D, t) is the lattice sum ( β(λ + r)d 1 + (β(t) D)d + β(t, λ + r) ). λ L/cL,d (Z/cZ) e c

13 suppose that is even dene (L) := ( 1) det(l) and write (L) = df, with d the discriminant of Q( (L)); set χ d ( ) := ( ) d Theorem (M., 017) When k is odd, E 0 0. When k and are even, E 0 has the Fourier expansion E 0(τ, z) = ϑ L,0(τ, z) + C 0(D, t)e ((β(t) D)τ + β(t, z)) and C 0(D, t) = (D,t) supp(l) D<0 ( 1) 4 ( D d ) ( ) fl 1 k +, χ d d f µ(d) k d L p(k 1) p DNt det(l) 1 χ (L) (p)p k 1. k χ d(d)σ 1 k+ ( f d )

14 Sketch of proof dene the representation numbers R(b) := #{λ L/bL : β(λ + t) D 0 mod b} set r = 0 in the formula for C r(d, t), introduce the Möbius function to remove the coprimality conditions in H L,c(0, D, t) and obtain k (π) i k ( D) C k,l,0 (D, t) = ( det(l) 1 Γ k reminder: k 1 ) ζ(k ) (1+( 1) k ) b 1 singular-term(e k,l,0 ) = 1 + ( 1)k ϑ L,0 the R(b)'s are multiplicative functions of b; the Dirichlet series L(s) := b 1 R(b)b s arises in Brunier & Kuss (001) it converges for R(s) > and that it can be continued meromorphically to R(s) > + 1, with a simple pole at s = R(b) b k 1

15 dene w p = 1 + ord p(n td) and the local Euler factor L p(s) := p wps R(p wp ) + (1 p wp 1 (s +1)) l=0 p ls R(p l ) using results of Siegel (1935) on representation numbers of quadratic forms modulo prime powers, we obtain ζ(s + 1) L(s) = ( ) L s + 1, χ (L) p DN t det(l) L p(s) 1 χ (L) (p)p (s +1) functional equations, Gauss sums, the duplication formula and Euler's reection formula for the Gamma function, values of the Riemann zeta function at positive even integers complete the proof

16 we have obtained: C 0(D, t) = ( 1) 4 ( D d ) ( ) fl 1 k +, χ d d f µ(d) k d L p(k 1) p DNt det(l) 1 χ (L) (p)p k 1 k χ d(d)σ 1 k+ ( f d ) Corollary The Fourier coecients of E 0 are rational numbers. use results of Zagier (1981) L( n, χ) = M n M ( ) l χ(l)b n+1 n + 1 M l=1

17 can we say something about the `bad' Euler factors? Cowan, Katz and White (017): set D := p Nt D p vp(n t D) gcd(p, det(l))=1 p N t D p det(l) L p(k 1) 1 χ L(p)p ( k ) = χ L(D)D ( ) k 1 the formulas in CKW were implemented in Sage by B. Williams: d D χ L(d)d k 1 Example If L is unimodular (L # = L), then C 0(D, t) = k B k This was also shown by Woitalla (018). σ (D). k 1

18 Fourier coecients of non-trivial Eisenstein series let x L # /L and dene the following Schrödinger representation σ x : Z 3 Span C {ϑ L,y : y L # /L}: σ x(λ, µ, ν)ϑ L,y := e (µβ(x, y) + (ν λµ)β(x)) ϑ L,y λx note that Z 3 has group law (λ, µ, t)(λ, µ, t ) = (λ + λ, µ + µ, t + t + λµ µλ ). every Jacobi form has a theta expansion: φ(τ, z) = h φ,y (τ)ϑ L,y(τ, z), y L # /L where h φ,y (τ) = D Q (D,y) supp(l) C(D, y)q D

19 Denition Dene the averaging operator at x in the following way: Av xφ(τ, z) := 1 N x (λ,µ) (Z/ZN x ) σ x(λ, µ, 0)φ(τ, z). dened for vector-valued modular forms by Williams (018) σ x is unitary and Av x maps J k,l to J k,l Proposition (M., 017) Suppose that k is even and let r be an element of Iso(D L). Then: E k,l,λr (τ, z) = h Ek,L,0,t+λr(τ) ϑ L,t(τ, z). λ Z/ZN r λ Z/ZN r t L # /L β(r,t) Z

20 Sketch of proof for any r in L # /L, Av re k,l,0 (τ, z) = λ Z/ZN r λβ(r) Z E k,l,λr (τ, z) insert denition of Av r and theta expansion of E k,l,0 on left-hand side and expand: Av re k,l,0 (τ, z) = N r h Ek,L,0,t(τ) t L # /L β(r,t) Z if β(r) Z, then trivially E k,l,λr (τ, z) = N r λ Z/ZN r λ Z/ZN r ϑ L,t λr λ Z/ZN r E k,l,λr (τ, z),

21 with this formula, we can compute Fourier coecients of E k,l,r for any r of order, 3, 4 or 6 Example Suppose that r in has order. Then: E k,l,0 (τ, z) + E k,l,r (τ, z) = t L # /L β(r,t) Z ( ) hek,l,0,t + h Ek,L,0,t+r (τ)ϑl,t(τ, z) and hence C k,l,r (D, t) = { C k,l,0 (D, t + r), C k,l,0 (D, t), if β(r, t) Z otherwise

22 Poincaré series: C k,l,f,r (D, t) :=δ L(F, r, D, t) + ( 1) k δ L(F, r, D, t) + πik det(l) 1 ( ) k ( ) D 4 1 4π(DF ) 1 J F k 1 c c 1 ( ) H L,c(F, r, D, t) + ( 1) k H L,c(F, r, D, t) c 1 Schwagenscheidt (018): Fourier coecients of E DL,r,χ Ran and Skoruppa (in progress): Fourier coecients of E k,l,r,χ Thank you!