Theta Operators on Hecke Eigenvalues Angus McAndrew The University of Melbourne
Overview 1 Modular Forms 2 Hecke Operators 3 Theta Operators 4 Theta on Eigenvalues 5 The slide where I say thank you
Modular Forms Definition (Modular Form) A modular form of weight k and level N is a holomorphic function f : H 1 C such that f ( ) az + b = (cz + d) k f (z) for cz + d ( ) a b Γ(N), c d where Γ(N) = ker ( SL 2 (Z) SL 2 (Z/NZ) ), and f is holomorphic at the cusps.
The Basic Theory In fact, modular forms are extremely explicit and nice to study. For example, when N = 1 we have the following classification. Example M (SL 2 (Z); C) = C[E 4, E 6 ], where E k (q) = 1 2k B k σ k 1 (n)q n, q = e 2πiz. n=1 We can also construct spaces M k (Γ(N); F p ) by tensoring (reducing mod p).
Siegel Modular Forms Definition (Siegel Modular Form) Let κ : GL g (C) GL m (C) be a rational representation. A Siegel modular form of degree g, weight κ and level N is a holomorphic function F : H g C m such that F ((Az + B)(Cz + D) 1 ) = κ(cz + D)F (z), for γ Γ g (N), where Γ g (N) = ker ( Sp 2g (Z) Sp 2g (Z/NZ) ) Note: When the weight κ = det k, we will just write k.
The Basic Theory Here things are more tricky. There are quite a few obstacles in even computing examples. There are still some nice qualities, like the following when κ = det k and g = 2, N = 1. Theorem (Igusa Generators) M (Sp 4 (Z); C) = C[φ 4, φ 6, χ 10, χ 12, χ 35 ]/R Note that we have q-expansions of the form f (q N ) = n a(n)q n N, qn N = e 2πi N Tr(nz).
The Slash Operator Definition (Weight κ slash operator) The weight κ slash operator κ gives an action of GSp 2g (Q) + on weight κ forms by (f κ )(z) = η(γ) λ i g(g+1)/2 κ(cz + D) 1 f (γz), where (λ i... λ g ) is the highest weight of κ. Note that if γ Γ g (N), then (f κ )(z) = f (z).
Hecke Operators The slash operator is well-defined on cosets Γ g (N)γ. Given a double coset Γ g (N)γΓ g (N), it can be expressed as a union of cosets as Γ g (N)γΓ g (N) = Γ g (N)γ j. j Definition (Hecke Operator) Let γ GSp 2g (Q) + have a coset expansion as above. The Hecke operator T γ is (T γ f )(z) = (f κ γ j )(z). j
Eigenforms and Eigenvalues Definition (Eigenform) An eigenform is a form f which is an simultaneous eigenvector for all the Hecke operators T, i.e. Ψ f (T ) F p such that Tf = Ψ f (T )f. Definition (Eigensystem) Let H be the algebra of Hecke operators and f an eigenform. The function Ψ f : H F p T Ψ f (T ) is called the Hecke eigensystem of f.
The Theta Operator The word holomorphic may lead one to think that modular forms should be differentiable. This turns out to be true (mod p), and for g = 1 we have the operator ϑ : M k (Γ(N); F p ) M k+p+1 (Γ(N); F p ) f 1 d 2πi dz f (z) = d dq f (q). One can construct this many ways, either using Rankin-Cohen brackets or algebraic geometry.
Siegel Theta Operators We d like a similar differential operator on Siegel modular forms. As it turns out, there are many generalisations of ϑ. ϑ BN from Boecherer-Nagaoka, using Rankin-Cohen-type brackets. ϑ FG from Flander-Ghitza, using algebraic geometry. Various operators from Yamauchi, using algebraic geometry in the special case g = 2.
Boecherer-Nagaoka A theorem of Eholzer-Ibukiyama gives us a differential bracket taking two forms F, G of weight k 1, k 2 in characteristic 0 to a form [F, G] of weight k 1 + k 2 + 2. There is a form called the Hasse invariant H which reduced (mod p) is the constant 1. So one defines ϑ 0 BNf = [F, H] and then splitting ϑ 0 BN = ϑ1 BN + pϑ2 BN, one defines ϑ BN = reduction of ( 1) g (g + 1)! ϑ1 BN (mod p).
Boecherer-Nagaoka Theorem The operator ϑ BN acts on Siegel modular forms by ϑ BN : M k (Γ g (N); F p ) M k+p+1 (Γ g (N); F p ) f (q) det( q )f (q), i.e. (ϑ BN f )(q) = 1 N g det(n)a(n)q n N. n
Flander-Ghitza Simplified construction: ϑ FG : E κ (H 1 dr )λ Important features: λ (H 1 dr ) Ω1 id KS 1 h (E E ) λ Sym 2 E ω (p 1) E κ ω (p 1) Sym 2 E. E is the Hodge bundle on A g,n. is the Gauss-Manin connection. KS is the Kodaira-Spencer isomorphism.
Flander-Ghitza Theorem The operator ϑ FG acts on Siegel modular forms by ϑ FG : M κ (Γ g (N); F p ) M κ det (p 1) Sym 2(Γg (N); F p ) f (q) (ϑ FG f )(q), where (ϑ FG f )(q) = n (n a(n))q n N.
Yamauchi The operator of Flander-Ghitza ends in a projection to make the weight irreducible. Working with g = 2, Yamauchi does this in a highly explicit way, achieveing multiple maps ϑ i. Further, if one begins with κ = det k and applies ϑ twice, one gets a new map Θ from k to k + p + 1, which is in fact the same as ϑ BN. We will thus discuss these operators in no further detail. Thus in g = 1 and g = 2 we have a direct relationship between the Rankin-Cohen bracket and algebro-geometric constructions.
Theta on Eigenvalues One of the most fascinating properties of the theta operator on modular forms its the following. Theorem Let f be an eigenform with eigensystem Ψ f. If ϑf 0, then ϑf is an eigenform and Ψ ϑf (T γ ) = det(γ)ψ f (T γ ). One proves this by demonstrating the commutation relation T γ ϑ = det(γ) ϑ T γ
Main Theorem Theorem Let f be an eigenform with eigensystem Ψ f. Then (1) If ϑ BN f 0, then ϑ BN f is an eigenform and Ψ ϑbn f (T γ ) = det(γ)ψ f (T γ ). (2) If ϑ FG f 0, then ϑ FG f is an eigenform and Ψ ϑfg f (T γ ) = η(γ)ψ f (T γ ). We will describe the proof for part (1).
Some lemmas Lemma A double coset Γ g (N)γΓ g (N) can be decomposed into right cosets of the form ( ) η(γ)(d Γ g ) 1 B (N). 0 D Lemma ( ) η(γ)(d ) 1 B If γ = then 0 D ( k+p+1 γ) ϑ 1 BN = det(γ) ϑ1 BN ( kγ).
The advantage of having a block upper triangular matrix is that we can do explicit calculations on q-expansions. Furthermore, splitting ϑ 0 BN = ϑ1 BN + pϑ2 BN allows us to avoid certain issues when coming to reduction (mod p).
Proof of second Lemma. We will omit all the actual effort and just show the big impressive equations. ((ϑ 1 BN F ) k+p+1γ)(q) = ( η(γ) g det(d) 1) p 1 η(γ) g 1 N g η(γ)(k+1)g g(g+1)/2 det(d) (k+2) n det(n)a(n)c(n)q n N, (ϑ 1 BN (F k))(q) = 1 N g η(γ)(k+1)g g(g+1)/2 det(d) (k+2) n det(n)a(n)c(n)q n N, where n = η(γ)d 1 n(d ) 1 and c(n) = e 2πi Tr(nBD 1 )/N.
Proof of Theorem part (1). By the first lemma, we can expand Γ g (N)γΓ g (N) = j Γ g (N)γ j where γ j is block upper triangular, and det(γ j ) = det(γ). By the second lemma, these each commute with the slash operator, up to det(γ). Thus, the result follows.
Some Further Questions 1 Relationship between theta constructions 2 Generalised Rankin-Cohen brackets 3 Theta cycles 4 Weight in Serre s Conjecture analogue
Thanks for listening!