Explicit methods in the theory of Jacobi forms of 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 � − 4 r 2 r � � 8 ζ ϑ ( τ, z ) = q 2 r r ∈ Z 1 1 2 − ζ − 1 2 � � 1 − q n �� 1 − q n ζ 1 − q n ζ − 1 � 8 � � �� = q ζ n > 0 Notation q = e 2 π i τ , ζ = e 2 π iz for τ ∈ H , z ∈ C Theorem 2 ( ε 3 ) . ϑ ∈ J 1 2 , 1 Nils Skoruppa Explicit methods September 29, 2015 2 / 23
Jacobi forms over Q : Basic example. II Automorphic Properties of ϑ ( τ, z ) � a b � For all A = in SL(2 , Z ) and all integers λ, µ : c d � − c 1 2 z 2 � 1 z � � 2 = ζ 8 ( A ) ϑ ( τ, z ) ϑ A τ, e ( c τ + d ) c τ + d c τ + d � 2 λ 2 + 2 λ 1 � � 2 ( λ + µ ) 2 � τ 1 1 � � ϑ τ, z + λτ + µ e 2 z = e 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 ≃ → M − J k , m − 2 k − 2 ( m ) , where M − � � 2 k − 2 ( m ) is a certain subspace of M 2 k − 2 Γ 0 ( 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 − 2 k − 2 ( m ), let � c φ ( n , r ) q n ζ r φ = 4 mn − r 2 ≥ 0 be its associated Jacobi form. Theorem (Waldpurger,Gross-Kohnen-Zagier) For every negative fundamental discriminant D ≡ � mod 4 m, say, D = r 2 − 4 mn, one has | c φ ( n , r ) | 2 = const ( k , m , D ) L ( f ⊗ χ D , k − 1) . | φ | 2 | 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 : y 2 + y = x 3 − x conductor = 37 . The associated newform f (so that L ( E , s ) = L ( f , s )) is f E = q − 2 q 2 − 3 q 3 + 2 q 4 − 2 q 5 + 6 q 6 − q 7 + 6 q 9 + O ( q 10 ) . The associated Jacobi form is a theta block: φ E = ϑ a ϑ b ϑ c ϑ d ϑ a + b ϑ b + c ϑ c + d ϑ a + b + c ϑ b + c + d ϑ a + b + c + d /η 6 , where ( a , b , c , d ) = (1 , 1 , 1 , 2), and where ϑ a ( τ, z ) = ϑ ( τ, az ). Recall � − 4 r 2 1 2 − ζ − 1 1 r � 2 � � 1 − q n ζ − 1 � � 8 ζ 2 = q 8 � � 1 − q n �� 1 − q n ζ �� ϑ ( τ, z ) = q ζ . r r ∈ Z n > 0 Nils Skoruppa Explicit methods September 29, 2015 7 / 23
Computation via periods Example The elliptic curve of congruent numbers: C : y 2 = x ( x − D )( x + D ). Associated Jacobi form φ C is in (spans) J cusp , + , 2 , 32 c φ ( n , r ) = ν + ( r 2 − 128 n , r ) − ν − ( r 2 − 128 n , r ), For D > 0, D ≡ r 2 mod 128: � ( a , b , c ) ∈ Z 3 : b 2 − 4 ac = D , b 2 < D , ǫ a > 0 , ν ǫ ( D , r ) = # a ≡ 3 b + r mod 32 , 3 c ≡ b − r � mod 32 . 2 2 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 2 β ( x , x ) integral. Definition J k , L ( k integral): space of holomorphic φ ( τ, z ) ( τ ∈ H , z ∈ C ⊗ Z L ) such that: τ − k = φ ( τ, z ), � � φ ( τ + 1 , z ) = φ ( − 1 /τ, z /τ ) e − β ( z ) /τ � � φ ( τ, z + τ x + 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 c φ ( n , r ) q n e � � � φ = β ( r , z ) . n ∈ Z , r ∈ L ♯ n ≥ β ( r ) ( L ♯ = { y ∈ Q ⊗ Z L : β ( y , L ) ⊆ Z } ) Proposition For fixed D ≤ 0 , the map C φ ( D , r ) := c φ ( β ( r ) − D , r ) for D ≡ β ( r ) mod Z , and C φ ( D , r ) := 0 otherwise, depends only on r + L. Remark Let Z (2 m ) := ( Z , ( x , y ) �→ 2 mxy ). Then J k , Z (2 m ) equals “classical” J k , m . Nils Skoruppa Explicit methods September 29, 2015 11 / 23
Examples A simple effective construction method Let α = ( α 1 , . . . , α m ) be an isometric embedding of L into Z m . Then, for sufficiently large (possibly negative) t ≡ − 3 m mod 24, the function ϑ ( τ, α 1 ( z )) · · · ϑ ( τ, α m ( z )) η t defines an element of J k , L . (If z j are coordinate functions with respect to a Z -basis of L , the α j ( z ) become linear forms in z j with integral coefficients.) Examples ϑ ( τ, z 1 ) ϑ ( τ, z 2 ) ϑ ( τ, z 1 + z 2 ) η 15 ∈ J 9 , A 2 , ϑ ( τ, z 1 ) ϑ ( τ, z 2 ) 3 ϑ ( τ, z 1 + z 2 ) η 9 ∈ J 7 , ” [ 2 1 1 4 ] ” , ϑ ( τ, z 1 ) ϑ ( τ, z 2 ) ϑ ( τ, z 1 − z 2 ) ϑ ( τ, z 1 + z 2 ) ϑ ( τ, z 1 + 2 z 2 ) ϑ ( τ, 2 z 1 + z 2 ) /η ( τ ) 4 ∈ J 1 , ” [ 8 4 4 8 ] ” . Nils Skoruppa Explicit methods September 29, 2015 12 / 23
Basic features of the theory What is known dim J k , L = explicit formula (for all k , including singular or critical) � k ∈ Z J k , L is finite free C [ E 4 , E 6 ]-module with explicit Hilber-Poincar´ e p ( x ) series (1 − x 4 )(1 − x 6 ) . ( p polynomial of weight < 12, coefficients give number of generators in a given weight.) Various methods for generating explicit closed formulas for Jacobi forms: ⊗ -products ( ≈ orthogonal sums of lattices), 1 pull-backs ( ≈ isometric maps between lattices), 2 Taylor expansion in around z = 0 yields J k , L as finite direct sum of 3 spaces of quasi-modular forms, Forms of singular ( k = n 2 ) and critical weight ( k = n +1 2 ) are in 1–1 4 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 n − 1 2 2 det( L ) , and let φ be a Jacobi form in J k , L . For a positive ∆ = ( − 1) integer ℓ , relatively prime to N, let � � � T ( ℓ ) φ := C T ( ℓ ) ( D , r ) e ( β ( r ) − D ) τ + β ( r , z ) , D ≤ 0 , r ∈ L ♯ D ≡ β ( r ) mod Z where a 2 k − n − 1 ρ ( D , a ) C φ ( ℓ 2 � a 2 D , ℓ a ′ r ) . C T ( ℓ ) φ ( D , r ) = a Here a is over all a | ℓ 2 , a 2 | ℓ 2 ND, a ′ a ≡ 1 mod N, and ρ ( D , a ) equals � D ∆ / f 2 � if gcd( ND , a ) = f 2 , and it equals 0 otherwise. a / f 2 The application φ �→ T ( ℓ ) φ defines an endomorphism of J k , L . Nils Skoruppa Explicit methods September 29, 2015 14 / 23
Hecke operators: even rank Theorem (Ajouz-S. 2015) n 2 det( L ) , Let L be a lattice of even rank n, level N, discriminant ∆ = ( − 1) and let φ be a Jacobi form in J k , L . For a positive integer ℓ , relatively prime to N, let � � � T ( ℓ ) φ := C T ( ℓ ) ( D , r ) e ( β ( r ) − D ) τ + β ( r , z ) , D ≤ 0 , r ∈ L ♯ D ≡ β ( r ) mod Z where � ∆ � C φ ( ℓ 2 � a k − n / 2 a 2 D , ℓ a ′ r ) . C T ( ℓ ) φ ( D , r ) = a a | ℓ 2 , ND The application φ �→ T ( ℓ ) φ defines an endomorphism of J k , L . Nils Skoruppa Explicit methods September 29, 2015 15 / 23
L -series Theorem J k , 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 gcd( ℓ, N )=1 λ ( ℓ ) ℓ − s . Then one has, for odd rank n, L ( φ, s ) = � (1 − λ ( p ) p − s + p 2 k − n − 2 − 2 s ) − 1 , � L ( φ, s ) = p �| N and, for even rank n, with λ ′ ( p ) = λ ( p ) − p k − n / 2 − 1 � � ∆ p � ∆ � L ( φ, s ) = L ( , s − k + n / 2 + 1) (1 − λ ′ ( p ) p − s + p 2( k − n / 2 − 1 − 2 s ) ) − 1 . � · ζ ( N ) (2 s − 2 k + n + 2) p �| N Nils Skoruppa Explicit methods September 29, 2015 16 / 23
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