Some remarks on the trace formula for Jacobi forms of prime power - PowerPoint PPT Presentation

Some remarks on the trace formula for Jacobi forms of prime power level Hiroshi SAKATA Waseda University Senior High School M¨ arz 27,2014(Donnerstag). Explicit Theory of Automorphic Forms Tongji University Shanghai 2014 Tongji University

1. Purpose of this talk ． 1. I tell the deep acknowledgement for being given a opportunity of my talk and many heartful supports on Shanghai life to the all organizers of this conference. 2. We describe the trace formulas for Jacobi forms of prime power level, and give some trace relations between them and the trace formulas for elliptic modular forms. Moreover we consider about the level-index changing operator (the ‘swapping’ operator) on Jacobi forms in the case of prime odd power level. 2. General Notation ． We put √ √ e n ( z ) = exp(2 π e( z ) = e 1 ( z ) = exp(2 π − 1 nz ) , − 1 z ) . Let S k ( N ) be the elliptic cusp form space of weight k and level N .

3. Recall of definition of (scalar-valued) Jacobi forms. Let k and m be positive integers and H ( R ) be the Heisenberg group on R 2 . The Jacobi group 2 ( R ) J = GL + GL + 2 ( R ) ⋉ H ( R ) acts on Hol( H × C ) as follows : � � � aτ + b � cz 2 lz (1) φ | k,m [ g ]( τ, z ) = ( cτ + d ) − k e ml − φ cτ + d, , cτ + d cτ + d (2) φ | k,m [(( λ, µ ) , κ )]( τ, z ) = e m � � λ 2 τ + 2 λz + λµ + κ φ ( τ, z + λτ + µ ) for any g ∈ GL + 2 ( R ) with det( g ) = l > 0 and (( λ, µ ) , κ ) ∈ H ( R ). Let Γ < SL 2 ( Z ) be a Fuchs group of finite index. We say a function φ ( τ, z ) of Hol( H × C ) is a (holomorphic) Jacobi cusp form of weight k and index m with respect to Γ J = Γ ⋉ H ( Z ) if it satisfies the following three conditions :

(1) φ | k,m [ g ] = φ for any g ∈ Γ . φ | k,m [( λ, µ )] = φ for any ( λ, µ ) ∈ Z 2 . (2) For any g ∈ GL + (3) 2 ( Z ), φ | k,m [ g ] has the Fourier expansion in the following � c g ( n, r )e( nτ + rz ) . φ | k,m [ g ]( τ, z ) = ( n,r ) ∈ Q 2 4 mn − r 2 > 0 We denote the space of Jacobi cusp forms of weight k and index m with respect to Γ J by J cusp k,m ( Γ J ). Especially, if Γ J = Γ 0 ( N ) J , we denote J cusp k,m ( Γ J ) by J cusp k,m ( N ). For a finite union ∆ of double Γ J -cosets in SL 2 ( Q ) J , we define an operator on J cusp k,m ( Γ ) as follows: � φ | H k,m,Γ (∆) = φ | k,m ξ. (This is a finite sum . ) ξ ∈ Γ J \ ∆ For a positive integer l with ( l, Nm ) = 1, we define the l -th Hecke operator T ( l ) on J cusp k,m ( N ) as follows: � � � � l 2 l ′− 1 � 0 φ | T ( l ) = l k − 4 Γ 0 ( N ) J Γ 0 ( N ) J l ′ φ | H k,m,Γ 0 ( N ) . l ′ 0 l ′| l l/l ′ = �

Theorem 1 ( S (2009,2010)). Suppose that k is an even natural number with k ≥ 4 , N is a positive odd squarefree integer and l is a natural number with ( l, N ) = 1 . Then we have the following � � � , � T ( l ) , S 2 k − 2 ( N ) T ( l ) , J cusp tr k, 1 ( N ) = tr � � � . � T ( l ) ◦ W n , S 2 k − 2 ( N ) T ( l ) ◦ W L ( n ) , J cusp tr k, 1 ( N ) = tr Here W n with n | N is the Atkin-Lehner operator on elliptic cusp forms. This theorem in the case of N = 1 was derived by Skoruppa-Zagier(1989). The proof of this theorem in the case of general squarefree level is given by obtaining generalizations of Skoruppa-Zagier’s methods.

4. Sketch of Proof of Theorem 1. We calculate the following two traces in this case explicitly and compare them. (they are very hardwork!) Trace formula for elliptic modular forms (Eichler,Selberg, Oesterl´ e,Yamauchi,Hijikata and Zagier) Let k be a positive integer with k ≥ 2 and ( n 1 , n 2 ) are positive integers pair of relative prime. Then we have � T ( l ) ◦ W n 1 ; S 2 k − 2 ( n 1 n 2 ) � tr �� � � � � � � � = − 1 H t ( s 2 − 4 ln ′ ) n 1 /n ′ n ′ , l µ p 2 k − 2 s/ 2 n ′| n 1 s 2 ≤ 4 ln ′ t | n 2 √ n 1 /n ′ = � n 2 /t ; squarefree n 1 n ′| s � � 2 k − 3 � � �� � � − 1 l ′ , l l ′ − l � √ n 1 � 2 δ ( n 1 = � ) ϕ min Q ( t ) , l ′ l ′ l ′| l t | n 2 � � n 2 /t ; squarefree √ n 1 | l ′ + l l ′ + δ ( k = 2) σ 1 ( l ) .

Remark 1. Here we put  ρ k − 1 − ρ ′ k − 1   of X 2 − sX + l = 0) if s 2 − 4 l � = 0 , ( ρ, ρ ′ ; the  roots  ρ − ρ ′ (1) p k ( s, l ) = � s � k − 2   if s 2 − 4 l = 0 .   ( k − 1) 2  � � ∆ /a 2 b 2    a 2 b H 1 (∆ /a 2 b 2 ) if a 2 b 2 | ∆ , n/a 2 b (2) H n (∆) =    0 if otherwise . where H 1 (0) = − 1 12 and H 1 (∆) for ∆ is the number of equivalence classes with respect to SL 2 ( Z ) of integral,positive definite, binary quadratic forms of discriminant ∆, counting forms equivalent to a multiple of x 2 + y 2 (resp. x 2 + xy + y 2 ) with multiplicity 1 2 (resp. 1 3 ). H 1 (∆) is called the Hurwitz-Kronecker class number.

Trace formula for Jacobi forms (Skoruppa-Zagier) Suppose that k > 2, Γ is a subgroup of SL 2 ( Z ) of finite index, m ≥ 1 and ∆ is a finite union of double Γ J -cosets in SL 2 ( Q ) J . Then we have � � � H k,m,Γ (∆); J cusp tr k,m ( Γ ) = I k,m,Γ ( A ) g m (∆ , A ) , A ∈ Pr(∆) / � m,Γ where Pr : SL 2 ( Q ) J → SL 2 ( Q ) denotes the canonical projection and � m,Γ is the equivalence relation defined by A � m,Γ B if and only if (1) A and B are Γ − conjugate , or (2) A and B are parabolic and GA is Γ − conjugate to B for some G ∈ Γ A ∩ Γ (4 m ) . ( Γ A = { G ∈ Γ | GAG − 1 = A } ) , Here I k,m,Γ ( A ) is the contribution term of modular element A , and g m (∆ , A ) is certain limit of exponential quadraric sum over lattice (This term changes class numbers which is represented by H t in many cases).

Corollary 1 ( S (2009,2010)). From the above Theorem 1, we obtain the following tr( T ( l ) , J cusp , new , + ··· + ( N )) = tr( T ( l ) , S new , + ··· + ( N )) , k, 1 2 k − 2 where J cusp , new , + ··· + ( N ) denotes the space of all forms φ ∈ J cusp , new ( N ) k, 1 k, 1 satisfying φ | W L ( n ) = φ ( n | N ) , and S new , + ··· + ( N ) denotes the space of 2 k − 2 all forms f ∈ S new 2 k − 2 ( N ) with f | W n = f ( n | N ) . Remark 2. For ( Q, N/Q ) = 1, W L ( Q ) is the Atkin-Lehner operator with respect to level on J k, 1 ( N ) as follows: � � � � � � φ | W L ( Q ) = 1 1 1 0 Qa b QΓ 0 ( N ) J Γ 0 ( N ) J Q 2 φ | H k, 1 ,Γ 0 ( N ) , 0 Q Nc Qd where a, b, c and d are integers with Q 2 ad − Nbc = Q .

Corollary 2 ( S (2009,2010)). Notation and assumptions are the same as above. Then, from Corollary 1 and the result of Skoruppa-Zagier, we get the following tr( T ( l ) , J cusp , new , + ··· + ( N )) = tr( T ( l ) , J cusp , new , + ··· + (1)) . k, 1 k,N Here J cusp , new , + ··· + (1) is the subspace of J cusp , new (1) consisting of all φ k,N k,N which satisfies φ | W I ( n ) = φ ( n | N ) . Remark 3. For n || m , W I ( n ) is the Atkin-Lehner operator with respect to index on J k,m (1) � � 1 � � 1 n Z 2 φ | W I ( n ) = n 2 φ | H k,m,Γ 0 ( N ) Γ 0 ( N ) ⋉ · µ n   � � r 2 − D   � �   = c φ ( D, λ n r )  e τ + rz ,    4 m r ∈ Z D ≦ 0 D ≡ r 2(mod 4 m ) where λ n is the modulo 2 m uniquely determined integer which satisfies λ n ≡ − 1 (mod 2 n ) and λ n ≡ +1 (mod 2 m/n ).

Theorem 2 ( S (2009,2010)). The notation and assumption is the same as above. For a squarefree odd N , we obtain the Lifting map from J cusp , new , + ··· + ( N )) to J cusp , new , + ··· + (1) as follows: k, 1 k,N � � � � r 2 − D r 2 − D � � c φ ( D ; r )e τ + rz → c φ ( D ; r )e τ + rz . 4 4 N 0 >D,r ∈ Z 0 >D,r ∈ Z D ≡ r 2 (mod4) D ≡ r 2 (mod4 N ) Remark 4. In the case of N being not squarefree, the coincidence of Hecke Traces does not work in the same form as Theorem 1. Therefore we have not the level-index changing phenomenon on these spaces. We construct the same isomorphism as them in the case of prime power level by restricting the special subspace.

Theorem 3 ( S (2013)). k ∈ 2 N ≥ 2 , p : odd prime and ( l, p ) = 1 . � � � � T ( l ) , J cusp k, 1 ( p 2 ) T ( l ) , S 2 k − 2 ( p 2 ) (1) tr elliptic − tr elliptic    ∆ ( s,l ) /p 2 � − 1 p 2 k − 2 ( s, l ) H 1 ( s 2 − 4 l )  , = 2 p s 2 < 4 l s � =0 � � � � T ( l ) , J cusp k, 1 ( p 4 ) T ( l ) , S 2 k − 2 ( p 4 ) (2) tr elliptic − tr elliptic � − 1 p 2 k − 2 ( s, l ) H 1 ( s 2 − 4 l ) = 2 s 2 < 4 l s � =0            ∆ ( s,l ) /p 2  ∆ ( s,l ) /p 2  ∆ ( s,l ) /p 4  + p   1 +     , × p p p � � � � T ( l ) , J cusp k, 1 ( p 6 ) T ( l ) , S 2 k − 2 ( p 6 ) (3) tr elliptic − tr elliptic � − 1 p 2 k − 2 ( s, l ) H 1 ( s 2 − 4 l ) = 2 s 2 < 4 l s � =0                ∆ ( s,l ) /p 2  ∆ ( s,l ) /p 2  ∆ ( s,l ) /p 4  ∆ ( s,l ) /p 6  + p 2  + p   1 +    1 +    . × p p p p � a � Here ∆ ( s,l ) = s 2 − 4 l and = 0 if a �∈ Z or ( a, b ) > 1 . b