Outline Part One Part Two Part Three Tables of Paramodular Forms Cris Poor David S. Yuen Fordham University Lake Forest College including work in progess with: Jeff Breeding, Fordham University Modular Forms and Curves of Low Genus: Computational Aspects ICERM, September 2015 Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three Computational Aspects of Modularity 1. Part I . The Paramodular Conjecture. 2. Part II . Using Fourier-Jacobi expansions to make rigorous tables of paramodular forms. (Joint work with J. Breeding and D. Yuen.) 3. Part III . Using Fourier-Jacobi expansions to make heuristic tables of paramodular forms. (Joint work with D. Yuen.) 4. Our paramodular website exists: math.lfc.edu/ ∼ yuen/paramodular Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three All elliptic curves E / Q are modular Theorem (Wiles; Wiles and Taylor; Breuil, Conrad, Diamond and Taylor) Let N ∈ N . There is a bijection between 1. isogeny classes of elliptic curves E / Q with conductor N 2. normalized Hecke eigenforms f ∈ S 2 (Γ 0 ( N )) new with rational eigenvalues. In this correspondence we have L ( E , s , Hasse ) = L ( f , s , Hecke ) . Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three All elliptic curves E / Q are modular Theorem (Wiles; Wiles and Taylor; Breuil, Conrad, Diamond and Taylor) Let N ∈ N . There is a bijection between 1. isogeny classes of elliptic curves E / Q with conductor N 2. normalized Hecke eigenforms f ∈ S 2 (Γ 0 ( N )) new with rational eigenvalues. In this correspondence we have L ( E , s , Hasse ) = L ( f , s , Hecke ) . Eichler (1954) proved the first examples L ( X 0 (11) , s , Hasse ) = L ( η ( τ ) 2 η (11 τ ) 2 , s , Hecke ) . Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three All elliptic curves E / Q are modular Theorem (Wiles; Wiles and Taylor; Breuil, Conrad, Diamond and Taylor) Let N ∈ N . There is a bijection between 1. isogeny classes of elliptic curves E / Q with conductor N 2. normalized Hecke eigenforms f ∈ S 2 (Γ 0 ( N )) new with rational eigenvalues. In this correspondence we have L ( E , s , Hasse ) = L ( f , s , Hecke ) . Eichler (1954) proved the first examples L ( X 0 (11) , s , Hasse ) = L ( η ( τ ) 2 η (11 τ ) 2 , s , Hecke ) . Shimura proved 2 implies 1. Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three All elliptic curves E / Q are modular Theorem (Wiles; Wiles and Taylor; Breuil, Conrad, Diamond and Taylor) Let N ∈ N . There is a bijection between 1. isogeny classes of elliptic curves E / Q with conductor N 2. normalized Hecke eigenforms f ∈ S 2 (Γ 0 ( N )) new with rational eigenvalues. In this correspondence we have L ( E , s , Hasse ) = L ( f , s , Hecke ) . Eichler (1954) proved the first examples L ( X 0 (11) , s , Hasse ) = L ( η ( τ ) 2 η (11 τ ) 2 , s , Hecke ) . Shimura proved 2 implies 1. Weil added N = N . Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three All abelian surfaces A / Q are paramodular Paramodular Conjecture (Brumer and Kramer 2009) Let N ∈ N . There is a bijection between 1. isogeny classes of abelian surfaces A / Q with conductor N and endomorphisms End Q ( A ) = Z , 2. lines of Hecke eigenforms f ∈ S 2 ( K ( N )) new that have rational eigenvalues and are not Gritsenko lifts from J cusp 2 , N . In this correspondence we have L ( A , s , Hasse-Weil ) = L ( f , s , spin ) . Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three Remarks The paramodular group of level N , ∗ N ∗ ∗ ∗ ∗ ∗ ∗ ∗ / N K ( N ) = ∩ Sp 2 ( Q ) , ∗ ∈ Z , ∗ N ∗ ∗ ∗ N ∗ N ∗ N ∗ ∗ Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three Remarks The paramodular group of level N , ∗ N ∗ ∗ ∗ ∗ ∗ ∗ ∗ / N K ( N ) = ∩ Sp 2 ( Q ) , ∗ ∈ Z , ∗ N ∗ ∗ ∗ N ∗ N ∗ N ∗ ∗ K ( N ) \H 2 is a moduli space for complex abelian surfaces with polarization type (1 , N ). Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three Remarks The paramodular group of level N , ∗ N ∗ ∗ ∗ ∗ ∗ ∗ ∗ / N K ( N ) = ∩ Sp 2 ( Q ) , ∗ ∈ Z , ∗ N ∗ ∗ ∗ N ∗ N ∗ N ∗ ∗ K ( N ) \H 2 is a moduli space for complex abelian surfaces with polarization type (1 , N ). K ( N ) is the stabilizer in Sp 2 ( Q ) of Z ⊕ Z ⊕ Z ⊕ N Z . Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three Remarks The paramodular group of level N , ∗ N ∗ ∗ ∗ ∗ ∗ ∗ ∗ / N K ( N ) = ∩ Sp 2 ( Q ) , ∗ ∈ Z , ∗ N ∗ ∗ ∗ N ∗ N ∗ N ∗ ∗ K ( N ) \H 2 is a moduli space for complex abelian surfaces with polarization type (1 , N ). K ( N ) is the stabilizer in Sp 2 ( Q ) of Z ⊕ Z ⊕ Z ⊕ N Z . New form theory for paramodular groups: Ibukiyama 1984; Roberts and Schmidt 2004, (LNM 1918). Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three Remarks The paramodular group of level N , ∗ N ∗ ∗ ∗ ∗ ∗ ∗ ∗ / N K ( N ) = ∩ Sp 2 ( Q ) , ∗ ∈ Z , ∗ N ∗ ∗ ∗ N ∗ N ∗ N ∗ ∗ K ( N ) \H 2 is a moduli space for complex abelian surfaces with polarization type (1 , N ). K ( N ) is the stabilizer in Sp 2 ( Q ) of Z ⊕ Z ⊕ Z ⊕ N Z . New form theory for paramodular groups: Ibukiyama 1984; Roberts and Schmidt 2004, (LNM 1918). Grit : J cusp k , N → S k ( K ( N )), the Gritsenko lift from Jacobi cusp forms of index N to paramodular cusp forms of level N is an advanced version of the Maass lift. Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three All abelian surfaces A / Q are paramodular The Paramodular Conjecture again after the remarks Paramodular Conjecture Let N ∈ N . There is a bijection between 1. isogeny classes of abelian surfaces A / Q with conductor N and endomorphisms End Q ( A ) = Z , 2. lines of Hecke eigenforms f ∈ S 2 ( K ( N )) new that have rational eigenvalues and are not Gritsenko lifts from J cusp 2 , N . In this correspondence we have L ( A , s , Hasse-Weil ) = L ( f , s , spin ) . Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three A glimpse of higher modularity? A sequence of discrete subgroups The L -groups of symplectic groups are orthogonal groups. Consider the following special, stable, integral orthogonal groups of spinor norm one. 1 + Γ 0 ( N ) ∼ = Sˆ O − 2 N Z 1 Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three A glimpse of higher modularity? A sequence of discrete subgroups The L -groups of symplectic groups are orthogonal groups. Consider the following special, stable, integral orthogonal groups of spinor norm one. 1 + Γ 0 ( N ) ∼ = Sˆ O − 2 N Z 1 1 1 + K ( N ) ∼ = Sˆ O − 2 N (Gritsenko, Nikulin 1998) Z 1 1 Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three A glimpse of higher modularity? 1 1 . . Q = Q 2 n +1 ( N ) = − 2 N . . 1 1 Q 2 n +1 ( N ) = M ′ M ; L = M Z 2 n +1 ; U is hyperbolic plane Q 2 n +1 ( N ) , Z 2 n +1 � ∼ = U n ⊕ ( − 2 N ) = ( · , L ) ∼ � + Sˆ O Z ( Q ) = { g ∈ SL 2 n +1 ( Z ) : g ′ Qg = Q , ∀ x ∈ L ∗ , gx − x ∈ L , sp . nm . ( g ) = 1 } Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three A glimpse of higher modularity? References 1. Pei-Yu Tsai, On Newforms for Split Special Odd Orthogonal Groups. (Harvard thesis: 2013) 2. Benedict Gross, On the Langlands correspondence for symplectic motives. (2015) Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three Definition of Siegel Modular Form Siegel Upper Half Space: H n = { Z ∈ M sym n × n ( C ) : Im Z > 0 } . � A B � Symplectic group: σ = ∈ Sp n ( R ) acts on Z ∈ H n by C D σ · Z = ( AZ + B )( CZ + D ) − 1 . Γ ⊆ Sp n ( R ) such that Γ ∩ Sp n ( Z ) has finite index in Γ and Sp n ( Z ) Slash action: For f : H n → C and σ ∈ Sp n ( R ), ( f | k σ ) ( Z ) = det( CZ + D ) − k f ( σ · Z ). Siegel Modular Forms: M k (Γ) is the C -vector space of holomorphic f : H n → C that are “bounded at the cusps” and that satisfy f | k σ = f for all σ ∈ Γ. Cris and David Tables of Paramodular Forms

Outline Part One Part Two Part Three Definition of Siegel Modular Form Cusp Forms: S k (Γ) = { f ∈ M k (Γ) that “vanish at the cusps” } Fourier Expansion: f ( Z ) = � T > 0 a ( T ; f ) e (tr( ZT )) For paramodular groups: n = 2; Γ = K ( N ); � Z 1 � 2 Z symmetric T ∈ 1 2 Z N Z Cris and David Tables of Paramodular Forms

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