Using Katsurada’s determination of the Eisenstein series to compute Siegel eigenforms in degree three Cris Poor David S. Yuen Fordham University Lake Forest College including slides from a book in progress with Jerry Shurman, Reed College Computational Aspects of L-functions ICERM, November 2015 Cris and David Katsurada and Eisenstein series ICERM 1 / 36
Computational Aspects of L-functions 1. Part I . What are Siegel modular forms? 2. Part II . What are examples of Siegel modular forms? 3. Part III . What good are Siegel modular forms? 4. Part IV . How are we going to compute Euler factors in degree three? 5. Part V . What Euler factors of Siegel modular forms have been seen? 6. You can see some data at: math.lfc.edu/ ∼ yuen/genus3 Cris and David Katsurada and Eisenstein series ICERM 2 / 36
Siegel Modular Forms � 0 I � R ⊆ R is a commutative subring and J = . − I 0 General Symplectic group n ( R ) = { σ ∈ GL 2 n ( R ) : ∃ ν ∈ R + : σ ′ J σ = ν J } GSp + n ( R ) → R + given by σ �→ n � Similitude: ν : GSp + det( σ ) Symplectic group Sp n ( R ) = ker( ν ) = { σ ∈ SL 2 n ( R ) : σ ′ J σ = J } Siegel Upper Half Space H n = { Ω ∈ M sym n × n ( C ) : Im Ω > 0 } Cris and David Katsurada and Eisenstein series ICERM 3 / 36
Siegel Modular Forms � A B ∈ GSp + � Action of σ = n ( R ) on Ω ∈ H n C D σ · Ω = ( A Ω + B )( C Ω + D ) − 1 Factor of Automorphy � A B n ( R ) × H n → C × given by j ( j : GSp + � , Ω) = det( C Ω + D ) C D Cocycle condition: j ( σ 1 σ 2 , Ω) = j ( σ 1 , σ 2 · Ω) j ( σ 2 , Ω) Slash action of group on functions f : H n → C ( f | k σ ) (Ω) = ν ( σ ) nk − n ( n +1) / 2 j ( σ, Ω) − k f ( σ · Ω) Cris and David Katsurada and Eisenstein series ICERM 4 / 36
Siegel Modular Forms Siegel modular group Γ n = Sp n ( Z ) Vector space of Siegel modular forms of weight k and level one. M k (Γ n ) = { holomorphic f : H n → C : ∀ γ ∈ Γ n , f | k γ = f , and ∀ Y o > 0 , f is bounded on { Ω : Im Ω > Y o }} Siegel Phi map Φ : M k (Γ n ) → M k (Γ n − 1 ) given by � Ω 0 � (Φ f )(Ω) = λ → + ∞ f lim 0 i λ Siegel modular cusp forms S k (Γ n ) = ker(Φ) = { f ∈ M k (Γ n ) : Φ f = 0 } Cris and David Katsurada and Eisenstein series ICERM 5 / 36
Fourier expansion Every Siegel modular form f ∈ M k (Γ n ) has a Fourier expansion � f (Ω) = a ( T ; f ) e ( � Ω , T � ) T : T ≥ 0 , 2 T even • Here, e ( z ) = e 2 π iz and � Ω , T � = tr(Ω T ). �� T 0 � � • a ( T ; Φ f ) = a ; f 0 0 Every Siegel modular cusp form f ∈ S k (Γ n ) has a Fourier expansion � f (Ω) = a ( T ; f ) e ( � Ω , T � ) T : T > 0 , 2 T even Cris and David Katsurada and Eisenstein series ICERM 6 / 36
Ways to make Siegel Modular Forms Eisenstein series Theta series Polynomials in the thetanullwerte Various lifts Specializations of symplectic embeddings. Generating functions, multiplication, differential operators,... Cris and David Katsurada and Eisenstein series ICERM 7 / 36
Siegel Eisenstein Series E ( n ) � = 1 | k σ ∈ M k (Γ n ) for k > n + 1. k P n , 0 ( Z ) σ ∈ P n , 0 ( Z ) \ Γ n � A B � P n , 0 ( Z ) = { ∈ Γ n } 0 D � E ( n ) � = E ( n − 1) � E (1) � Φ ; Φ = 1 k k k k > n + 1 ensures absolute convergence on compact sets Remarkably, the Fourier coefficients of an Eisenstein series, � � T ; E ( n ) a , depend only upon the genus of the index T . k The algorithmic computation of the Fourier coefficients of Siegel � � T ; E ( n ) Eisenstein series a began with C. Siegel in the 1930s and was k completed by Hidenori Katsurada in 1999. Cris and David Katsurada and Eisenstein series ICERM 8 / 36
Example in degree n = 2 Weight 4 Eisenstein series of degree 2: z ∈ H 2 E (2) 4 ( z ) = 1 + 240 e ( z 22 ) + 2160 e (2 z 22 ) + 6720 e (3 z 22 ) + 17520 e (4 z 22 ) + · · · 1 1 1 1 � � � � 2 , z � + 30240 e � ( 1 0 2 + 13440 e � 0 1 ) , z � + 138240 e � , z � 1 1 2 1 2 2 1 1 � � + 181440 e � ( 1 0 0 2 ) , z � + 604800 e � ( 2 1 2 1 2 ) , z � + 362880 e � , z � 1 2 3 2 1 � � 2 , z � + 497280 e � ( 1 0 0 3 ) , z � + 1239840 e � ( 2 0 + 967680 e � 0 2 ) , z � 1 2 2 + 1814400 e � ( 2 1 1 3 ) , z � + · · · omitting GL 2 ( Z )-equivalent terms Cris and David Katsurada and Eisenstein series ICERM 9 / 36
Type II lattices Unimodular self-dual even lattices Definition A lattice Λ in a euclidean space V is Type II means Λ is even: For all u ∈ Λ , � u , u � ∈ 2 Z . Λ is self-dual: Λ = Λ ∗ = { u ∈ V : ∀ v ∈ Λ , � u , v � ∈ Z } . For a fixed rank, necessarily a multiple of 8, the even unimodular lattices form a single genus. rank = 8, genus = { E 8 } . rank = 16, genus = { E 8 ⊕ E 8 , D + 16 } . (Witt) rank = 24, genus = { 24 Niemeier lattices } . (Niemeier) rank = 32, | genus | > 80 million. Cris and David Katsurada and Eisenstein series ICERM 10 / 36
An infinite family of Type II lattices n The checkerboard lattice: D n = { v ∈ Z n : � v i ≡ 0 mod 2 } . j =1 D n is even but [ D ∗ n : D n ] = 4 . The glue vector: [1] = (1 2 , 1 2 , . . . , 1 2) ∈ Q n . The Type II lattice: D + n = D n ∪ ([1] + D n ) . (In fact, D + 8 = E 8 .) Cris and David Katsurada and Eisenstein series ICERM 11 / 36
Theta series of Type II lattices Theorem Let Λ be a Type II lattice of rank 2 k. The degree n theta series of Λ � 1 � ϑ ( n ) � 2 � LL ′ , Ω � Λ (Ω) = e ∈ M k (Γ n ) L ∈ Λ n is a Siegel modular form of weight k and degree n. � � ϑ ( n ) = ϑ ( n − 1) Φ Λ Λ ϑ ( n ) E 8 = E ( n ) ∈ M 4 (Γ n ) = C ϑ ( n ) E 8 , (Duke and Imamo¯ glu) 4 ϑ ( n ) E 8 ⊕ E 8 = ϑ ( n ) 16 if and only if n ≤ 3 , (Problem of Witt) D + J (4) = ϑ (4) E 8 ⊕ E 8 − ϑ (4) 16 ∈ S 8 (Γ 4 ) is the 1888 Schottky form. (Igusa) 8 D + The Wall: � 32 n =1 dim S 16 (Γ n ) > 80 , 000 , 000. Cris and David Katsurada and Eisenstein series ICERM 12 / 36
Fourier coefficients of Theta series Generating functions for lattice counts are Siegel modular forms a ( T ; ϑ Λ ) | Aut Z ( T ) | = Number of sublattices ˜ Λ ⊆ Λ with Gram matrix 2 T . � 1 2 [ 2 1 � = 13 440 = 12 · 1120 = | Aut Z [ 2 1 Example: a 1 2 ]; ϑ E 8 1 2 ] | 1120 There are 1120 sublattices ˜ Λ ⊆ E 8 with a basis ( v 1 , v 2 ) that satisfies � � v 1 , v 1 � � v 1 , v 1 � � � 2 1 � = . � v 2 , v 1 � � v 2 , v 2 � 1 2 There are 1120 sublattices of type A 2 inside E 8 . This motivates the whole theory of Siegel modular forms. Cris and David Katsurada and Eisenstein series ICERM 13 / 36
Siegel’s Theorem Theorem (Siegel) Let k be divisible by 4 and satisfy k > n + 1 . We have 1 1 E ( n ) | Aut Z Λ | ϑ ( n ) � � = Λ , k | Aut Z Λ | [Λ] [Λ] where the sum is over isomorphism classes of Type II latices. 1 We can get a similar theorem for k ≡ 2 mod 4 by attaching pluriharmonic polynomials Q to the theta series � 1 ϑ ( n ) 2 � LL ′ , Ω � Λ , Q (Ω) = � � L ∈ Λ n Q ( L ) e but let’s skip the details. 2 The Eisenstein series is naturally associated to the Type II genus. Cris and David Katsurada and Eisenstein series ICERM 14 / 36
What good are Siegel Modular Forms? Uses in Algebraic Geometry Satake Compactification: S (Γ n \H n ) = proj ( ⊕ ∞ k =0 M k (Γ n )) Smooth Compactification: ˆ S (Γ n \H n ) = proj (valuation subring) The Schottky form J (4) = ϑ (4) E 8 ⊕ E 8 − ϑ (4) 16 ∈ S 8 (Γ 4 ) has the Jacobian locus D + 8 as its zero divisor in degree n = 4. C 4 / (Ω Z 4 + Z 4 ) is the limit of Jacobians of compact Riemann surfaces of genus 4 if and only if J (4) 8 (Ω) = 0 Cris and David Katsurada and Eisenstein series ICERM 15 / 36
What good are Siegel Modular Forms? They make L -functions Both M k (Γ n ) and S k (Γ n ) have a basis of Hecke eigenforms. But how are we going to compute these spaces in order to make our L -functions? The difficulty is in getting enough Fourier coefficients to break the space into eigenspaces and to compute Euler factors. Cris and David Katsurada and Eisenstein series ICERM 16 / 36
The Witt Map A particular symplectic embedding. The embedding W ij : H i × H j → H i + j � � Ω 1 0 (Ω 1 , Ω 2 ) �→ 0 Ω 2 pulls back the the Witt Map W ∗ ij : M k (Γ i + j ) → M k (Γ i ) ⊗ M k (Γ j ) � � Ω 1 0 (Ω �→ f (Ω)) �→ ((Ω 1 , Ω 2 ) �→ f ) 0 Ω 2 The Witt map takes cusp forms to cusp forms. Cris and David Katsurada and Eisenstein series ICERM 17 / 36
Properties of the Witt Map Fourier coefficients of W ∗ ij f ∈ M k (Γ i ) ⊗ M k (Γ j ) in terms of f ∈ M k (Γ i + j ): �� � � T 1 × T 2 ; W ∗ � T 1 R � � a ij f = a ; f R ′ T 2 R ∈ 1 2 M i × j ( Z ) 1. Pay attention to the fact that R has ij entries. Looping over these entires is the cost of evaluating the Witt map. ij ϑ ( i + j ) = ϑ ( i ) Λ ⊗ ϑ ( j ) 2. W ∗ Λ Λ The theta series have a beautiful decomposition under the Witt map. The decomposition of the Eisenstein series under the Witt map is also beautiful but more subtle. Cris and David Katsurada and Eisenstein series ICERM 18 / 36
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