Quadratic twists of Siegel modular forms of paramodular level: Hecke operators and Fourier coefficients

Transcription

1 Quadratic twists of Siegel modular forms of paramodular level: Hecke operators and Fourier coefficients Jennifer Johnson-Leung University of Idaho October, 205 JJL Twisted Paramodular Forms October, 205 / 30

2 Joint work with Brooks Roberts JJL Twisted Paramodular Forms October, / 30

3 Recap Let L be a local field, (π, V ) a representation of GSp(4, L), χ a quadratic character of L. JJL Twisted Paramodular Forms October, / 30

4 Recap Let L be a local field, (π, V ) a representation of GSp(4, L), χ a quadratic character of L. In the last talk we constructed the twisting map T χ (v) : V (0) V (4, χ). JJL Twisted Paramodular Forms October, / 30

5 Recap Let L be a local field, (π, V ) a representation of GSp(4, L), χ a quadratic character of L. In the last talk we constructed the twisting map T χ (v) : V (0) V (4, χ). In this talk we consider the induced map on Siegel modular forms T χ : S k (Γ para (N)) S k (Γ para (Np 4 )). JJL Twisted Paramodular Forms October, / 30

6 Siegel upper half plane Let J =. Define the algebraic group GSp(4) to be the set of all g GL(4) such that t gjg = λ(g)j for some λ GL(). The group GSp(4, R) + (where λ(g) > 0) acts on the Siegel upper half-plane ( ) z τ H 2 := {Z = τ z Mat(2 2, C) : I(Z) > 0} by transformations g < Z >= (AZ + B)(CZ + D) ( ) A B where g =. C D JJL Twisted Paramodular Forms October, / 30

7 Paramodular Group For N a positive integer we define Z Z N Z Z Γ para (N) = Sp(4, Q) NZ Z Z Z NZ NZ Z NZ NZ Z Z Z Notice that Γ para (N) Γ para (Np). JJL Twisted Paramodular Forms October, / 30

8 Paramodular Forms A Paramodular form, is a holomorpic function on H 2 such that for every ( ) A B g = Γ para (N), C D F k g(z) = λ(g) k det(cz + D) k F (g < Z >) = F (Z). S k (Γ para (N)) be the space of Siegel modular cusp forms of weight k, degree two, and paramodular level N. JJL Twisted Paramodular Forms October, / 30

9 Fourier Coefficients F has a Fourier expansion F (Z) = S A(N) + a(s)e 2πitr(SZ) for Z H 2. Here, A(N) + is the set of all 2 2 matrices S of the form: [ α β S = β γ Let, α NZ, γ Z, β Z, 2 α > 0, det [ A g = [ α β = αγ β 2 > 0. β γ ta, A Γ 0 (N). Since g Γ para (N), we have F k (g) = F and so deduce that a( t ASA) = det(a) k a(s). JJL Twisted Paramodular Forms October, / 30

10 Fourier Coefficients Since for ( ) A B g = Γ para (N), C D F k g(z) = λ(g) k det(cz + D) k a χ (S)e 2πitr(S(AZ+B)(CZ+D) ) S A(N) + JJL Twisted Paramodular Forms October, / 30

11 Fourier Coefficients Since for ( ) A B g = Γ para (N), C D F k g(z) = λ(g) k det(cz + D) k a χ (S)e 2πitr(S(AZ+B)(CZ+D) ) S A(N) + To have hope for a formula T χ (F )(Z) = S A(Np 4 ) + a χ (S)e 2πitr(SZ) with the a χ (S) given in terms of the coefficients a(s), we would like to have C = 0. JJL Twisted Paramodular Forms October, / 30

12 Iwasawa decomposition The Iwasawa decomposition asserts that GSp(4, L) = B GSp(4, o) where B is the Borel subgroup of upper-triangular matrices in GSp(4, L) and o is the ring of integers of L. JJL Twisted Paramodular Forms October, / 30

13 Iwasawa decomposition The Iwasawa decomposition asserts that GSp(4, L) = B GSp(4, o) where B is the Borel subgroup of upper-triangular matrices in GSp(4, L) and o is the ring of integers of L. If v is invariant under GSp(4, o), then it is possible to obtain a formula for T χ (v) involving only upper-triangular matrices. JJL Twisted Paramodular Forms October, / 30

14 Iwasawa decomposition The Iwasawa decomposition asserts that GSp(4, L) = B GSp(4, o) where B is the Borel subgroup of upper-triangular matrices in GSp(4, L) and o is the ring of integers of L. If v is invariant under GSp(4, o), then it is possible to obtain a formula for T χ (v) involving only upper-triangular matrices. This is also possible in principle for v V (). JJL Twisted Paramodular Forms October, / 30

15 Iwasawa decomposition The Iwasawa decomposition asserts that GSp(4, L) = B GSp(4, o) where B is the Borel subgroup of upper-triangular matrices in GSp(4, L) and o is the ring of integers of L. If v is invariant under GSp(4, o), then it is possible to obtain a formula for T χ (v) involving only upper-triangular matrices. This is also possible in principle for v V (). For the remainder of the talk, we fix an odd prime p N and we let χ be the nontrivial quadratic Dirichlet character modulo p. JJL Twisted Paramodular Forms October, / 30

16 Local Twisting Operator aϖ bϖ 2 zϖ 4 T χ (v) = q 3 χ(ab)π( bϖ 2 x aϖ )τv da db dx dz o o o o aϖ bϖ 2 zϖ 4 + q 3 χ(ab)π( bϖ 2 y aϖ )τv da db dy dz o p o o aϖ bϖ 2 zϖ 3 + q 2 χ(ab)π(t 4 bϖ 2 x aϖ )τv da db dx dz o o o o aϖ bϖ 2 zϖ 3 + q 2 χ(ab)π(t 4 bϖ 2 y aϖ )τv da db dy dz. o p o o JJL Twisted Paramodular Forms October, / 30

17 Plan of attack Try to directly provide an Iwasawa identity g = bk where g GSp(4, F ), b B, and k GSp(4, o). JJL Twisted Paramodular Forms October, 205 / 30

18 Plan of attack Try to directly provide an Iwasawa identity g = bk where g GSp(4, F ), b B, and k GSp(4, o). Use the formal matrix identity [ [ [ x x = x x [ [ x. JJL Twisted Paramodular Forms October, 205 / 30

19 Plan of attack Try to directly provide an Iwasawa identity g = bk where g GSp(4, F ), b B, and k GSp(4, o). Use the formal matrix identity [ [ [ x x = x x [ [ x. It is necessary to decompose the domains of integration, and this leads to a proliferation of terms. JJL Twisted Paramodular Forms October, 205 / 30

20 Example of a lemma Lemma aϖ bϖ 2 zϖ 3 q 2 χ(ab)π(t 4 bϖ 2 x aϖ )τv da db dx dz o o o o xϖ 2 aϖ 2 bϖ 3 =q 2 χ(b)ηπ( aϖ 2 xϖ 2 )v da db dx dz o o o o zϖ 3 xϖ 2 aϖ 2 bϖ 3 + q χ(b)ηπ( aϖ 2 xϖ 2 )v da db dx dz o o o o zϖ 4 xϖ 2 aϖ 2 bϖ 3 + χ(b)ηπ( aϖ 2 xϖ 2 )v da db dx dz o o o o zϖ 5 xϖ 2 aϖ 2 bϖ 3 + q χ(b)ηπ( aϖ 2 xϖ 2 )v da db dx dz o o o o zϖ 6 JJL Twisted Paramodular Forms October, / 30

21 The slash formula Theorem Let N and k be positive integers, p a prime with p N, and χ a nontrivial quadratic Dirichlet character mod p. Then, the local twisting map T χ : S k (Γ para (N)) S k (Γ para (Np 4 )). If F S k (Γ para (N)), then this map is given by the formula where the T i are given below. 4 T χ (F ) = F k Tχ, i i= JJL Twisted Paramodular Forms October, / 30

22 Slash formula T χ = p T 2 χ = p T 3 χ = p 6 T 4 χ = p 0 T 5 χ = p 9 a,b,x (Z/p 3 Z) z Z/p 4 Z a,b (Z/p 3 Z) x,y (Z/p 3 Z) x,y (p) a (Z/p 2 Z) b (Z/p 3 Z) z (Z/pZ) z (p) a Z/p 4 Z b (Z/p 3 Z) x (Z/p 4 Z) a,b (Z/p 3 Z) x Z/p 3 Z [ zp 4 bp χ(ab)u( 2 bp 2 x p )A( [ (a + xb)p ), [ ab( ( y) χ(abxy)u( x)p 3 ap 2 ap 2 ab ( x) p )A( [ bp 3 ap χ(b( z))u( 2 [ p ap 2 a 2 b zp )A( [ (ax bp)p 4 ap χ(b)u( 2 ap 2 [ (ax b)p 3 ap χ(b)u( 2 ap 2 )A( A( [ p xp 2 [ p xp ) ), ), [ p bp ), JJL Twisted Paramodular Forms October, / 30

23 Slash formula T 6 χ = p 6 T 7 χ = p 7 T 8 χ = p 9 T 9 χ = p 6 a,b (Z/p 2 Z) x (Z/pZ) a (Z/p 3 Z) b (Z/pZ) z Z/p 4 Z a,b,z (Z/p 3 Z) z (p) a (Z/p 2 Z) b,x (Z/pZ) Tχ 0 = p 6 a (Z/p 2 Z) b (Z/pZ) [ b( + xp)p 2 ap χ(bx)u( 2 [ p 2 ap 2 a 2 b p 2 )A( [ zp 4 bp χ(ab)u( bp )A( [ ap ap p ), [ ab( z)p 3 ap χ(abz( z))u( A( [ bp χ(b)u( [ bp χ(b)u( xp [ p 2 a )A( ), p [ p 2 a )A( ), p 2 ), [ p bp p ), JJL Twisted Paramodular Forms October, / 30

24 Slash formula Tχ = p 0 a (Z/p 2 Z) b (Z/p 4 Z) x Z/p 3 Z z Z/p 4 Z Tχ 2 = p 2 y Z/p 4 Z a (Z/p 4 Z) b,z (Z/p 3 Z) z (p) Tχ 3 = p 6 a (Z/p 2 Z) b (Z/p 3 Z) x (Z/pZ) Tχ 4 = p 6 a (Z/p 2 Z) b (Z/pZ) x Z/p 4 Z [ χ(ab)u( zp 4 (ap + xb)p 3 (ap + xb)p 3 [ bp 2 )A( ), xp 2 p [ a(y b( z)p)p 4 yp χ(abz( z))u( 3 yp 3 a (y + bp)p 2 )A( [ b( + x)p ap χ(bx)u( 2 [ p 2 ap 2 a 2 b p 3 )A( [ bp ap χ(b)u( 2 [ p 2 )A( ap 2 xp 4 p 2 ). p ), [ p ap 2 p ), JJL Twisted Paramodular Forms October, / 30

25 How do you know that this is correct? JJL Twisted Paramodular Forms October, / 30

26 How do you know that this is correct? How do you know there is not something simpler? JJL Twisted Paramodular Forms October, / 30

27 Method of Calculating Fourier Coefficients (F k T χ )(Z) = p ( z Z/p 4 Z e a(s) S A(N) + 2πitr(S zp 4 bp 2 a,b,x (Z/p 3 Z) χ(ab) bp 2 x p ) 2πitr(S[ ) e (a + xb)p Z). Then the inner sum is calculated as z Z/p 4 Z e 2πi( γp x 2bβp 2 +αzp 4) = After simplification and rearrangement we obtain (F k T χ )(Z) = p S A(Np 4 ) + p 2β a,b (Z/pZ) a(s[ { p 4 e 2πi(γpx +2bβ)p 2 if p 4 α, 0 otherwise. [ (a + b)p ) χ(ab)w (χ, γ + 2βp a)e 2πitr(SZ). JJL Twisted Paramodular Forms October, / 30

28 Fourier Coefficient Theorem The twist of F has the Fourier expansion T χ (F )(Z) = W (χ)a χ (S)e 2πitr(SZ) S A(Np 4 ) + where W (χ) = [ α β b (Z/pZ) χ(b)e2πibp and a χ (S) for S = is β γ given as follows: JJL Twisted Paramodular Forms October, / 30

29 Fourier Coefficient Theorem The twist of F has the Fourier expansion T χ (F )(Z) = W (χ)a χ (S)e 2πitr(SZ) S A(Np 4 ) + where W (χ) = [ α β b (Z/pZ) χ(b)e2πibp and a χ (S) for S = is β γ given as follows: If p 2β and p 4 α, then [ a χ (S) = p k bp χ(2β) ). p b (Z/pZ) χ(b)a(s[ JJL Twisted Paramodular Forms October, / 30

30 Fourier Coefficient Theorem If p 2β and p 4 α, then a χ (S) is p a,b (Z/pZ) χ + p a,z (Z/pZ) z (p) [ p χ(αp 4 p 2 )a(s[ [ + p k 2 yp 2 χ( γ)a(s[ ( ab(2βp a γ) ) (a + b)p a(s[[ ) χ ( az( z)(azαp 4 2βp ) ) [ p ap a(s[ 2 p p p 2 ) + p k 2 χ( αp 4 )a(s[ ), where xαp 4 2βp (p 2 ) and y2βp γ(p 2 ). [ p xp 3 ) ) JJL Twisted Paramodular Forms October, / 30

31 Fourier Coefficient Theorem If p 2β and p 5 α, then a χ (S) is ( χ ab(2βp a γ) ) (a + b)p a(s[[ ) a,b (Z/pZ) [ + p k 2 yp 2 χ( γ)a(s[ p ) [ p χ(2βp p ap ) 2 ), p p a (Z/pZ) χ(a)a(s[ where y (Z/p 2 Z) is such that 2βp y γ(p 2 ). JJL Twisted Paramodular Forms October, / 30

32 If p 2 2β and p 4 α, then a χ (S) = a χ(s) + a χ(s) + a χ (S) () where [ a χ(s) = ( p )χ(γ)a(s) p bp χ(γ) a(s[ ) b (Z/pZ) + p k 3 b,x,y (Z/pZ) x,y (p) χ ( y( x)(αp 4(y x y ) b 2 + 2βp 2 b γx ) ) p bp a(s[[ 2 ) + p k 3( χ(γ) + pχ( 4 det(s)p 4 γ) χ( αp 4 )W (, 2βp 2 ) ) [ p a(s[ ) [ + p k 3 χ( αp 4 ) W (, 2βp 2 xαp 4 p xp )a(s[ 2 ) x Z/pZ + p 2k 4 χ ( z( z)(γ zαp 4 b 2 ) ) [ p bp a(s[ 3 p b (Z/p 2 Z) z (Z/pZ) p 2 (αp 4 b 2 2βp 2 b+γ) z (p) [ p χ(αp 4 p ap ) a(s[ 2 ) p a (Z/pZ) [ + p k 3 χ(αp 4 )(pχ( 4 det(s)p 4 p 2 ) W (, γ))a(s[ ) p [ + ( p )χ(αp 4 p 2 )a(s[ p 2 ), ) JJL Twisted Paramodular Forms October, / 30

33 [ a p 2k 4 χ(αp 4 )W (, 4 det(s)p 5 p 2 )a(s[ ) if p 5 det(s) and χ(γαp 4 ) = χ(s) = 0 otherwise, and a χ (S) = p 3k 5 χ(αp 4 )W (, 4p 6 det(s))a(s[ [ and p 8 4 det(s) ( χ(αp 4 ) p 3k 5 p 2 ap 3 (p )a(s[ p ) if p 8 4 det(s) [ +p 4k 6 p 2 bp 4 ) a(s[ p 2 ) 0 otherwise, [ p 2 ap 3 ) if p 6 4 det(s) p where a (Z/pZ) satisfies 2αp 4 a 2βp 2 (p) and b (Z/p 2 Z) satisfies 2αp 4 b 2βp 2 (p 2 ). (If a or b does not exist, then the corresponding term is 0.) JJL Twisted Paramodular Forms October, / 30

34 If p 2 2β and p 5 α, then a χ (S) = ( p )χ(γ)a(s) p χ(γ) p k 3 b (Z/pZ) x (Z/pZ) x (p) b (Z/pZ) a(s[ [ bp ) χ ( ( x)(2βbp 2 γx ) ) [ p bp a(s[ 2 [ + p k 3 χ(γ)( p + pχ( 4 det(s)p 4 p ))a(s[ ) [ p 2k 4 p bp χ( γ) a(s[ 3 b (Z/p 2 Z) p 2 (αp 4 b 2 2βp 2 b+γ) p ) ). (2) JJL Twisted Paramodular Forms October, / 30

35 Perhaps the first formula is enough Theorem (Saha) Let F S k (Sp(4, Z)) be such that a(s) = 0 for all but finitely many matrices S such that 4 det(s) is odd and squarefree. Then F = 0. Theorem (Marzec) Let F be a nonzero paramodular cuspidal newform of square-free level N and even weight k > 2. Then F has infinitely many nonzero fundamental Fourier coefficients. JJL Twisted Paramodular Forms October, / 30

36 Vanishing on the Mass Space Corollary Let the notation be as in our Theorem and assume that k is even and N =. If F is in the Maass space as defined in then T χ (F ) = 0. This is another rather hairy calculation, but the upshot is that we have non-trivial cancellations between the fourteen terms. Further, the terms that cancel with eachother vary with S. JJL Twisted Paramodular Forms October, / 30

37 LMFDB tells us that it is nonzero in general! N = p = 3 F = Υ20 in LMFDB. [ 8 22 a χ ( 22 6 ) = 3 9 χ(44) ( a( = 3 9( [ 0 a( 0 8 [ [ ) a( [ 2 0 ) a( ) ) 0 9 ) ) = 3 9 ( ) = JJL Twisted Paramodular Forms October, / 30

38 Compatibility with Hecke Operators Theorem For every prime l p we have the following commutation relations for the paramodular Hecke operators and Atkin-Lehner operator: for F S k (Γ para (N)). T (,, l, l)t χ =χ(l)t χ T (,, l, l), T (, l, l, l 2 )T χ =T χ T (, l, l, l 2 ), T χ (F ) k U l =χ(l) val l(n) T χ (F k U l ), JJL Twisted Paramodular Forms October, / 30

39 What about this example? Recall that Chris Poor showed us the pair of examples of conductor 954 that seem to be quadratic twists. JJL Twisted Paramodular Forms October, / 30

40 What about this example? Recall that Chris Poor showed us the pair of examples of conductor 954 that seem to be quadratic twists. The problem is that 954 = so it is not covered by our theory. JJL Twisted Paramodular Forms October, / 30

41 Thank You! JJL Twisted Paramodular Forms October, / 30