Eigevalues of Ikeda Lifts Rodey Keato Abstract I this paper we compute explicit formulas for the Hecke eigevalues of Ikeda lifts These formulas, though complicated, are obtaied by purely elemetary techiques 1 Itroductio It is well-kow i the settig of elliptic ewforms that the Hecke eigevalues ecode importat arithmetic iformatio For example, if we attach a Galois represetatio to our ewform (for a survey of this material see [5]), the the Hecke eigevalues at p determie the characteristic polyomial of the image of the Frobeius elemet at p Similarly, by the work of Laumo i [] ad Weissauer i [9], we have Galois represetatios attached to degree Siegel eigeforms, ad just as i the elliptic settig, the Hecke eigevalues at p determie the characteristic polyomial of the image of the Frobeius elemet at p I the settig of higher geus Siegel eigeforms, it is reasoable to assume that similar arithmetic iformatio is cotaied i the Hecke eigevalues Note, i this settig we are hidered by the lack of Galois represetatios attached to higher geus Siegel eigeforms We will cosider the Ikeda lift, ie, a map which seds a elliptic eigeform to a eve degree Siegel eigeform I this settig, we derive explicit formulas for the Hecke eigevalues at p of the lifted form i terms of the Hecke eigevalues at p of the origial elliptic eigeform Our paper is orgaized as follows I Sectio we give the ecessary backgroud o Siegel modular forms I Sectio 3 we preset the properties of the Satake parameters ad Hecke operators which we will eed Sectio gives a brief itroductio the Ikeda lift ad some required results Fially, i Sectio 5 we derive the followig expressios for the Hecke eigevalues of Ikeda lifts The expressio for λ(p; I (f)) is p k (+1) j ( ) j r j( 1) r p c r j=1 r=0 +j λ(p; f) j r + p j (j r) 8 (1) 1
ad the expressio for λ r (p ; I (f)) is r (+1) k p (+1) k + p a a a=1 b=r c=0 b a (mod ) b=r b 0 (mod ) a( 1) c( ) a c c p d m b (r) λ(p; f) a c +a+b a b +a b a+b (a c) m b (r)p 5b, +b b b +b () where d = a +5b +( k+1)(a r) Siegel Modular Forms Give a rig R we set M (R) to be the set of by matrices with etries i R Defie the degree symplectic group to be GSp := {g GL : t gj g = µ (g)j m, µ (g) GL 1 }, ( ) 0 1 where J = We defie Sp 1 0 to be the kerel of µ Throughout we will deote Sp (Z) by Γ The geus Siegel upper half-plae is give by h = {Z M (C : t Z = Z, Im(Z) > 0} We have a actio of GSp + (R) := {g GSp (R) : µ (g) > 0} o h give by gz = (az + b)(cz + d) 1, ( ) a b for g = c d For g GSp + (R) ad Z h, we set j(g, Z) = det(cz + d) Let k be a positive iteger For a fuctio F : h C we defie the weight k slash operator o F by (F k g ( Z)) = µ (g) k j(g, Z) k f(gz) We say that F is a geus Siegel modular form of weight k if F is holomorphic ad (F k γ)(z) = F (Z) for all γ Γ Note, if = 1 the we must also require F to be holomorphic at the cusps Deote the space of all such forms by M k (Γ ) It is a basic result from the theore of Siegel modular forms that every F M k (Γ ) has a Fourier expasio of the form F (Z) = Λ a F (T )e(tr(t Z)),
where Λ is the set of by half-itegral positive semi-defiite symmetric matrices ad e(w ) := e πiw If a F (T ) = 0 uless T is strictly positive defiite we say that F is a cusp form, ad we deote the space of all such cusp forms by S k (Γ ) 3 Hecke Operators ad Satake Parameters Let g GSp (Q), the we write T (g) to deote the double coset Γ gγ The, we call T (g) a Hecke operator ad we have a actio of T (g) o F M k (Γ ) give by T (g)f = F k g i, where we are summig over a set of coset represetatives for Γ \Γ gγ Let p be a prime ad defie T = T (diag(1, p1 )), ad for i = 1,, defie T i (p ) = T (diag(1 i, p1 i, p 1 i, p1 i )) Note, we will drop the from the superscript whe the geus is clear from cotext It is well kow that these operators geerate the local Hecke algebra at p We call F M k (Γ ) a eigeform if F is a simultaeous eigevector for each of the Hecke operators Let F S k (Γ ) be a eigeform The, there is a set of complex umbers α i (p; F ) for 0 i, called the Satake p-parameters of F, which completely determie the eigevalues of F with respect to T ad T i (p ) for 1 i, which we deote by λ(p; F ) ad λ i (p ; F ), respectively I fact, from [7], we have the followig relatioships betwee the eigevalues of F, ad the Satake parameters of F, λ(p; F ) = α 0 (p; F )(1 + σ 1 + + σ ), (3) λ r (p ; F ) = α 0 (p; F ) i j r i,j 0 m i j (r)p (i j+1 ) σi σ j, () where m h (r) := #{M M h (F p ) : t A = A, corak(a) = r}, σ i is the degree i elemetary symmetric fuctio i α 1 (p; F ),, α (p; F ), ad 1 i Note, we have ormalized our Satake parameters to satisfy α 0 (p; F ) (+1) k α 1 (p; F ) α (p; F ) = p Suppose f S k (Γ 1 ) is a ormalized eigeform, ie, a f (1) = 1 The we will make use of the followig relatioship as well, α(p; f) + α(p; f) 1 = p 1 k λ(p; f), (5) 3
where we simply use α(p; f) to deote α 0 (p; f) Usig these Satake parameters we ca associate two L-fuctios to a eigeform F S k (Γ ) First, i order to defie the stadard L-fuctio of F we eed the followig local L-factor L p (s, F ; st) = (1 p s ) (1 α i (p; F )p s )(1 α i (p; F ) 1 p s ) i=1 The, the stadard L-fuctio associated to F is give by the followig Euler product L(s, F ; st) = L p (s, F ; st) 1 p Secod, i order to defie the spior L-fuctio associated to F we eed the followig local L-factor L p (s, F ; spi)=(1 α 0 (p; F )p s ) (1 α 0 (p; F )α i1 (p; F ) α ij (p; F )p s ) i=1 1 i 1 i j The, the spior L-fuctio associated to F is give by L(s, F ; spi) = p L p (s, F ; spi) 1 Note, whe F is of geus 1, this is simply the usual Dirichlet series associated to F The Ikeda Lift I this sectio we itroduce the Ikeda lift as preseted i [3] Let T > 0 be i Λ Set D T to be the determiat of T, T to be the absolute value of the discrimiat of Q( D T ), χ T to be the primitive Dirichlet character associated to Q( D T )/Q, ad f T to be the ratioal umber satisfyig D T = T f T Let S (R) deote the set of symmetric matrices over a rig R For a ratioal prime p, let ψ p : Q p C x be the uique additive character give by ψ p (x) = exp( {x} p ), where {x} p Z[ 1 p ] is the p-adic fractioal part of x The Siegel series for T is defied to be b p (T, s) := ψ p (Tr(T S)) ν(s) p p s, for Re(s) >> 0, S S (Q p)/s (Z p) where ν(s) := det(s 1 ) Z p, ad S 1 is from the factorizatio S = S 1 1 S for a symmetric coprime pair of matrices S 1, S We have a factorizatio of the Siegel series give by b p (T, s) = γ p (T, p s )F p (T, p s ),
where γ p (T, X) = / 1 X (1 p i X ), 1 p χ T X ad F p (T, X) Z[X] has costat term 1 ad deg(f p (T, X)) = ord p (f T ) Usig this polyomial F p (T, X) we defie i=1 F p (T, X) := X ordp(f T ) F p (T, p 1 X) We will make use of this F p (T, X) later Let f(τ) S k (Γ 1 )) be a ormalized eigeform ad let h(τ) = c(n)q N S + (Γ k + 1 0 ()) m>0 ( 1) k m 0,1() be a eigeform correspodig to f via the Shimura correspodece For each T > 0 i Λ, defie +1 k a(t ) = c( T )f F p (T, α(p; f)), (6) ad form the followig series T p I (f)(z) = T a(t )e(tr(t z)) We have the followig theorem Theorem 1 ([3, Thm 3, 33]) The series I (f)(z), referred to as the Ikeda lift of f, is a eigeform i S k (Γ ) whose stadard L-fuctio factors as L(s, F ; st) = ζ(s ) L(s + k i, f) i=1 Note, if = the this L-fuctio factorizatio agrees with that of the Saito-Kurokawa lift From the factorizatio of the stadard L-fuctio give i the previous theorem, we are able to obtai a expressio for α i (p; I (f)) i terms of α(p; f) for 1 i However, i order to determie the eigevalue of I (f) we must also have iformatio about α 0 (p; I (f)) This ca be achieved by determiig the factorizatio of the spior L-fuctio of I (f) This factorizatio ca be foud i [6] I summary, we obtai the followig relatios betwee Satake parameters α 0 (p; I (f)) = p k (+1) α(p; f), (7) (+1) j α j (p; I (f)) = p α(p; f), for j = 1,, (8) Note, we our Satake parameters are ormalized differetly tha the Satake parameters i [6] 5
5 Eigevalues of the Ikeda lift I this sectio we will derive Equatios 1 ad We begi by determiig a ice expressio for the symmetric polyomials i Equatios 3 ad Usig Equatio 8 we obtai the followig expressio, σ j = p j(+1) α(p; f) j (+1) q (i, j)p i, (9) where we are usig q (i, j) to deote the umber of partitios of i ito j distict parts where each part is o greater tha ad our symmetric polyomial is i α 1 (p; I (f)),, α (p; I (f)) Combiig Equatio (73) ad Theorem 8 of [1] we have the followig geeratig series for this partitio fuctio q (i, j)z j 1 zi = j=0 = (1 + z 1 z) l l=1 (z ) j j z j(j+1) z j 1, (z ) (z ) j=0 where (x) := m=1 (1 xm ) If we fix a j o the left had side ad evaluate at z = p we obtai (+1) q (i, j)p i z j 1, where we eed oly take the summatio to ( + 1)/ sice we have restricted to j Matchig coefficiets we have (+1) q (i, j)p i = p j(j+1) (10) j j The followig lemma follows immediately from this equality Lemma For ay j, p j(+1) (+1) q (i, j)p i = p ( j)(+1) (+1) q (i, j)p i 6
I summary we have σ j = p j(j ) α(p; f) j j j We are ow prepared to compute λ(p; I (f)) Combiig Equatios 3, 9, ad 10 ad pairig our terms usig Lemma we obtai the followig expressio for λ(p; I (f)), p k (+1) j=1 p ( j)( +j) +j j ( α j + α j) + p 8 Applyig a theorem of Waterso from [8] alog with Equatio 5 we have j ( ) α j + α j = ( 1) r j j r ( j r p k+1 λ(p; f)) j r r r=0 Isertig this ito the previous equatio ad rearragig we obtai the followig expressio for λ I(f) p k (+1) j ( ) j r j( 1) r p c r j=1 r=0 +j λ(p; f) j r + p 8 j, (j r) where c = ( j)( +j)+j( k+1)+r(k 1) I order to compute a similar formula for λ r (p ; I (f)) for r <, we first eed the followig expressio for m h (r) from [] m h (r) = p ) h h (h (1 p 1 l ) l=0 h r r Applyig this formula alog with similar techiques used above we have the followig expressio for λ r (p ; I (f)), (+1) k p i j r i,j 0 p j(j )+i(i )+(i j) i j i j l=0 (1 p 1 l ) α(p; f) i j r r j j i i i+j 7
By makig the substitutios a = i + j ad b = i j ad simplifyig we obtai (+1) k = p (+1) k + p a b r a=r b=r b a (mod ) r a p a +5b b r a=+1 b=r b a (mod ) b b (1 p 1 l ) l=0 α(p; f) a r a b a b a+b a+b p 5b b b (1 p 1 l ) l=0 α(p; f) a r a b a b a+b a+b Just as above, we ca shift each summatio, pair appropriately, ad the apply the theorem of Waterso to obtai our fial expressio for λ r (p ; I (f)) r (+1) k = p (+1) k + p a a a=1 b=r c=0 b a (mod ) b=r b 0 (mod ) a( 1) c( ) a c c p d m b (r) λ(p; f) a c +a+b a b +a b a+b (a c) m b (r)p 5b +b b b +b where d = a +5b +( k+1)(a r) Refereces [1] G Adrews ad Eriksso K Iteger Partitios Cambridge Uiversity Press, 00 [] R Bret ad B McKay O determiats of radom symmetric matrices over Z m Ars Combiatoria, 6A:57 6, 1988 [3] T Ikeda O the liftig of elliptic modular forms to Siegel cusp forms of degree A of Math, 15:61 681, 001 [] G Laumo Foctios zéta des variétés de Siegel de dimesio trois Astérisque, 30:1 66, 005 [5] K Ribet The l-adic represetatios attached to a eigeform with ebetypus: a survey Lecture Notes i Mathematics, 601:17 5, 1977 [6] R Schmidt O the spi L-fuctios of Ikeda lifts Commet Math Uiv St Paul, 5:1 6, 003 8
[7] G va der Geer Siegel modular forms ad their applicatios I 1--3 of Modular Forms, Lectures at a Summer School i Nordfjordeid, Norway, pages 181 6 Uiversitext, 008 [8] A Waterso A expasio for x + y Ediburgh Mathematical Notes, 3:1 15, 19 [9] R Weissauer Four dimesioal Galois represetatios Astérisque, 30:67 150, 005 9