Eigenvalues of Ikeda Lifts
Rodney Keaton
Abstract
In this paper we compute explicit formulas for the Hecke eigenvaluesof Ikeda lifts. These formulas, though complicated, are obtained by purely
elementary techniques.

1Introduction

It is well-known in the setting of elliptic newforms that the Hecke eigenvalues
encode important arithmetic information. For example, if we attach a Galois
representation to our newform (for a survey of this material see [5]), then theHecke eigenvalues at p determine the characteristic polynomial of the image of
the Frobenius element at p.
Similarly, by the work of Laumon in [4] and Weissauer in [9], we have Galois
representations attached to degree 2 Siegel eigenforms, and just as in the elliptic
setting, the Hecke eigenvalues at p determine the characteristic polynomial of
the image of the Frobenius element at p.
In the setting of higher genus Siegel eigenforms, it is reasonable to assume
that similar arithmetic information is contained in the Hecke eigenvalues. Note,in this setting we are hindered by the lack of Galois representations attached tohigher genus Siegel eigenforms.
We will consider the Ikeda lift, i.e., a map which sends an elliptic eigenformto an even degree Siegel eigenform. In this setting, we derive explicit formulas
for the Hecke eigenvalues at p of the lifted form in terms of the Hecke eigenvalues
at p of the original elliptic eigenform.
Our paper is organized as follows. In Section 2 we give the necessary background on Siegel modular forms. In Section 3 we present the properties of the
Satake parameters and Hecke operators which we will need. Section 4 gives abrief introduction the Ikeda lift and some required results. Finally, in Section 5
we derive the following expressions for the Hecke eigenvalues of Ikeda lifts. The
expression for λ(p; In (f )) isnn
Y2 Yjnn 2 b 2 c j − r j(−1)r pc (p)p− 8(p) n(n+1) X Xnk
j−2r
p2− 4 λ(p;f)+nnn+j−jr j=1 r=0
222YYY2(p)(p)(j − r)(p)(1)
1and the expression for λr (p2 ; In (f )) ispnk−n(n+1)
2n−rXn−aXab2cXnY da(−1)c a−cpm(r)(p)2bcn+a+b2

n−a−b2n+a−b2n−a+b2

λ(p; f )a−2cYYYY
a=1 b=r c=0b≡a (mod 2)(a − c)
(p)
(p)
(p)
(p)
+ pnk−n(n+1)
2nXmb (r)pn+b2

5b2 −n24n−b(2)nY(p)2n−bn+b,

222YYYY
b=rb≡0 (mod 2)
(p)
(p)
(p)
(p)
222

where d = a +5b −n +2(n−2k+1)(a−2r).
42

Siegel Modular Forms

Given a ring R we set Mn (R) to be the set of n by n matrices with entries inR.
Define the degree n symplectic group to beGSp2n := {g ∈ GL2n : t gJn g = µn (g)Jm , µn (g) ∈ GL1 },
0n −1nwhere Jn =. We define Sp2n to be the kernel of µn . Throughout
1n 0n
we will denote Sp2n (Z) by Γn .
The genus n Siegel upper half-plane is given by
hn = {Z ∈ Mn (C : t Z = Z, Im(Z) > 0}.
nWe have an action of GSp+
2n (R) := {g ∈ GSp2 n(R) : µn (g) > 0} on h given by

gZ = (aZ + b)(cZ + d)−1 ,a b.
c dnFor g ∈ GSp+2n (R) and Z ∈ h , we set j(g, Z) = det(cZ + d). Let k bea positive integer. For a function F : hn → C we define the weight k slash
operator on F by
for g =nk(F |k g( Z)) = µn (g) 2 j(g, Z)−k f (gZ).
We say that F is a genus n Siegel modular form of weight k if F is holomorphicand (F |k γ)(Z) = F (Z) for all γ ∈ Γn . Note, if n = 1 then we must also
require F to be holomorphic at the cusps. Denote the space of all such forms
by Mk (Γn ). It is a basic result from the theore of Siegel modular forms thatevery F ∈ Mk (Γn ) has a Fourier expansion of the form
XF (Z) =aF (T )e(Tr(T Z)),
Λn2

where Λn is the set of n by n half-integral positive semi-definite symmetricmatrices and e(W ) := e2πiW . If aF (T ) = 0 unless T is strictly positive definite
we say that F is a cusp form, and we denote the space of all such cusp forms
by Sk (Γn ).

3Hecke Operators and Satake Parameters

Let g ∈ GSp2n (Q), then we write T (g) to denote the double coset Γn gΓn . Then,
we call T (g) a Hecke operator and we have an action of T (g) on F ∈ Mk (Γn )given byXT (g)F =F |k gi ,where we are summing over a set of coset representatives for Γn \Γn gΓn . Let p
be a prime and define
T n (p) = T (diag(1n , p1n )),and for i = 1, . . . , n defineTin (p2 ) = T (diag(1n−i , p1i , p2 1n−i , p1i )).
Note, we will drop the n from the superscript when the genus is clear fromcontext. It is well known that these operators generate the local Hecke algebra
at p. We call F ∈ Mk (Γn ) an eigenform if F is a simultaneous eigenvector foreach of the Hecke operators.
Let F ∈ Sk (Γn ) be an eigenform. Then, there is a set of complex numbers
αi (p; F ) for 0 ≤ i ≤ n, called the Satake p-parameters of F , which completely
determine the eigenvalues of F with respect to T (p) and Ti (p2 ) for 1 ≤ i ≤ n,which we denote by λ(p; F ) and λi (p2 ; F ), respectively.
In fact, from [7], we have the following relationships between the eigenvaluesof F , and the Satake parameters of F ,λ(p; F ) = α0 (p; F )(1 + σ1 + · · · + σn ),λr (p2 ; F ) = α0 (p; F )2nX

mi−j (r)p(i−j+12(3))σ σ ,i j(4)i−j≥ri,j≥0where mh (r) := #{M ∈ Mh (Fp ) : t A = A, corank(A) = r}, σi is the degree ielementary symmetric function in α1 (p; F ), . . . , αn (p; F ), and 1 ≤ i ≤ n. Note,
we have normalized our Satake parameters to satisfy
α0 (p; F )2 α1 (p; F ) . . . αn (p; F ) = pnk−n(n+1)
2.

Suppose f ∈ Sk (Γ1 ) is a normalized eigenform, i.e., af (1) = 1. Then we will
make use of the following relationship as well,1−kα(p; f ) + α(p; f )−1 = p 2 λ(p; f ),3(5)where we simply use α(p; f ) to denote α0 (p; f ).
Using these Satake parameters we can associate two L-functions to an eigenform F ∈ Sk (Γn ). First, in order to define the standard L-function of F we need
the following local L-factor
Lp (s, F ; st) = (1 − p−s )nY(1 − αi (p; F )p−s )(1 − αi (p; F )−1 p−s ).

i=1Then, the standard L-function associated to F is given by the following Euler
productYL(s, F ; st) =Lp (s, F ; st)−1 .
pSecond, in order to define the spinor L-function associated to F we need thefollowing local L-factor
Lp (s, F ; spin) = (1−α0 (p; F )p−s )nYY(1−α0 (p; F )αi1 (p; F ) . . . αij (p; F )p−s ).

i=1 1≤i1 ≤···≤ij ≤nThen, the spinor L-function associated to F is given byY
L(s, F ; spin) =Lp (s, F ; spin)−1 .
pNote, when F is of genus 1, this is simply the usual Dirichlet series associatedto F .

4The Ikeda Lift

In this section we introduce the Ikeda lift as presented in [3].
Let T > 0 be in Λn . Set DT to be√the determinant of 2T , ∆T to be theabsolute value of the discriminantof Q( DT ), χT to be the primitive Dirichlet√character associated to Q( DT )/Q, and fT to be the rational number satisfying
DT = ∆T f2T .
Let Sn (R) denote the set of symmetric n × n matrices over a ring R. For arational prime p, let ψp : Qp → Cx be the unique additive character given byψp (x) = exp(− {x}p ),where {x}p ∈ Z[ p1 ] is the p-adic fractional part of x. The Siegel series for T is
defined to beX
bp (T, s) :=ψp (Tr(T S))|ν(S)|p p−s , for Re(s) >> 0,S∈Sn (Qp )/Sn (Zp )where ν(S) := det(S1 ) · Zp , and S1 is from the factorization S = S1−1 S2 for asymmetric coprime pair of matrices S1 , S2 . We have a factorization of the Siegel
series given by
bp (T, s) = γp (T, p−s )Fp (T, p−s ),4whereγp (T, X) =n/2Y1−X(1 − p2i X 2 ),n1 − p 2 χT (p)X i=1and Fp (T, X) ∈ Z[X] has constant term 1 and deg(Fp (T, X)) = 2 ordp (fT ).
Using this polynomial Fp (T, X) we define
1nF̃p (T, X) := X − ordp (fT ) Fp (T, p− 2 − 2 X).
We will make use of this F̃p (T, X) later.
Let f (τ ) ∈ S2k−n (Γ1 )) be a normalized eigenform and let
X+h(τ ) =c(N )q N ∈ Sk−(Γ0 (4))
n+12m>0(−1)k m≡0,1(4)2

be an eigenform corresponding to f via. the Shimura correspondence.
For each T > 0 in Λn , defineYk− n+1a(T ) = c(|∆T |)fT 2F̃p (T, α(p; f )),(6)pand form the following series
In (f )(z) =Xa(T )e(Tr(T z)).

TWe have the following theorem.
Theorem 1. ([3, Thm. 3.2, 3.3]) The series In (f )(Z), referred to as the Ikeda
lift of f , is an eigenform in Sk (Γn ) whose standard L-function factors as
L(s, F ; st) = ζ(s − n)nYL(s + k − n − i, f ).

i=1Note, if n = 2 then this L-function factorization agrees with that of theSaito-Kurokawa lift. From the factorization of the standard L-function givenin the previous theorem, we are able to obtain an expression for αi (p; In (f )) interms of α(p; f ) for 1 ≤ i ≤ n. However, in order to determine the eigenvalue ofIn (f ) we must also have information about α0 (p; In (f )). This can be achieved
by determining the factorization of the spinor L-function of In (f ). This factorization can be found in [6]. In summary, we obtain the following relations
between Satake parameters
nkα0 (p; In (f )) = p 2 −αj (p; In (f )) = pj−(n+1)
2n(n+1)4nα(p; f )− 2 ,(7)α(p; f ), for j = 1, . . . , n.

(8)Note, we our Satake parameters are normalized differently than the Satakeparameters in [6].
5

5

Eigenvalues of the Ikeda lift

In this section we will derive Equations 1 and 2.
We begin by determining a nice expression for the symmetric polynomials
in Equations 3 and 4. Using Equation 8 we obtain the following expression,

σj = p−j(n+1)2n(n+1)2jα(p; f )Xq≤n (i, j)pi ,(9)i=0where we are using q≤n (i, j) to denote the number of partitions of i into j distinct
parts where each part is no greater than n and our symmetric polynomial is in
α1 (p; In (f )), . . . , αn (p; In (f )).
Combining Equation (7.3) and Theorem 8 of [1] we have the following generating
series for this partition function∞∞ X
Xq≤n (i, j)z1j z2i=i=0 j=0nY(1 + z1 z2` )`=1nY(z2 )=∞Xj=0 n−jY(z2 )

jY(z2 ) j(j+1) jz 2 z ,1
 2

QnQn
where(x) := m=1 (1 − xm ). If we fix a j ≤ n on the left hand side andevaluate at z2 = p we obtain n(n+1)2Xq≤n (i, j)pi  z1j ,i=0where we need only take the summation to n(n + 1)/2 since we have restricted
to j ≤ n. Matching coefficients we haven(n+1)
2Xi=0pj(j+1)
2q≤n (i, j)pi = n−j
YnY(p)jY(p) (p).

The following lemma follows immediately from this equality.
Lemma 2. For any j ≤ n,p−j(n+1)
2n(n+1)
2

Xiq≤n (i, j)p = p−(n−j)(n+1)
2i=0n(n+1)
2

X
i=06q≤n (i, n − j)pi .

(10)

In summary we havepj(j−n)2σj = n−jYnY(p)jY(p) (p)α(p; f )j .

We are now prepared to compute λ(p; In (f )). Combining Equations 3, 9, and10 and pairing our terms using Lemma 2 we obtain the following expression forλ(p; In (f )),nnn −j)( n +j) YY−(2nn22 22(p)(p)  p− 8n(n+1) X pnk−j−j24pα(p)+α(p)+.
nnn+j−j j=1
222YYY
2(p)(p)(p)Applying a theorem of Waterson from [8] along with Equation 5 we havej−jjα(p) + α(p)=b2cX(−1)rr=0
j−2r
jj − r  n−2k+1.
p 2 λ(p; f )j−rr

Inserting this into the previous equation and rearranging we obtain the following
expression for λIn (f ) (p)nnY2 Yjnn
 2 b 2 c j − r j(−1)r pc (p)(p) p− 8n(n+1) X Xnk
j−2r
p2− 4 λ(p;f)+,
nnn+j−jr j=1 r=0222YY
Y(p)2(j − r) (p)(p)−( n −j)( n +j)+j(n−2k+1)+2r(2k−n−1)22where c =.
2In order to compute a similar formula for λr (p2 ; In (f )) for r < n, we firstneed the following expression for mh (r) from [2]hhp( 2 )b2chYY(p)(1 − p1−2` )mh (r) =`=0h−rY.

r
Y(p) (p)Applying this formula along with similar techniques used above we have thefollowing expression for λr (p2 ; In (f )),i−jpnk−n(n+1)
2nXi−j≥ri,j≥0pj(j−n)+i(i−n)+2(i−j)22i−jn2 cYY bY2(p)(p)(1 − p1−2` )`=0i−j−r
Yn−jjrn−iiYYYYY(p) (p)(p) (p)(p) (p)7α(p; f )i+j−n .

By making the substitutions a = i + j and b = i − j and simplifying we obtainbn(n+1)
nk−2=pnXaX

a=r
b=rb≡a (mod 2)

pb2cnbYY
Y(1 − p1−2` )(p)2 (p)a2 +5b2 −n24`=0b−rYn− a−b2a−b2a+ba+br2
YY n−Y2YY(p) (p)(p)(p)(p)(p)α(p; f )a−nbn(n+1)
nk−2+p2n−rX

p2n−aX5b2 −n24b2cnbYYY(p)2 (p)(1 − p1−2` )`=0a=n+1b=rb≡a (mod 2)

b−r
Ya+ba−b2n− a−b2a+br2
YY n−Y2YY(p)
(p)
(p)
(p)
(p) (p)α(p; f )a−n

Just as above, we can shift each summation, pair appropriately, and then apply
the theorem of Waterson to obtain our final expression for λr (p2 ; In (f ))n−rXn(n+1)nk−2=pn−aXab2cXnY
 d(p)2a(−1)c a−cc p mb (r)n+a+b2n−a−b2

5b2 −n24nY(p)2n+a−b2n−a+b2

λ(p; f )a−2cYYYY
a=1 b=r c=0b≡a (mod 2)(a − c)(p)(p)(p)(p)+ pnk−nXn(n+1)
2mb (r)pn+b2n−bn−bn+b.

222YYYY
b=rb≡0 (mod 2)
(p)
(p)
(p)
(p)

222

where d = a +5b −n +2(n−2k+1)(a−2r).
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St. Paul, 52:1–46, 2003.
8

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