The Spinor L -Function and Böcherer’s conjecture for oldforms

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φ_p if p-rst, φ˜_p,1 if p|r, φ˜p,2 if p|s, φ˜_p,3 if p|t.

Then, ˜δ_r,s,t is a mapping from S_k^p2qpmq^new,SK to S_k^p2qprstmq^SK and we have an or-thogonal direct decomposition

S_k^p2qpNq^SK“ à

r,s,t,m rstm“N

δ˜_r,s,t

´

S_k^p2qpmq^new,SK

¯

. (21)

Form ‰N, the right hand side is S_k^p2qpNq^old,SK.

In the following section, we compute Böcherer’s relation for members of these bases, (19) and (21).

2.2 The Spinor L-Function and Böcherer’s conjecture for oldforms

As in the previous section, we assume thatF PS_k^p2qpNq^new generates an irreducible representation and is an eigenform ofT₂ppq if it is not a Saito-Kurokawa lift. The former implies that F is an eigenform of the Hecke algebra for all p - N. Let

α_p, β_p, γ_p denote the Satake parameter of F. Then, we define the local spin L -factors at p-N by

Lpps, Fq “ ˆ

1´ α_p p^s

˙´1ˆ

1´ α_pβ_p p^s

˙´1ˆ

1´α_pγ_p p^s

˙´1ˆ

1´α_pβ_pγ_p p^s

˙´1

for <s sufficiently large. Furthermore, we set L₈ “ Γ_Cps`1{2qΓ<sub>C</sub>ps`k´3{2q, where ΓCpsq “2p2πq^´sΓpsq. For F PS_k^p2qpNq^new,T, we only considerF that satisfy w_F,N ‰ 0. (In this case the forms F have a corresponding Bessel model.) This implies that for p|N, the type of π_F,N is either IIIa or VIb. To distinguish these two cases, we consider the eigenvalues µ_p under the T₂ppq operator. If µ_p “ ˘p, then F is of type VIb. As in [51], we set forp|N

L_pps, Fq^´1 “ p1´µ_pp^´3{2´sqp1´µ^´1_p p^1{2´sq for µ_p ‰ ˘p pπ_F,p of type IIIaq, L_pps, Fq^´1 “ p1´µ_pp^´3{2´sq² otherwise pπ_F,p of type VIbq. Then, the completed L-function Λps, Fq “ ś

pY8L_pps, Fq has meromorphic con-tinuation to the whole complex plane and satisfies the functional equation

Λps, Fq “N^1´2sΛp1´s, Fq. (22) However, this result is conditional on a nice L-function theory for GSp4 as in [49, 3.14]. The associated representation to F PS_k^p2qpNq^new,T is cuspidal and non-CAP⁴ and thus not in one of the classes (Q),(P) and (B) in the notion of [52]. For representations belonging to the other two possible classes, (G) and (Y), a nice L-function theory is known, cf. [52, Lemma 1.2 and 1.3].

In a standard way [27, Theorem 5.3], we get the following approximate function Lp1{2, F ˆχ_qq “ 2ÿ

n

A_Fpnqχ_qpnq n^1{2 W

ˆ n N|q|²

˙

, (23)

where Lps, Fq “ ś

pL_pps, Fq “ř

nA_Fpnqn^´s and Wpxq “ 1

2πi ż

p2q

L₈ps`1{2q

L₈p1{2q p1´s²qx^´sds s .

By shifting the contour, we see immediately that the integral satisfies for allA ą0 the bound

Wpxq !_Ap1`xq^´A. (24)

4A representation ofGpAqis said to be CAP if it is nearly equivalent to a global induced representation of a proper parabolic subgroup ofGpAqand otherwise non-CAP. In our setting,Fis a Saito-Kurokawa lift if and only if the associated representation is CAP.

A key role plays the following formula from Andrianov L^Nps, F ˆχ_qqa_FpIq “L^Nps`1{2, χ<sub>q</sub>qL<sup>N</sup>ps`1{2, χ_´4qq

ÿ

pm,Nq“1

a_FpmIqχ_qpmq m^s ,

(25) cf. [1, Theorem 4.3.16] with l “a“1, η “χ“trivial. We denote by

rpnq “r_qpnq “ χ_qpnq n^1{2

ÿ

d|n

χ_´4pdq (26)

the Dirichlet coefficients of Lps`1{2, χ<sub>q</sub>qLps`1{2, χ_´4qq. For q“1, the latter is the Dedekind zeta functionζ_Qpiqps`1{2q.

For f P S_2k´2pNq with L-function Lps, fq, the partial L-function of the corre-sponding Saito-Kurokawa lift F is given by, cf. [41],

L^Nps, Fq “ ζ^Nps´1{2qζ^Nps`1{2qL^Nps, fq. (27) At primes p|N, we define the local spinL-factors for F P S_k^p2qpNq^new,SK as in [11].

LetF PS_k^p2qpNqbe an eigenform of the Hecke algebra at allp-N with eigenval-ues λ_pp^k´3{2. Then, the eigenvalues satisfy λ_p — p¹² if F is a Saito-Kurokawa lift, and due to Weissauer λ_p ! p otherwise. Furthermore, it holds by [61, Theorem 1.1] that a_FpmpIq “ λ_papmIq `|apmIq|Opp^´1{2q, where the Fourier coefficients a_FpmIq are normalized as in (4). It follows forpm, Nq “ 1 that

a_FpmIq !ma_FpIq for F PS_k^p2qpNq^T,

a_FpmIq ! m^1{2a_FpIq for F PS_k^p2qpNq^SK. (28) For the proof of Theorem 1, we need to bound the coefficientsA_Fppqof the spinor L-function at ramified primes p:⁵

Lemma 7. Let F PS_k^p2qpNq^new,T with a_FpIq ‰0. Then, it holds for p|N that

|A_Fppq|!p^´1{11.

Proof. Since aFpIq ‰ 0, we know that πF,p is of type IIIa or VIb. For latter we directly see |A_Fppq| ! p^´12. For type IIIa, π_F,p “ χ¸σStGSpp2q, where χ, σ are unramified characters of Q^ˆp with χσ² “1 and StGSpp2q is the Steinberg represen-tation. Let ω P Zp be a generator of the maximal ideal pZp. By [49, Table 2], it holds that

L_pps, Fq^´1 “ p1´σpωqp^´1{2´sqp1´σχpωqp^´1{2´sq. (29)

5The argument presented in this lemma was communicated to the author by Ralf Schmidt.

The key to bound σpωq, σχpωq is to transfer π_F to a cuspidal automorphic rep-resentation of GLp4,Aq. For representations in the general class (G) in the notion of [52] such a lift is possible. Since F is a non-CAP form, π_F cannot be in one of the classes (Q), (P) and (B). Furthermore, [46, Table 16] states all possible representation types for (Y) and IIIa is not one of them, so π_F is in (G).

For π_F,p “χ¸σStGSpp2q, the attached Langlands L-parameter is pρ, Nq with ρ:W_Qp ÑGSp4pCq

ω ÞÑ

¨

˝

ν^1{2χσpωq

ν^´1{2χσpωq

ν^1{2σpωq

ν^´1{2σpωq

˛

‚ and N “ ˆ_{0 1}

00 ´1 0

˙ , cf. [49, p. 266], where W_Qp is the Weil group of Q^p. We map this parameter into GL4pCq and apply the local Langlands correspondence for GL4. In this way, we obtain the representation χσStGLp2q ˆσStGLp2q of GL4. Since χ, σare unramified characters, this corresponds in the notation of [8] to the representation induced from StGLp2qrepχσqs bStGLp2qrepσqs, where e denotes the exponent of a character defined by|σ| “ |.|^epσq. By applying [8, Theorem 1], we obtainepσq, epσχq ď9{22.

In other words, |σpωq|,|σχpωq|ďp^9{22.

The proof of Böcherer’s conjecture in [17] and [20] is obtained via local com-putations. In the introduction, we already stated relations for newforms; now we present similar results for members of the oldspace basis constructed in the previ-ous section. Recall that if F P S_k^p2qpNq is an oldform, there is at least one p | N for which π_F,p is of type I or IIb . For these types, we define the local standard L-factor by

Lps, π_p,Stdq^´1 “ p1´p^´sqp1´α_pp^´sqp1´α^´1_p p^´sqp1´β_pp^´sqp1´β_p^´1p^´sq, where we use the following notation for the Satake parameter α_p, β_p, γ_p: If π_p “ χ₁ˆχ₂¸σis a type I representation, we setα_p “χ₁pωq, β_p “χ₂pωqandγ_p “σpωq, while for πp “χ1GLp2q¸σ, we set αp “ p^´1{2χpωq, βp “ p^1{2χpωq and γp “σpωq.

In both cases, it holds that αpβpγ_p² “1. For elements in the basis (19), a proof of Böcherer’s conjecture has been obtained in [17, Theorem 3.9]:

Lemma 8. Let F PS_k^p2qpNq^T. Assume thatF “δ_a,b,c,dpGq, where abcde“N and G is a newform in S_k^p2qpeq^T. Let π “ b¹_vπ_v denote the representation attached to G (or equivalently to F). Then

w_F,N “ 2^sπ⁵p1´N^´4qΓp2k´4qLp1{2, FqLp1{2, F ˆχ_´4q N³Γp2k´1qLp1, π_F,Adq

ź

p|N

J_p

p1`p<sup>´1</sup>qp1`p^´2q,

where s “6 if F is a weak Yoshida lift and 7 otherwise and

J_p “

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Lp1, π_p,Stdqp1´p^´4q if p|bc pcannot occur if F newformq, Lp1, π_p,Stdqp1´p^´4qp^´1 if p|ad pcannot occur if F newformq, p1`p<sup>´2</sup>qp1`p^´1q if p|e and π_p is of type IIIa,

2p1`p<sup>´2</sup>qp1`p^´1q if p|e and πp is of type VIb,

0 otherwise.

Analogously, we compute a relation for our basis (21) of Saito-Kurokawa lifts : Lemma 9. Letg PS_2k´2pmqdenote a newform of squarefree levelmthat generates an irreducible representation π₀. Let π :“ SKpπ₀q denote the lift of π₀ and G the corresponding Siegel newform. Let r, s, t denote squarefree, mutually coprime numbers with prst, mq “ 1 and consider F “ δ˜_r,s,tpGq P S_k^p2qpNq^SK where N “ rstm. Then:

w_F,N “ 3¨2⁶π⁷Γp2k´4q N³Γp2k´1q

Lp1{2, f ˆχ_´4q Lp3{2, fqLp1, π₀,Adq

ź

p|N

J_p

p1`p<sup>´1</sup>qp1`p^´2q, (30) where

J_p “

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Lp1, π_p,Stdqp1´p^´4qp^´1 if p|r pcannot occur if F newformq, Lp1, π_p,Stdqp1´p^´4qpp^´1`1q if p|s pcannot occur if F newformq, Lp1, π<sub>p</sub>,Stdqp1´p<sup>´4</sup>q if p|t pcannot occur if F newformq, 2p1`p^´2qp1`p^´1q if p|m.

Proof. By [17, (105) and (106)] we have that

|a_FpIq|rSp4pZq: Γ0pNqs

xF, Fy “3¨2^1`4kLp1{2, π₀ˆχ_´4qLp1, χ_´4q² Lp3{2, π₀qLp1, π₀,Adq

ÿ

p|N

`2^´1J^˚pφ_pq˘ , where φ_p P π_p is aP₁ppq fixed vector. If φ_p is not spherical, then it is of type VIb and we apply the results from [17]. Ifφ_p is spherical, then by construction of ˜δ_r,s,t it is one the three vectors from (20). We have

J^˚pφ_pqq “MpπqJ₀pφ_pq, where

J₀pφ_pq “1´ pq`1qq<sup>´3</sup>λpφq `q^´2µpφq, Mpπq “ Lp1, π_F,Adqp1`p^´1q

p1´p^´2qp1´p^´4qLp1{2, πqLp1{2, πˆχ_´4q,

cf. [17, §2.5], and λpφq, µpφqare given with respect to (17) by λpφq “ xd₁e₁e₀e₁φ, φy

xφ, φy , µpφq “ xd₁e₀e₁e₀φ, φy xφ, φy .

The action of the local operators e₀, e₁ are given as a 8ˆ8 matrix with respect to (16) in [49, Lemma 2.1.1]. Here, d₁ “ pq`1q<sup>´1</sup>pe`e₁q is the so-called Siege-lization and maps onto the space of P₁-fixed vectors. Recall thatπ_p^P1ppq is spanned by ˜φ_p,1,φ˜_p,2,φ˜_p,3 and ˜φ_p,1 `φ˜<sub>p,2</sub> `φ˜_p,3 is the GpZ^pq fixed vector. If we write d₁e₁e₀e₁φ˜_p,i “ ř

j“1c_ijφ˜_p,j and d₁e₀e₁e₀φ˜_p,i “ ř

j“1˜c_ijφ˜_p,j, then λpφ˜_p,iq “ c_ii and µpφ˜_p,iq “c˜_ii. A straightforward but lengthy calculation shows

λpφ˜_p,1q “ pp´1qp², λpφ˜_p,2q “ p´1

p`1p², λpφ˜_p,3q “0, µpφ˜_p,1q “ pp´1qp², µpφ˜_p,2q “p², µpφ˜_p,3q “0. As a consequence, we get

J₀pφ˜_p,1q “p^´1, J₀pφ˜₂q “1`p^´1, J₀pφ˜₃q “ 1.

It remains to compute Mpπq. The local L-factors are computed in [2, §3.1.2] and [49, Table 2] and it holds forπ of type IIb that

Lp1, π,Adq

Lp1{2, πqLp1{2, πˆχ´4q “ p1´p^´1qLp1, π,Stdq.

Im Dokument Arithmetic and analytical aspects of Siegel modular forms (Seite 24-29)