Theta series and representation of quadratic forms

γpmq PGSp4pQq

are integers. Assume there is t ą 1 such that t | l and t - m. For λpMq “ pc₃, c₄, d₃, d₄q we have

λpmM κplqκpmq^´1q ” λ ˆ

M

ˆ ´mI lI

˙˙

” pld₃, ld₄,´mc₃,´mc₄q pmodt²q.

Consequently, we that d₃, d₄ ”0pmodtq as l is squarefree. As t | N, every entry in the lower row of M is divisible by t. However, this contradicts the assumption that M PΓ^p2q0 pNq.

For t-l and t|m we obtain

λpmM κplqκpmq^´1q ” pmd₃, md₄,´lc₃,´lc₄q pmodt²q and the very same contradiction.

Consider F P M_k^p2qpNq for squarefree N and set f_l “ΦlpFqand c_i “Φ²γipFq for a set of representatives γ_i of 0-cusps. Then

FpZq ´ÿ

l|N

Epf_l, Zq ´ÿ

i

c_iEγipZq PS_k^p2qpNq.

Hence, we get the following decomposition:

Corollary 29. For squarefree N we have

M_k^p2qpNq “ S_k^p2qpNq ‘ xE_lpf, Zq |f PS_k^p1qpNq, l |Ny

‘ xEγpZq |γ PΓ0pNqzSp4pZq{Γ8y.

For arbitrary level, a set of 1-cusp representatives and the corresponding decom-position of the Klingen-Eisenstein space can be found in [54, Corollary 3.3 and 3.4].

4.1.2 Theta series and representation of quadratic forms

As the theta series satisfiesθpQ, Zq “θpU^TQU, ZqforU PGLmpZqit only depends on the class of Q. As in the case of elliptic modular forms we approximate the

theta series by a weighted average over the classes in the genus. Recall that opQq “ #tU PGL2mpZq|Q“U^TQUu is finite. We set

θpgenQ, Zq “ ˆ

ÿ

RPgenpQq

1 opRq

˙´1

ÿ

RPgenpQq

θ_RpZq opRq “

ÿ

TPS

rpgenQ, TqeptrT Zq.

The Fourier coefficients were explicitly computed by Siegel:

rpgenQ, Tq “ π^2k´12pdetTq^k´32 ΓpkqΓpk´1qdetQ

ź

p

βppQ, Tq (76) where the p-adic densities are given by

β_ppQ, Tq “ lim

tÑ8p^tp3´4kq#tGPM_m,npZ{p^tZq |G^TQG”T pmod p^tqu. (77) Our first aim is to show that θpQ, Zq ´θpQ¹, Zq vanishes at all 0-cusps if Q and Q¹ are in the same genus. By the results above, this enables us to express the difference θpQ, Zq ´θpQ¹, Zq as a linear combination of Klingen-Eisenstein series and cusp forms.

To this end, we introduce the generalized theta series for A, B PM_2,2kpCq θ_pA,BqpQ, Zq “ ÿ

XPM_2k,2pZq

e

´1

2trppX`Aq<sup>T</sup>QpX`Aq `2B^TXq

¯

that satisfies the transformation formula

θ_pA,BqpQ^´1, Z^´1q “ ep´trpA^TBqqdetQpdetZ{iq^kθ_pB,´AqpQ, Zq, (78) cf. [19, Satz 0.13]. Note that θ_pA,BqpQ, Zq “ θ_p´A,´BqpQ, Zq.

Before we compute the value at each 0-cusp, we need to determine a set of suitable representatives:

Lemma 30. In every coset class of Γ0pNqzSp4pZq{Γ8 there is M “

ˆA B C D

˙

with detC ‰0.

Proof. Consider M PSp4pZqwith detC “0. By left and right multiplication with Γ8, we can assume that

C “

ˆc₁ 0 0 0

˙ . Ifc₁ “0 then detA‰0 and left multiplication by

K “

ˆ˚ ˚ C˜ D˜

˙

PΓ0pNq

with det ˜C ‰0 gives a matrix with the desired properties. Forc₁ ‰0 we still have a₂ ‰0 or a₄ ‰0 byA^TD´B^TC “I. In the former case we choose ˜C “

ˆ0 0

˜ c₃ 0

˙

with ˜c3 ‰0 and ˜D“

ˆd˜₁ ˚ d˜₃ ˚

˙

with ˜d1 ‰0. Then,

detpCA˜ `DCq “˜ det

ˆ d˜₁c₁ 0

˜

c₃a₁`d˜₃c₁ ˜c₃a₂

˙

“d˜₁c₁˜c₃a₂ ‰0. The latter case works analogously, we choose ˜C “

ˆ0 0 0 ˜c₄

˙

and ˜Das before. This yields

det ˜CA`DC˜ “d˜₁c₁˜c₄a₄ ‰0.

For a representative with detC‰0, we decompose M into ˆA B

C D

˙

“

ˆI AC^´1 I

˙ ˆ ´C^´T C

˙ ˆI C^´1D I

˙ .

This decomposition allows us to get a similar transformation formula for the action of a 0-cusp representative on the theta series as in the case n“1:

Lemma 31. Consider the decomposition of M with detC‰0 as above. Then θpQ, Zq|rMs “

ÿ

XPM2k,2pZq

αpX, Q, Mqe ˆ1

2X^TQ^´1XZ

˙

with

αpX, Q, Mq “ ω

pdetCq^kdetQe

´1 2tr`

X^TQ^´1XC^´1D˘¯

ˆ ÿ

X₁^TPM2,2kpZq{CM2,2kpZq

e

´1 2tr`

2C^´1X₁^TX`AC^´1X₁^TQX₁˘¯ , where ω is either 1 or -1.

Proof.

θpQ, Zq|

„ˆI AC^´1 I

˙ ˆ ´C^´T C

˙

“ pdetCZq^´k ÿ

XPM2k,2pZq

e

´1 2tr`

X^TQX`

C^´Tp´Z^´1qC^´1`AC^´1˘˘¯ .

We putX^T “X₁^T`CX₂^T whereX₁^T runs over matrices inX₁^T P M_2,2kpZq{CM_2,2kpZq and X₂ over matrices in M_2k,2pZq. Since C^TAC^´1 “A^T is integral, the previous display equals

pdetCZq^´k ÿ

X₁^TPM2,2kpZq{CM2,2kpZq

e

´1 2tr`

X₁^TQX₁AC^´1˘¯

ˆ

ÿ

X2PM2k,2pZq

e

´1 2tr`

ppX₁C^´Tq^T `X<sub>2</sub><sup>T</sup>qQpX<sub>1</sub>C<sup>´T</sup> `X₂qp´Zq^´1˘¯

“ pdetCZq^´k ÿ

X₁^TPM2,2kpZq{CM2,2kpZq

e

´1 2tr`

X₁^TQX₁AC^´1˘¯

θ_pX1C^´T_,0qpQ, Z^´1q.

By applying (78) we have that

θ_pX1C^´T_,0qpQ, Z^´1q “ p´1q^kpdetZq^kθ_p0,X1C^´T_qpQ^´1, Zq

detQ ,

where

θ_p0,X1C^´T_qpQ^´1, Zq “ ÿ

XPM2k,2pZq

e

´1 2

´trpX^TQ^´1XZ`2C^´1X₁^TXq

¯¯

.

The claim follows now by taking the third matrix in the decomposition of M into account.

Corollary 32. Let Q, Q¹ be in the same genus. Then θpQ, Zq ´θpQ¹, Zq vanishes at all 0-cusps.

Proof. The value of θpQ, Zq at the cusp i8I is obviously 1. For all other cusps, we choose a representative with detC ‰ 0 and apply Lemma 31. Consequently, the value of θpQ, Zq at the cusp represented byM^´1 is given by

ω pdetCq^kdetQ

ÿ

X₁^TPM2,2kpZq{CM2,2kpZq

e ˆ1

2tr`

X₁^TQX₁AC^´1˘

˙ .

This term only depends on the discriminant and the congruence class of the un-derlying quadratic form, thus, it is genus invariant.

By Corollary 29 and Corollary 32 we obtain the following decomposition for squarefree level:

θpQ, Zq “ θpgenQ, Zq `ÿ

l|N

E_lpf_l, Zq `GpZq,

where GpZq is a cusp form of degree two and f_lpzq are cusp forms of degree one.

To determine f_l we apply the Φl operator which yields f_lpzq “_tÑ8lim

ˆ

pθpQ,˚q|rκplq^´1sq ˆz

it

˙

´ pθpgenQ,˚q|rκplq^´1sq ˆz

it

˙ ˙ . On the level of Fourier coefficients, this corresponds to

rpQ, Tq “ rpgenQ, Tq ` ÿ

l|N

A_lpTq `SpTq, (79) where A_lpTq, gpTq are the Fourier coefficients of E_lpf_l, Zq and GpZq. For T ą0, the followings bounds for A_lpTq and gpTq are known by Kitaoka, cf. [30] and [31],

A_lpTq !N,kpdetTq^k´

3

2 minpTq

1´k

2 gpTq !N,k pdetTq

k 2´¹₄.

Furthermore, Kitaoka [29], [32] has shown that the product over the p-adic densi-ties for 2kě7 is bounded from below by a constant depending only on Qas long as T is locally represented by Q.⁹ By means of (76) this gives a lower bound for rpgenQ, Tq and hence, an asymptotic formula forrpQ, Tq with respect toT.

The goal of this thesis is to make these estimates uniform in the level N. The treatment of the main term is rather straightforward, since we can apply a formula from Yang [63] to evaluate the local densities at odd primes. The evaluation of the error term, however, is very elaborate. To estimate the Fourier coefficients of the Klingen-Eisenstein series we modify Kitaoka’s work [34]. The main challenge lies in bounding the Fourier coefficients of the cusps form GpZq. To do so, one expresses the cusp form as a linear combination of Poincaré series and then applies the Kitaoka-Petersson formula to evaluate the Fourier coefficients of the latter. This way, Chida, Katsurada and Matsumoto [15] obtained the following bound:

gpTq ! a

xG, Gy`

pdetTq^{k2´14`</sup>N<sup>´12`}`1˘

. (80)

To compute the inner product we make use of the fundamental domain of the full modular group Sp4pZq. This implies that we need to bound the Fourier coefficients of G|rMswhere M runs over a set of representatives in Γ0p2q

pNqzSp4pZq.

For the treatment of the error term, we assume for simplicity that the level N is prime. However, an extension of the proof to squarefree level is sketched out in Section 4.4.

Im Dokument Arithmetic and analytical aspects of Siegel modular forms (Seite 64-68)