Integers represented by positive-definite quadratic forms and Petersson inner products

Abstract

Let Q be a positive-definite quaternary quadratic form with integer coefficients. We study the problem of giving bounds on the largest positive integer n that is locally represented by Q but not represented. Assuming that n is relatively prime to D(Q), the determinant of the Gram matrix of Q, we show that n is represented provided thatHere N (Q) is the level of Q. We give three other bounds that hold under successively weaker local conditions on n.These results are proven by bounding the Petersson norm of the cuspidal part of the theta series, which is accomplished using an explicit formula for the Weil representation due to Scheithauer.

Arithmetic and analytical aspects of Siegel modular forms

“…det Q. Then, qpxq " n is soluble provided that q satisfies (SLC) for n and n " ´N 2 pn, N q `min ´pn, N qpdet Qq 2This is a significant improvement of previous results as for example[45, Theorem 1]. For quadratic forms in m variables with m ě 5 , we get a similar result: Corollary 26.…”

“…If this strong local solubility condition (SLC) holds, they obtain a lower bound of size pdet Qq 2 hpQq 8` where hpQq is the size of the largest coefficient (in absolute values) which satisfies pdet Qq 1 4 ď hpQq ď det Q. A much broader array of results for m " 4 is obtained via the theory of modular forms and theta series by Rouse [45]. Let N denote the level of Q.…”

“…The key difference is our estimation of the Petersson inner product of the cuspidal part of the theta series. By rather elementary methods that are based on [4], we obtain a considerable sharper bound than [45,Theorem 3].…”

“…To evaluate the p-adic densities, we follow the work of [24] and [45]. If n is (primitively) locally represented by Q for all primes this yields…”

“…For p " 2, we consider q 0 p x 0 q `2q 1 p x 1 q `22 q 2 p x 2 q `23 q 3 p x 3 q where q 0 , q 1 , q 2 are of discriminant coprime to 2 and each q i consists of forms as in (62). For m " 4 Rouse [45] shows that the existence of one good solution implies that there at least 2 7 good solutions modulo 8 by enumerating all possibilities of the 2-adic Jordan decomposition. As a direct consequence, we get β 2 pQ, nq ě 1 16 if dim q 0 `dim q 1 ď 4 and there is a good solution.…”