Estimates of Petersson's inner squares of cusp forms and arithmetic applications

Abstract

We give estimates for Petersson's inner squares of cusp forms in the case of increasing levels and weights. The considered cusp forms are normalized eigenfunctions of all Hecke operators or cusp forms occurring in the theta series associated with positive definite quadratic forms. We give arithmetic applications of the obtained estimates in the problems of the representability of numbers by quaternary quadratic forms of a special kind, in the investigation of the eigenvalues of Hecke operators with small indices, in the study of the uniform distribution of integral points on ellipsoids, as well as in other problems.

Applications of the petersson formula for a bilinear form in fourier coefficients of cusp forms

“…The proofs of Theorems 6.1)-6.3), 7, and 8 [5] depend on a statement which is equivalent to Theorem 2 of the present paper. In [5] its proof was based on an analogue of the Siegel theorem. Here we give a new proof, which does not depend on the existence of a Siegel zero.…”

“…where the <<-constant depends only on e. The cusp form generated by the theta-series of the quaternary quadratic form x~ + x~ + px 2 + px~ gives a counterexample to this conjecture (see [5] for details). Moreover, this example shows that Theorem 2 cannot be sharpened with respect to level, in other words, any statement of the form…”

“…In Theorem 2 of [5] we announced an analogue of the Siegel theorem for the Rankin L-function LF, F($), F(z) being a normalized newform from S2k(r0(N)). We also obtained some corollaries of this statement.…”

“…So some remarks concerning results in [5] which depend on Theorem 2 [5] should be made. The proofs of Theorems 6.1)-6.3), 7, and 8 [5] depend on a statement which is equivalent to Theorem 2 of the present paper. In [5] its proof was based on an analogue of the Siegel theorem.…”

“…It was already used in [5] for studying the uniform distribution of lattice points on ellipsoids in an even number l _> 4 of variables and obtaining very precise and uniform in level asymptotic formulas for the number of representations of positive integers by positive definite quaternary quadratic forms of a special type. Since [5] contains some gaps. in the present paper we give a new very simple proof of Theorem 2.…”