PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

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

Volume 187 / 2019

Jeremy Rouse Acta Arithmetica 187 (2019), 81-100 MSC: Primary 11E20; Secondary 11E25, 11F27, 11F30. DOI: 10.4064/aa180220-26-9 Published online: 23 November 2018


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 that \[ n \gg \max \{ N(Q)^{3/2 + \epsilon} D(Q)^{5/4 + \epsilon}, N(Q)^{2 + \epsilon} D(Q)^{1 + \epsilon} \}.\] Here $N(Q)$ 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.


  • Jeremy RouseDepartment of Mathematics and Statistics
    Wake Forest University
    Winston-Salem, NC 27109, U.S.A.

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image