Bilinear forms in Weyl sums for modular square roots and applications
Abstract
Let $q$ be a prime, $P \geq 1$ and let $N_q(P)$ denote the number of rational primes $p \leq P$ that split in the imaginary quadratic field $\mathbb{Q}(\sqrt{q})$. The first part of this paper establishes various unconditional and conditional (under existence of a Siegel zero) lower bounds for $N_q(P)$ in the range $q^{1/4+\varepsilon} \leq P \leq q$, for any fixed $\varepsilon>0$. This improves upon what is implied by work of Pollack and BenliPollack. The second part of this paper is dedicated to proving an estimate for a bilinear form involving Weyl sums for modular square roots (equivalently Salié sums). Our estimate has a power saving in the socalled P{ó}lyaVinogradov range, and our methods involve studying an additive energy coming from quadratic residues in $\mathbb{F}_q$. This bilinear form is inspired by the recent automorphic motivation: the second moment for twisted $L$functions attached to Kohnen newforms has recently been computed by the first and fourth authors. So the third part of this paper links the above two directions together and outlines the arithmetic applications of this bilinear form. These include the equidistribution of quadratic roots of primes, products of primes, and relaxations of a conjecture of ErdosOdlyzkoSarkozy.
 Publication:

arXiv eprints
 Pub Date:
 August 2019
 arXiv:
 arXiv:1908.10143
 Bibcode:
 2019arXiv190810143D
 Keywords:

 Mathematics  Number Theory
 EPrint:
 56 pages. v2 Bryce Kerr added as coauthor. Improvements to T1.7 and applications. Minor revisions from referee report included. v3 Further typos corrected, final version to appear in Adv. Math