Bilinear forms with Kloosterman sums and applications

By Emmanuel Kowalski, P. Michel, W. Sawin. 2015

80 citations recorded by the source corpus.

We prove non-trivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the Polya-Vinogradov range. We then derive applications to the second moment of holomorphic cusp forms twisted by characters modulo primes, and to the distribution in arithmetic progressions to large moduli of certain Eisenstein-Hecke coefficients on $\GL_3$. Our main tools are new bounds for certain complete sums in three variables over finite fields, proved using methods from algebraic geometry, especially $\ell$-adic cohomology and the Riemann Hypothesis.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/72a9faf4587ca94c2938adf4d4c9649328253149

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In Algebra and Number Theory

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Citations in the corpus

On moments of twisted L-functions (2014, unknown): “e L(f ⊗χ,1/2)L(g ⊗χ,1/2). Using the computation of the root number in Lemma 2.1 and the invariance of the parity χ(−1) under complex conjugation, we ﬁnd that Mf,g,σ(q) = 1 +ε(f,g,σ) 2 Mf,g,σ(q), 4See [24] for further discussions on this conjecture. 33 where ε(f,g,σ) is the root number ε(f,g,χ) for a”
The orbit method and analysis of automorphic forms (2018, background): “g of families of automorphic forms and L-functions. It has been successfully undertaken in many low-degree cases, where obtaining strong error estimates remains an active area of research (see, e.g., [19, 20, 54, 17] and references). We aim here to explore some ﬁrst cases of arbitrarily large degree”