Publication

Bilinear forms with Kloosterman sums and applications

Philippe Michel
2017
Journal Articles

Abstract

We prove nontrivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the range controlled by Fourier-analytic methods (Polya-Vinogradov range). We then derive applications to the second moment of 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 l-adic cohomology and the Riemann Hypothesis.

Official source

https://infoscience.epfl.ch/entities/publication/044d50ac-a345-461e-a978-86d2d460e380

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Philippe Michel
2017
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/044d50ac-a345-461e-a978-86d2d460e380

About this result

Ontological neighbourhood

Mathematics

Number theory: Topics in number theory

Related concepts (32)

Bilinear form

In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The dot product on is an example of a bilinear form.

Bilinear map

In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Let and be three vector spaces over the same base field . A bilinear map is a function such that for all , the map is a linear map from to and for all , the map is a linear map from to In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed.

Degenerate bilinear form

In mathematics, specifically linear algebra, a degenerate bilinear form f (x, y ) on a vector space V is a bilinear form such that the map from V to V∗ (the dual space of V ) given by v ↦ (x ↦ f (x, v )) is not an isomorphism. An equivalent definition when V is finite-dimensional is that it has a non-trivial kernel: there exist some non-zero x in V such that for all A nondegenerate or nonsingular form is a bilinear form that is not degenerate, meaning that is an isomorphism, or equivalently in finite dimensions, if and only if for all implies that .

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Bilinear forms with Kloosterman sums and applications

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