We study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight k, subject to a specific weighting which allows us to apply ...results concerning Bessel models and a variant of Petersson’s formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the spinor L-functions (for restricted test functions) gives global evidence for a well-known conjecture of Böcherer concerning the arithmetic nature of Fourier coefficients of Siegel cusp forms.
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 (Pólya-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 GL₃. 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.
Among the modern methods used to study prime numbers, the 'sieve' has been one of the most efficient. Originally conceived by Linnik in 1941, the 'large sieve' has developed extensively since the ...1960s, with a recent realisation that the underlying principles were capable of applications going well beyond prime number theory. This book develops a general form of sieve inequality, and describes its varied applications, including the study of families of zeta functions of algebraic curves over finite fields; arithmetic properties of characteristic polynomials of random unimodular matrices; homological properties of random 3-manifolds; and the average number of primes dividing the denominators of rational points on elliptic curves. Also covered in detail are the tools of harmonic analysis used to implement the forms of the large sieve inequality, including the Riemann Hypothesis over finite fields, and Property (T) or Property (tau) for discrete groups.
Let
be a subvariety defined over a number field
and let
be a special point not
contained in a positive-dimensional special subvariety of
. We
show that if a coordinate
corresponds to an order not
...contained in a single exceptional Siegel–Tatuzawa imaginary
quadratic field
, then the associated discriminant
is bounded by an effective constant depending only
on
and
. We derive analogous effective results for
the positive-dimensional maximal special subvarieties.
From the main theorem we deduce various effective results of
André–Oort type. In particular, we define a genericity condition on
the leading homogeneous part of a polynomial, and give a fully
effective André–Oort statement for hypersurfaces defined by
polynomials satisfying this condition.
We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of
GL
(
N
) Maass cusp forms for all
N
≥
2
, satisfy a ...central limit theorem in a suitable range, generalizing the case
N
= 2 treated by Fouvry et al. (Commentarii Math Helvetici,
2014
). Such universal Gaussian behaviour relies on a deep equidistribution result of products of hyper-Kloosterman sums.
Explicit Growth and Expansion for SL2 Kowalski, Emmanuel
International mathematics research notices,
01/2013, Letnik:
2013, Številka:
24
Journal Article
A study in sums of products Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe
Philosophical transactions - Royal Society. Mathematical, Physical and engineering sciences/Philosophical transactions - Royal Society. Mathematical, physical and engineering sciences,
04/2015, Letnik:
373, Številka:
2040
Journal Article
Recenzirano
Odprti dostop
We give a general version of cancellation in exponential sums that arise as sums of products of trace functions satisfying a suitable independence condition related to the Goursat-Kolchin-Ribet ...criterion, in a form that is easily applicable in analytic number theory.
We consider, in the special case of certain one-parameter families of Jacobians of curves defined over a number field, the problem of how the property that the generic fiber of such a family is ...absolutely simple ‘spreads’ to other fibers. We show that this question can be approached using arithmetic geometry or with more analytic methods based on sieve theory. In the first setting, non-trivial group-theoretic information is needed, while the version of the sieve we use is also of independent interest.
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