Mathematics

We propose a refined version of the Sato-Tate conjecture about the spacing distribution of the angle determined for each prime number. We also discuss its implications on L -function associated with elliptic curves in the relation to random matrix theory.

Let K be a multiquadratic extension of Q and let Cl + (K) be its narrow class group. Recently, the authors \cite{KP} gave a bound for | Cl + (K)[2]| only in terms of the degree of K and the number of ramifying primes. In the present work we show that this bound is sharp in a wide number of cases. Furthermore, we extend this to ray class groups.

Let f(z)=q+ ??n?? a(n) q n be a weight k normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in \cite{AH} for k=2 by ruling out or locating all odd prime values |?�|<100 of their Fourier coefficients a(n) when n satisfies some congruences. We also study the case of odd weights k?? newforms where the nebentypus is given by a real quadratic Dirichlet character.

We prove a theorem that generalizes Schmidt's Subspace Theorem in the context of metric diophantine approximation. To do so we reformulate the Subspace theorem in the framework of homogeneous dynamics by introducing and studying a slope formalism and the corresponding notion of semistability for diagonal flows.

Let V 1 , V 2 , V 3 be a triple of even dimensional vector spaces over a number field F equipped with nondegenerate quadratic forms Q 1 , Q 2 , Q 3 , respectively. Let Y⊂ ∏ i=1 V i be the closed subscheme consisting of ( v 1 , v 2 , v 3 ) such that Q 1 ( v 1 )= Q 2 ( v 2 )= Q 3 ( v 3 ) . The first author and B. Liu previously proved a Poisson summation formula for this scheme under suitable assumptions on the functions involved. In the current work we extend the formula to a broader class of test functions which necessitates the introduction of boundary terms. This is the first time a summation formula with boundary terms has been proven for a spherical variety that is not a Braverman-Kazhdan space. As an application, we prove that the Fourier transform on Y, previously defined as a correspondence, descends to an isomorphism of the Schwartz space of Y.

The fifteen supersingular primes, see this https URL, appear in the theory of the moduli of abelian surfaces. This short expository note explains why.

In 1994, P.-G. Becker and W. Bergweiler listed all the differentially algebraic solutions of three famous functional equations: the Schr{ö}der's, B{ö}ttcher's and Abel's equations. The proof of this theorem combines various domains of mathematics. This goes from the theory of iteration, which gave birth to these equations, to the algebro-differential notion of coherent families developed by M. Boshernitzan and L. A. Rubel. This survey is an excursion into the history of these equations, in order to enlighten the different pieces of mathematics they bring together and how these parts fit into the result of P.-G. Becker and W. Bergweiler.

We survey some recent developments in the analytic theory of multiple Dirichlet series with arithmetical coefficients on the numerators.

A triangular field of rational numbers is characterized, with relations to Stirling numbers 2nd, Hyperbolic functions, and centered Binomial distribution. A Generating function is given.

If A and B are abelian varieties over a number field K such that there are non-trivial geometric homomorphisms of abelian varieties between reductions of A and B at most primes of K, then there exists a non-trivial (geometric) homomorphism from A to B defined over an algebraic closure of K.