相关论文: Generalized Sato-Tate and quadratic residues

Generalised Ito formulae are proved for time dependent functions of continuous real valued semi-martingales. The conditions involve left space and time first derivatives, with the left space derivative required to have locally bounded…

Using a relative trace formula approach, we prove the twisted global Gan-Gross-Prasad conjecture for $\operatorname{U}(V) \subseteq \operatorname{GL}(V)$, as well as its refinement, under some unramifiedness assumptions and local conditions…

n this paper, we state several conjectures regarding distribution of primes and of pairs of primes represented by irreducible homogeneous polynomial in two variables $f(a,b)$. We formulate conjectures with respect to the slope $t=b/a$ for…

The $t$-dependence of generalized parton distrbutions for $x\to 1$ is discussed. We argue that constituent quark models, where the $t$-dependence for $x\to 1$ is through the product $(1-x)t$, are inconsistent. Instead we suggest a leading…

We give a reformuation of the Tate conjecture for a surface over a finite field in terms of suitable affine open subsets. We then present three attempts to prove this reformulation, each of them falling short. Interestingly, the last two…

Extending a method of D. Wolke, we establish a general result on the large sieve with sparse sets S of moduli which are in a sense well-distributed in arithmetic progressions. We then apply our result to the case when S consists of sqares.…

Several results about the union-closed sets conjecture are presented.

Some generalized multi-sum Chu-Vandermonde identities are presented and proved, generalizing some known multi-sum Chu-Vandermonde identities from literature and adding some quadratic and cubic examples of these identities. Some other…

We survey recent developments on generalizing the Gross--Zagier formula to high dimensional Shimura varieties, with an emphasis on the Arithmetic Gan--Gross--Prasad conjecture and the relative trace formula approach.

We prove a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms. Our results are inspired by and analogous to recent results for diagonal quadratic forms due to Bourgain.

An approach is described to calculate Generalized Parton Distributions (GPDs) in Constituent Quark Models (CQM). The GPDs are obtained from wave functions to be evaluated in a given CQM. The general relations linking the twist-two GPDs to…

The classical theorems relating integral binary quadratic forms and ideal classes of quadratic orders have been of tremendous importance in mathematics, and many authors have given extensions of these theorems to rings other than the…

We prove an analogue of the Sato-Tate conjecture for Drinfeld modules. Using ideas of Drinfeld, J.-K. Yu showed that Drinfeld modules satisfy some Sato-Tate law, but did not describe the actual law. More precisely, for a Drinfeld module…

In this paper we establish several results concerning the generalized Ramanujan primes. For $n\in\mathbb{N}$ and $k \in \mathbb{R}_{> 1}$ we give estimates for the $n$th $k$-Ramanujan prime which lead both to generalizations and to…

We prove many non-generic cases of the Sato-Tate conjecture for abelian surfaces as formulated by Fite, Kedlaya, Rotger and Sutherland, using the potential automorphy theorems of Barnet-Lamb, Gee, Geraghty and Taylor.

We propose a generalization of the Elliott-Halberstam conjecture concerning the distribution of prime pairs in arithmetic progressions. This conjecture, which we call the Generalized Elliott-Halberstam Conjecture for Shifted Convolutions…

A number of new terminating series involving $\sin(n^2/k)$ and $\cos(n^2/k)$ are presented and connected to Gauss quadratic sums. Several new closed forms of generic Gauss quadratic sums are obtained and previously known results are…

Let $p\equiv1\pmod 4$ be a prime. In this paper, with the help of Jacobsthal sums, we study some permutation problems involving biquadratic residues modulo $p$.

In the first part of this article, we review a formalism of local zeta integrals attached to spherical reductive prehomogeneous vector spaces, which partially extends M. Sato's theory by incorporating the generalized matrix coefficients of…

We introduce the notion of {\em iterated residue} to study generalized Bott manifolds. When applying the iterated residues to compute the Borisov-Gunnells toric form and the Witten genus of certain toric varieties as well as complete…