Graded structures and differential operators on nearly holomorphic and quasimodular forms on classical groups

We wish to use graded structures [KrVu87], [Vu01] on dffierential operators and quasimodular forms on classical groups and show that these structures provide a tool to construct p-adic measures and p-adic L-functions on the corresponding non-archimedean weight spaces. An approach to constructions of automorphic L-functions on uni-tary groups and their p-adic avatars is presented. For an algebraic group G over a number eld K these L functions are certain Euler products L(s, $\pi$, r, $\chi$). In particular, our constructions cover the L-functions in [Shi00] via the doubling method of Piatetski-Shapiro and Rallis. A p-adic analogue of L(s, $\pi$, r, $\chi$) is a p-adic analytic function L p (s, $\pi$, r, $\chi$) of p-adic arguments s $\in$ Z p , $\chi$ mod p rPresented in a talk for the INTERNATIONAL SCIENTIFIC CONFERENCE "GRADED STRUCTURES IN ALGEBRA AND THEIR APPLICATIONS" dedicated to the memory of Prof. Marc Krasner on Friday, September 23, 2016, International University Centre (IUC), Dubrovnik, Croatia

Hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions

In this paper, we investigate the relationships among hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions through some families of `hypergeometric' algebraic varieties that are higher dimensional analogues of Legendre curves.Comment: 25 page

L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields

The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of L-functions over function fields vanishing to high order at the center point of their functional equation. The main application is that for every prime p and every integer g>0 there are absolutely simple abelian varieties of dimension g over Fp(t) for which the BSD conjecture holds and which have arbitrarily large rank.Comment: To appear in Inventiones Mathematica

Graded structures and differential operators on nearly holomorphic and quasimodular forms on classical groups

International audienceWe wish to use graded structures [KrVu87], [Vu01] on dffierential operators and quasimodular forms on classical groups and show that these structures provide a tool to construct p-adic measures and p-adic L-functions on the corresponding non-archimedean weight spaces. An approach to constructions of automorphic L-functions on uni-tary groups and their p-adic avatars is presented. For an algebraic group G over a number eld K these L functions are certain Euler products L(s, π, r, χ). In particular, our constructions cover the L-functions in [Shi00] via the doubling method of Piatetski-Shapiro and Rallis. A p-adic analogue of L(s, π, r, χ) is a p-adic analytic function L p (s, π, r, χ) of p-adic arguments s ∈ Z p , χ mod p rPresented in a talk for the INTERNATIONAL SCIENTIFIC CONFERENCE "GRADED STRUCTURES IN ALGEBRA AND THEIR APPLICATIONS"dedicated to the memory of Prof. Marc Krasner on Friday, September 23, 2016, International University Centre (IUC), Dubrovnik, Croatia

Constants for Artin like problems in Kummer and division fields

We apply the character sums method of Lenstra, Moree, and Stevenhagen, to explicitly compute the constants in the Titchmarsh divisor problem for Kummer fields and for division fields of Serre curves. We derive our results as special cases of a general result on the product expressions for the sums in the form $$\sum_{n=1}^{\infty}\frac{g(n)}{\#G(n)}$$ in which $g(n)$ is a multiplicative arithmetic function and $\{G(n)\}$ is a certain family of Galois groups. Our results extend the application of the character sums method to the evaluation of constants, such as the Titchmarsh divisor constants, that are not density constants

Dynamically affine maps in positive characteristic

We study fixed points of iterates of dynamically affine maps (a generalisation of Latt\`es maps) over algebraically closed fields of positive characteristic $p$. We present and study certain hypotheses that imply a dichotomy for the Artin-Mazur zeta function of the dynamical system: it is either rational or non-holonomic, depending on specific characteristics of the map. We also study the algebraicity of the so-called tame zeta function, the generating function for periodic points of order coprime to $p$. We then verify these hypotheses for dynamically affine maps on the projective line, generalising previous work of Bridy, and, in arbitrary dimension, for maps on Kummer varieties arising from multiplication by integers on abelian varieties.Comment: Lois van der Meijden co-authored Appendix B. 31 p

What is the best approach to counting primes?

As long as people have studied mathematics, they have wanted to know how many primes there are. Getting precise answers is a notoriously difficult problem, and the first suitable technique, due to Riemann, inspired an enormous amount of great mathematics, the techniques and insights permeating many different fields. In this article we will review some of the best techniques for counting primes, centering our discussion around Riemann's seminal paper. We will go on to discuss its limitations, and then recent efforts to replace Riemann's theory with one that is significantly simpler.Comment: To appear in a volume dedicated to the MAA Centennial in 201