Kloosterman sum is one of the most famous exponential sums
in number theory. It is defined using a prime p (and another number).
How do these sums vary with p? Ron Evans has made several conjectures
relating the moment of Kloosterman sums for p to the p-th Fourier
coefficient of certain modular forms. We sketch a proof of his
The Gross-Kudla-Schoen modified diagonal cycle on the triple product of the modular curve with itself provides a wealth of arithmetic information about modular forms, including derivatives of complex L-functions, special values of p-adic L-functions, and points on elliptic curves, known as Chow-Heegner points. In this talk, I will discuss a formula expressing the p-adic logarithm of a Chow-Heegner point in terms of the coefficients of the ordinary projection of certain p-adic modular forms.
In this talk, we will look at how congruences between Hecke eigensystems of modular forms affect the Iwasawa invariants of their anticyclotomic p-adic L-functions. It can be regarded as an application of Greenberg-Vatsal's idea on the variation of Iwasawa invariants to the anticyclotomic setting. As an application, we obtain examples of the anticyclotomic main conjecture for modular forms not treated by Skinner-Urban's work. An explicit example will be given.
We prove a B-SD conjecture for elliptic curves (for the p^infinity Selmer groups with arbitrary rank) a la Mazur-Tate and Darmon in anti-cyclotomic setting, for certain primes p. This is done, among other things, by proving a conjecture of Kolyvagin in 1991 on p-indivisibility of (derived) Heegner points over ring class fields. Some applications follow, for example, the p-part of the refined B-SD conjecture in the rank one case.
We first briefly review Dwork's trace formula and Wan's decomposition theorems. As an application, we consider a family of Laurent polynomials which is a generalization of the Laurent polynomials appeared in Iwaniec's work, and determine $p$-adic valuations for all the roots of the $L$-functions associated to an Zariski open dense subset of the space of Laurent polynomials. For lower dimension cases, we represent the Zariski open subset explicitly by computing an explicit Hasse polynomial.
We give a Chabauty-like method for finding p-adic approximations to
integral points on hyperelliptic curves when the Mordell-Weil rank of
the Jacobian equals the genus. The method uses an interpretation of
the component at p of the p-adic height pairing in terms of iterated
Coleman integrals. This is joint work with Amnon Besser and Steffen
In this talk, we show how to explicitly determine the zeta functions of
hyperelliptic curves of the form $y^2 = x^p-ax-b$ defined over a finite
field $GF(p^s}$ where $p$ is a prime. Joint work with Hui Xue and Lin
Given an ordinary differential equation whose coefficients are
meromorphic functions of a complex variable, the only obstruction to
convergence of local solutions in a disc is the presence of
singularities within the disc. It was observed decades ago that this
fails if one replaces "complex" by "p-adic", e.g., consider the
exponential function. In recent work of Baldassarri, Poineau, Pulita,
and the speaker, it has emerged that the convergence properties of such
solutions in the p-adic case can be described quite simply in terms of
Berkovich analytic geometry. We will give this description (without
assuming any prior familiarity with Berkovich's theory) and mention some
applications to studying wild ramification of covers of p-adic curves.
Let G be a reductive group satisfying the Harish-Chandra condition defined over a totally real field F, E/F a finite cyclic extension of fields. With further assumptions on G, by constructing an explicit morphism between eigenvarities, we prove that every p-adic family of p-adic automorphic representations of G over F can be lifted to a family of p-adic automorphic representations of G over E such that , at every classical point, the lifting is just the classical weak base change lifting. The key ingredients in the theory are: (1) a twisted p-adic trace formula for G/E ; (2) a p-adic fundamental lemma and an equation between p-adic trace formula and twisted p-adic trace formula; (3) a second construction of a twisted eigenvariety of G/E.
The local Langlands correspondence is a relationship between representations of the Galois group of a p-adic field F and the rerepresentations of GL_n(F). Understanding the behavior of the local Langlands correspondence as one varies Galois representations in families is an important ingredient in Emerton's recent proof of many cases of the Fontaine-Mazur conjecture. I will explain this question, and its connection to questions involving the Bernstein center, an algebra that acts naturally on a category of representations of GL_n(F).