We will introduce the notion of slope filtration through examples, including the Harder-Narasimhan filtration on finite flat group schemes due to Fargues. We will then introduce Kisin modules, a certain generalization of finite flat group schemes, and describe a slope filtration on Kisin modules. This is joint work with Brandon Levin.
A theorem of Eichler and Shimura says that the space of cusp forms with complex coefficients appears as a direct summand of the cohomology of the compactified modular curve. Ohta has proven an analog of this theorem for the space of ordinary p-adic cusp forms with integral coefficients. Ohta's result has important applications in the Iwasawa theory of cyclotomic fields.
We discuss a new proof of Ohta's result using the geometry of Hecke correspondences at places of bad reduction.
For any irregular prime p, one has a Hida family of cuspidal eigenforms of level 1 whose residual Galois representations are all reducible. This family has already played a starring role in Wiles’ proof of Iwasawa’s main conjecture for totally real fields. In this talk, we instead focus on the Iwasawa theory of these modular forms in their own right. We will discuss new phenomena that occur in this residually reducible case including the fact that analytic mu-invariants are unbounded in this family and directly related to the p-adic zeta-function. This is a joint work with Joel Bellaiche.
The general Ramanujan Conjectures for congruence subgroups of arithmetic groups, and approximations that have been proven towards them, are central to many diophantine applications. Recently analogous results have been established for quite general subgroups of GL(n,Z) called "thin groups ". We will describe some of these and review some of their applications (mainly diophantine) as well as the ubiquity of thin groups.
A curve is a one dimensional space cut out by polynomial equations. In particular, one can consider curves over finite fields, which means the polynomial equations should have coefficients in some finite field and that points on the curve are given by values of the variables in the finite field that satisfy the given polynomials. A basic question is how many points such a curve has, and for a family of curves one can study the distribution of this statistic. We will give concrete examples of families in which this distribution is known or predicted, and give a sense of the different kinds of mathematics that are used to study different families.
In their attempt to mimic the proof of Fermat's Last Theorem for GL(n), Clozel, Harris, and Taylor, were led to a conjectural analogue of Ihara's lemma -- which is still open for n>2. In this talk we will revisit their conjecture from a more modern point of view, and reformulate it in terms of local Langlands in families, as currently being developed by Emerton and Helm. At the end, we hope to hint at potential applications.
Let $\chi$ be a totally odd character of a totally real number field. In 1981, B. Gross formulated a p-adic analogue of a conjecture of Stark which expresses the leading term at s=0 of the p-adic L-function attached to $\chi\omega$ as a product of a regulator and an algebraic number. Recently, Dasgupta-Darmon-Pollack proved Gross' conjecture in the rank one case under two assumptions: that Leopoldt's conjecture holds for F and p, and a certain technical condition when there is a unique prime above p in F. After giving some background and outlining their proof, I will explain how to remove both conditions, thus giving an unconditional proof of the conjecture. If there is extra time I will explain an application to the Iwasawa Main Conjecture which comes out of the proof, and make a few remarks on the higher rank case.
The Mumford-Tate conjecture is a deep conjecture which relates the arithmetic and the geometry of abelian varieties defined over number fields. The results of Moonen and Zarhin indicate that this conjecture holds for almost all absolutely simple abelian fourfolds. The only exception is when the abelian varieties have no nontrivial endomorphism. In this talk we will begin with an introduction to the Mumford-Tate conjecture and a brief summary of known results towards it. Then we sketch a proof of this conjecture in the above 'missing' case for abelian fourfolds.
After defining exterior powers of \pi-divisible modules, we prove that the exterior powers of \pi-divisible modules of dimension at most one over any base scheme exist and their construction commute with arbitrary base change