Let K be a p-adic field, R the valuation ring of K, P the maximal ideal of R and q the cardinality of the residue field R/P . Let f be a polynomial over R in n > 1 variables and let χ be a character of R×. Let Mi(u) be the number of solutions of f = u in (R/P i)n for i ∈ Z≥0 and u ∈ R/P i. These numbers are related with Igusa’s p-adic zeta function Zf,χ(s) of f . We explain the connection betwe...

We prove a commutative algebra result which has consequences for congruences between automorphic forms modulo prime powers. If C denotes the congruence module for a fixed automorphic Hecke eigenform π0 we prove an exact relation between the p-adic valuation of the order of C and the sum of the exponents of p-power congruences between the Hecke eigenvalues of π0 and other automorphic forms. We a...

Gross–Stark units, Stark–Heegner points, and class fields of real quadratic fields by Samit Dasgupta Doctor of Philosophy in Mathematics University of California, Berkeley Professor Kenneth Ribet, Chair We present two generalizations of Darmon’s construction of Stark–Heegner points on elliptic curves defined overQ. First, we provide a lifting of Stark–Heegner points from elliptic curves to cert...

We explicitly compute all the slopes of the Hecke operator U2 acting on overconvergent 2-adic level 1 cusp forms of weight 0: the nth slope is 1 + 2v((3n)!/n!), where v denotes the 2-adic valuation. We formulate an explicit conjecture about what these slopes should be for weight k forms.

These are lecture notes from a graduate course on p-adic and motivic integration (given at BGU). Themain topics are: Quantifier elimination in the p-adics, rationality of p-adic zeta functions and their motivic analogues, basic model theory of algebraically closed valued fields, motivic integration following Hrushovski and Kazhdan, application to the Milnor fibration. Background: basic model th...

coming from the cokernels of the rst column. These are cyclic (pro-cyclic in the limit) and are generated by the image of q t. So, lim A(K t)=NA(K n;t) = ?=< (q t ; L t =K t) > where (q t ; L t =K t) is the norm residue symbol of q t for the extension L t =K t. We write Q t for the image of (q t ; L t =K t) under the isomorphism ? ! Z p induced by sending to 1. Note that although Q t is not can...

Let R be a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field, K the fraction field of R. Suppose G is a Barsotti-Tate group (p-divisible group) defined over K which acquires good reduction over a finite extension K′ of K. We prove that there exists a constant c ≥ 2 which depends on the absolute ramification index e(K′/Qp) and the height of G such that G ...

For certain algebraic Hecke characters χ of an imaginary quadratic field F we define an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal automorphic forms of GL2/F . By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the special L-...

In this paper we study the multiplicities and asymptotic behaviour of numbers totients in strata given by 2-adic valuation.

This work is a part of my research in the area of model theory. First we study a particular L-theory of rings whose models are called p-convexly valued domains (the first-order language L is the language of rings equipped with a linear divisibility predicate and predicates for the nth powers). It consists of a p-adic counterpart of Becker’s convexly oredered valuation rings. Then we are interes...