For a prime number p > 2, we give a direct proof of Breuil’s classification of killed by p finite flat group schemes over the valuation ring of a p-adic field with perfect residue field. As application we prove that the Galois modules of geometric points of such group schemes and of their characteristic p analogues coming from Faltings’s strict modules can be identified via the Fontaine-Wintenb...

Throughout this paper, let p be an odd prime number. The symbol, p, p , and p denote the ring of p-adic integers, the field of p-adic rational numbers, the complex number field and the completion of algebraic closure of p , respectively. Let be the set of natural numbers and ∪ {0}. Let νp be the normalized exponential valuation of p with |p|p p−νp p 1/p. Note that p {x | |x|p ≤ 1} lim← N /p p. ...

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...

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-...

Many mathematicians have studied Euler numbers and Euler polynomials( see [1-11]). Euler polynomials posses many interesting properties and arising in many areas of mathematics and physics. In this paper we introduce the generalized q-Euler numbers and polynomials with weak weight α. Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Q de...

Let p be a fixed odd prime. Throughout this paper Zp, Qp, C, and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field, and the completion of algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp with |p|p = p −vp(p) = p and let q be regarded as either a complex number q ∈ C or a p-adic number q...

Let p be an odd prime number. Throughout this paper Zp, Qp, C, and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field, and the completion of algebraic closure of Qp. The normalized valuation in Cp is denoted by | · |p with |p|p = 1 p . We say that f is a uniformly differentiable function at a point a ∈ Zp and denote...

The purpose of this paper is to study the arithmetic function f : Z+ → Q ∗ + defined by f(2l) = l (∀k, l ∈ N, l odd). We have, for example, f(1) = 1, f(2) = 1, f(3) = 3, f(12) = 1 3 , f(40) = 1 25 , . . . , so it is clear that f(n) is not always an integer. However, we will show in what follows that f satisfies the property that the product of the f(r) for 1 ≤ r ≤ n is always an integer, and it...

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...