The second Women in Numbers workshop (WIN2) was held November 6-11, 2011, at the Banff International Research Station (BIRS) in Banff, Alberta, Canada. During the workshop, group leaders presented ...open problems in various areas of number theory, and working groups tackled those problems in collaborations begun at the workshop and continuing long after.
This volume collects articles written by participants of WIN2. Survey papers written by project leaders are designed to introduce areas of active research in number theory to advanced graduate students and recent PhDs. Original research articles by the project groups detail their work on the open problems tackled during and after WIN2. Other articles in this volume contain new research on related topics by women number theorists.
The articles collected here encompass a wide range of topics in number theory including Galois representations, the Tamagawa number conjecture, arithmetic intersection formulas, Mahler measures, Newton polygons, the Dwork family, elliptic curves, cryptography, and supercongruences.
WIN2 and thisProceedingsvolume are part of the Women in Numbers network, aimed at increasing the visibility of women researchers' contributions to number theory and at increasing the participation of women mathematicians in number theory and related fields.
The main goal of this paper is to calculate the index of any number field K generated by a root of an irreducible trinomial x4+ax+b∈Zx. Our approach is based on Engstrom’s results and the ...factorization of 2ZK and 3ZK. In particular, we reformulate the Davis and Spearman’s results. Namely, for every prime integer p, we evaluate νp(i(K)). The existence of a common index divisor of K guaranties the non monogenity of K.
For thirty years, the biennial international conference AGC$^2$T (Arithmetic, Geometry, Cryptography, and Coding Theory) has brought researchers to Marseille to build connections between arithmetic ...geometry and its applications, originally highlighting coding theory but more recently including cryptography and other areas as well.This volume contains the proceedings of the 16th international conference, held from June 19-23, 2017.The papers are original research articles covering a large range of topics, including weight enumerators for codes, function field analogs of the Brauer-Siegel theorem, the computation of cohomological invariants of curves, the trace distributions of algebraic groups, and applications of the computation of zeta functions of curves. Despite the varied topics, the papers share a common thread: the beautiful interplay between abstract theory and explicit results.
In this paper, for any septic number field K generated by a root α of a monic irreducible trinomial , we describe all prime power divisors of the index of K answering Problem 22 of Narkiewicz 26. In ...particular, if , then K is not mongenic. We illustrate our results by some computational examples.
M. Ram Murty has had a profound impact on the development of number theory throughout the world. To honor his mathematical legacy, a conference focusing on new research directions in number theory ...inspired by his most significant achievements was held from October 15-17, 2013, at the Centre de Recherches Mathematiques in Montreal.This proceedings volume is representative of the broad spectrum of topics that were addressed at the conference, such as elliptic curves, function field arithmetic, Galois representations, $L$-functions, modular forms and automorphic forms, sieve methods, and transcendental number theory.
The construction of the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbf{Q}_p)$ uses in an essential way Fontaine's theory of cyclotomic $(\varphi ,\Gamma )$-modules. Here ...cyclotomic means that $\Gamma = \mathrm {Gal}(\mathbf{Q}_p(\mu_{p^\infty})/\mathbf{Q}_p)$ is the Galois group of the cyclotomic extension of $\mathbf Q_p$. In order to generalize the $p$-adic local Langlands correspondence to $\mathrm{GL}_{2}(L)$, where $L$ is a finite extension of $\mathbf{Q}_p$, it seems necessary to have at our disposal a theory of Lubin-Tate $(\varphi ,\Gamma )$-modules. Such a generalization has been carried out, to some extent, by working over the $p$-adic open unit disk, endowed with the action of the endomorphisms of a Lubin-Tate group. The main idea of this article is to carry out a Lubin-Tate generalization of the theory of cyclotomic $(\varphi ,\Gamma )$-modules in a different fashion. Instead of the $p$-adic open unit disk, the authors work over a character variety that parameterizes the locally $L$-analytic characters on $o_L$. They study $(\varphi ,\Gamma )$-modules in this setting and relate some of them to what was known previously.
Let be a number field generated by a complex root α of a monic irreducible trinomial In this paper, for every prime integer p, we give necessary and sufficient conditions on a and b so that p is a ...common index divisor of K. In particular, if any one of these conditions holds, then K is not monogenic.
This volume contains the proceedings of the international conference ``Around Langlands Correspondences'', held from June 17-20, 2015, at Universite Paris Sud in Orsay, France.The Langlands ...correspondence (nowadays called the usual Langlands correspondence), conjectured by Robert Langlands in the late 1960s and early 1970s, has recently seen some new mysterious generalizations: the modular Langlands correspondence, the $p$-adic Langlands correspondence, and the geometric Langlands correspondence, the last of which seems to share deep connections with the Baum-Connes conjecture.The aim of this volume is to present, through a mix of research and expository articles, some of the fascinating new directions in number theory and representation theory arising from recent developments in the Langlands program. Special emphasis is placed on nonclassical versions of the conjectural Langlands correspondences, where the underlying field is no longer the complex numbers.
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