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Current contacts: Vasily Dolgushev, Ed Letzter or Martin Lorenz.
The Seminar usually takes place on Mondays at 1:30 PM in Room 617 on the sixth floor of Wachman Hall. Click on title for abstract.
This is the organizational meeting of the Algebra Seminar.
Aniruddha Sudarshan, Temple University
This talk will give an overview of the main ingredients for Ribet's proof of the converse to Herbrand theorem. We will start with the statement of Herbrand's theorem, followed by reformulations of its converse using class field theory and Galois cohomology. Then, we venture into the world of congruences between modular forms, and Galois representations to prove a crucial lemma from which the converse to Herbrand's theorem follows.
Gilbert Moss, University of Maine
Let $G$ be a connected reductive algebraic group, such as $GL_n$, and let $F$ be a nonarchimedean local field, such as the p-adic numbers $\mathbb{Q}_p$. The local Langlands program describes a connection, which has been established in many cases, between irreducible smooth representations of $G(F)$ and Langlands parameters, which are described in terms of the absolute Galois group of $F$. The local Langlands correspondence "in families" is concerned with an aspect of the local Langlands program that seeks to upgrade this connection beyond irreducible representations to a smoothly varying morphism between natural moduli spaces of $G(F)$ representations and Langlands parameters. We will describe a precise conjecture in this direction and summarize past work establishing the conjecture for $GL_n(F)$, as well as ongoing work toward establishing it for classical groups.
Vasily Dolgushev, Temple University
This is an overview of the series of talks on Galois theory for infinite algebraic extensions. I will introduce the set-up and formulate the main theorem of the Galois theory for infinite algebraic extensions (the theorem is due to W. Krull). I will formulate the Nikolov-Segal theorem on finitely generated profinite groups and talk about examples of non-open subgroups of finite index in the absolute Galois group of rational numbers. If time permits, I will also formulate the Shafarevich conjecture.
Chathumini Kondasinghe, Temple University
This is a brief introduction to topological groups. I will define topological groups, give several examples and prove selected statements. This is a part of the series on talks on Galois theory for infinite algebraic extensions.
Wissam Raji, American University of Beruit
We consider the period polynomials $r_f (z)$ associated with cusp forms $f$ of weight $k$ on all of $SL_2(\mathbb{Z})$, which are generating functions for the critical L-values of the modular L-function associated to f. In 2014, El-Guindy and Raji proved that if $f$ is an eigenform, then $r_f (z)$ satisfies a “Riemann hypothesis” in the sense that all its zeros lie on the natural boundary of its functional equation. We show that this phenomenon is not restricted to eigenforms, and we provide large natural infinite families of cusp forms whose period polynomials almost always satisfy the Riemann hypothesis. For example, we show that for weights $k ≥ 120$, linear combinations of eigenforms with positive coefficients always have unimodular period polynomials.
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Frauke Bleher, University of Iowa
The discriminant d_F of a number field F is a basic invariant of F. The smaller d_F is relative to [F:Q], the more elements there are in the ring of integers O_F of F that have a given bounded size. This is relevant, for example, to cryptography using elements of O_F. In 2007, two cryptographers (Peikert and Rosen) asked whether one could give an explicit construction of an infinite family of number fields F having d_F^{1/[F:Q]} bounded by a constant times [F:Q]^d for some d < 1. By an explicit construction we mean an algorithm requiring time bounded by a polynomial in log([F:Q]) for producing a set of polynomials whose roots generate F. In this talk I will describe work with Ted Chinburg showing how this can be done for any d > 0. The proof uses the group theory of profinite 2-groups as well as recent results in analytic number theory.
William Chen, Rutgers University
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