Tag - Galois theory

Manjul Bharğava: Galois groups of random integer polynomials

21st April, 2022

Of the (2H+1)n monic integer polynomials f(x)=xn+a1xn−1+⋯+an with max{|a1|,…,|an|}≤H, how many have associated Galois group that is not the full symmetric group Sn? There are clearly ≫Hn−1 such polynomials, as may be obtained by setting an=0. In 1936, van der Waerden conjectured that O(Hn−1) should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees n≤4, due to work of van der Waerden and Chow and Dietmann. In this talk, we will describe a proof of van der Waerden's Conjecture for all degrees n.

Ashwin Iyengar: Local (ℓ=p) Galois Deformation Rings

10th February, 2022

I will present joint work with V. Paškūnas and G. Böckle concerning deformation rings for mod p Galois representations of p-adic local fields. After giving a short introduction to the subject, I will explain our main result which says that framed local deformation rings are complete intersections of the "expected dimension", and which gives a classification of their irreducible components in terms of a determinant map. I will explain some of the ingredients that go into our proof, which involves work on pseudo-deformations by Böckle-Juschka, and moduli spaces of representations with fixed pseudo-character. If time permits I will discuss an application to density of crystalline points in deformation spaces.

Sam Chow: Galois groups of random polynomials

19th July, 2021

How often is a quintic polynomial solvable by radicals? We establish that the number of such polynomials, monic and irreducible with integer coefficients in [-H, H], is O(H3.91). More generally, we show that if n ≥ 3 and n ≠ 7, 8, 10 then there are O(Hn-1.017) monic, irreducible polynomials of degree n with integer coefficients in [-H, H] and Galois group not containing An. Save for the alternating group and degrees 7, 8, 10, this establishes a 1936 conjecture of van der Waerden, that irreducible non-Sn polynomials are substantially rarer than reducible polynomials.

Sam Mundy: Eisenstein series, p-adic deformations, Galois representations, and the group G2

1st April, 2021

I will explain some recent work on special cases of the Bloch-Kato conjecture for the symmetric cube of certain modular Galois representations. Under certain standard conjectures, this work constructs non-trivial elements in the Selmer groups of these symmetric cube Galois representations; this works by p-adically deforming critical Eisenstein series in a generically cuspidal family of automorphic representations, and then constructing a lattice in the associated family of Galois representations, all for the exceptional group G2. While I will touch on all of these aspects of the construction, I will mainly focus on the Galois side in this talk.

Alexander Smith: Selmer groups and a Cassels-Tate pairing for finite Galois modules

25th February, 2021

I will discuss some new results on the structure of Selmer groups of finite Galois modules over global fields. Tate's definition of the Cassels-Tate pairing can be extended to a pairing on such Selmer groups with little adjustment, and many of the fundamental properties of the Cassels-Tate pairing can be reproved with new methods in this setting. I will also give a general definition of the theta/Mumford group and relate it to the structure of the Cassels-Tate pairing, generalizing work of Poonen and Stoll.

Florian Breuer: Realizing general linear groups as Galois groups

23rd November, 2020

I will show how to construct field extensions with Galois groups isomorphic to general linear groups (with entries in various rings and fields) from the torsion of elliptic curves and Drinfeld modules. No prior knowledge of these structures is assumed.

Mladen Dimitrov: A geometric view on Iwasawa theory

14th May, 2020

We will investigate the geometry of the p-adic eigencurve at classical points where the Galois representation is locally trivial at p, and will give applications to Iwasawa and Hida theories.

Koji Shimizu: A p-adic monodromy theorem for de Rham local systems

27th February, 2020

Every smooth proper algebraic variety over a p-adic field is expected to have a semistable model after passing to a finite extension. This conjecture is open in general, but its analogue for Galois representations, the p-adic monodromy theorem, is known. In this talk, we will explain a generalization of this theorem to étale local systems on a smooth rigid analytic variety.

Jared Weinstein: Moduli spaces of shtukas over function fields

13th February, 2020

We present some work in progress, on moduli spaces of Drinfeld shtukas. These spaces are the function field analogous to Shimura varieties. In fact they are more versatile; there are r-legged versions for any r. Tate's conjecture predicts some interesting relations between shtuka spaces and function field arithmetic. For instance, there should be a notion of modularity for the r-fold product of an elliptic curve. We verify these predictions in a few cases.

This is partly joint work with Noam Elkies.