Tag - Galois theory
The Fontaine-Mazur conjecture (proved by Kisin and Emerton) says that (under certain technical hypotheses) a Galois representation ρ : Galℚ → GL2(ℚ̅p) is modular if it is unramified outside finitely many places and de Rham at p. I will talk about what this means, and I will discuss an analogous modularity result for Galois representations ρ : Galℚ → GL2(L) when L is instead a non-archimedean local field of characteristic p.
Random polynomials with integer coefficients tend to be irreducible and to have a large Galois group with high probability. This was shown more than a century ago in the large box model, where we choose the coefficients uniformly from a box and let its size go to infinity, while only recently there are results in the restricted box model, when the size of the box is bounded and its dimension (i.e., the degree of the polynomial) goes to infinity. In this talk, we will discuss an important class of random polynomials: additive polynomials, which have coefficients in the polynomial ring over a finite field. In this case, the roots form a vector space, hence the Galois group is naturally a subgroup of GLn. While we prove that the Galois group is the full matrix group both in the large box model, and in the large finite field limit, our main result is in the restricted box model: under some necessary condition the Galois group is large (in the sense that it contains SLn) asymptotically almost surely, as the degree goes to infinity. The proof relies crucially on deep results on subgroups of GLn by Fulman and Guralnick, combined with tools from algebra and number theory.
Given a finite group G, one is interested in the number of Galois extensions of a global field with Galois group G and bounded discriminant. We consider a refinement of this problem where the discriminant is required to have the smallest possible number of (distinct) prime factors. We will discuss existing results and conjectures over number fields, and present some recent results over function fields.
We discuss some recent progress towards the strong form of Malle’s conjecture. Even for nilpotent extensions, only very few cases of this conjecture are currently known. We show how equidistribution of Frobenius elements plays an essential role in this problem and how this can be used to make further progress towards Malle’s conjecture. We will also discuss applications to the Massey vanishing conjecture and to lifting problems. This is joint work with Carlo Pagano.
The successive minima of an order in a degree n number field are n real numbers encoding information about the Euclidean structure of the order. How many orders in degree n number fields are there with almost prescribed successive minima, fixed Galois group, and bounded discriminant? In this talk, I will address this question for n = 3,4,5. The answers, appropriately interpreted, turn out to be piecewise linear functions on certain convex bodies. If time permits, I will also discuss function field analogues of this problem.
Let P(X) be a monic, quartic, irreducible polynomial of ℤ[X] with cyclic or dihedral Galois group. We prove that there exists cP >0, such that for a positive proportion of integers n, P(n) has a prime factor bigger than n1+cP.
Let E be an elliptic curve defined over ℚ. The ℚ̅-points of E form an abelian group on which the Galois group Gℚ=Gal(ℚ̅/ℚ) acts. The usual Galois representation associated to E captures the action of Gℚ on the points of finite order. However, one could also look at the action of Gℚ on the free part of E(ℚ̅). This infinite-dimensional representation encodes a great deal of interesting arithmetic information. I will state a conjecture concerning this other Galois representation and present supporting evidence from probability theory, Ramsey theory, algebraic geometry, and number theory.
Cohen, Lenstra, and Martinet gave conjectural distributions for the class group of a random number field. Since the class group is the Galois group of the maximum abelian unramified extension, a natural generalization would be to give a conjecture for the distribution of the Galois group of the maximal unramified extension. Previous work (joint with Liu and Zurieck-Brown) produced a plausible conjecture for the part of this Galois group relatively prime to the number of roots of unity in the base field. There is a deep analogy between number fields and 3-manifolds. Thus, an analogous question would be to describe the distribution of the profinite completion of the fundamental group of a random 3-manifold. In this talk, I will explain how Will Sawin and I answered this question for a model of random 3-manifolds defined by Dunfield and Thurston, and how the techniques we used should allow us, in future work, to prove large q limit theorems in the function field analogue and give a general conjecture in the number field case, taking into account roots of unity in the base field. This is part two of a series of two talks on joint work, some in progress, with Will Sawin. Both talks should be understandable on their own.
The Cohen-Lenstra heuristics give predictions for the distribution of the class groups of a random quadratic number field. Cohen and Martinet generalized them to predict the distribution of the class groups of random extensions of a fixed base field, but Malle pointed out that these predictions have errors arising from the roots of unity in the base field. We give amended predictions that account for the influence of roots of unity.
Our predictions are based on a result which produces a formula for the distribution of a random finite abelian group given its moments, i.e., the expected number of surjections onto a fixed group. This result is very general and we expect it to have further applications in arithmetic statistics.
This is part one of a series of two talks on joint work, some in progress, with Melanie Matchett Wood. Both talks should be understandable on their own.
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