Seminars in Algebraic Number Theory

Gene Kopp: The Shintani-Faddeev Modular Cocycle

9th November, 2023

We ask the question, "how does the infinite q-Pochhammer symbol transform under modular transformations?" and connect the answer to that question to the Stark conjectures. The infinite q-Pochhammer symbol transforms by a generalized factor of automorphy, or modular 1-cocycle, that is analytic on a cut complex plane. This "Shintani–Faddeev modular cocycle" is an SL2(ℤ)-parametrized family of functions generalizing Shintani’s double sine function and Faddeev’s non-compact quantum dilogarithm. We relate real multiplication values of the Shintani-Faddeev modular cocycle to exponentials of certain derivative L-values, conjectured by Stark to be algebraic units generating abelian extensions of real quadratic fields.

Rebecca Bellovin: Modularity of Trianguline Galois Representations

12th October, 2023

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.

Amalendu Krishna: Ramified class field theory of curves over local fields

18th July, 2023

In this talk, I will present some results on the class field theory of smooth projective curves over a local field where one allows arbitrary ramification along a proper closed subset. We shall derive these results using some new results on the class field theory of 2-local fields and a duality theorem. This is based on a joint work with Subhadip Majumder.

Lior Bary-Soroker: Random additive polynomials

1st June, 2023

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.

Mark Shusterman: Counting Minimally Ramified Global Field Extensions

18th May, 2023

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.

Peter Koymans: Counting nilpotent extensions

4th May, 2023

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.

Pierre Le Boudec: 2-torsion in class groups of number fields

27th April, 2023

It is well-known that the class number of a number field K of fixed degree n is roughly bounded by the square root of the absolute value of the discriminant of K. However, given a prime number p, the cardinality of the p-torsion subgroup of the class group of K is expected to be much smaller. Unfortunately, beating the trivial bound mentioned above is a hard problem. Indeed, this task had only been achieved for a handful of pairs (n,p) until Bhargava, Shankar, Taniguchi, Thorne, Tsimerman and Zhao managed to do so for any degree n in the case p=2. In this talk we will go through their proof and we will present new bounds which depend on the geometry of the lattice underlying the ring of integers of K.

Jeffrey Lagarias: The Factorial Function and Generalizations, Revisited

11th April, 2023

In 1996 Manjul Barghava introduced a notion of P-orderings for arbitrary sets S of a Dedekind domain, with respect to a prime ideal P, which defined associated invariants called P-sequences. He combined these invariants to define generalized factorials and binomial coefficients associated to the subset S. These factorials were used in characterizing rings of polynomials that are integer-valued on S. Further generalizations of P-orderings by Bhargava in 2009 (with more parameters) have many applications. This talk defines analogous invariants for all proper ideals B of a Dedekind domain, called B-sequences, and extends the notion of generalized factorials and binomial coefficients to this setting. (This is joint work with Wijit Yangjit (U. Michigan))

Rene Schoof: Greenberg’s 𝜆=0 conjecture

11th April, 2023

Recent and not so recent computations by Mercuri and Paoluzi have verified Greenberg’s 𝜆=0 conjecture in Iwasawa theory in many cases. We discuss the conjecture and the computation

Alexandra Shlapentokh: Defining integers using unit groups

2nd March, 2023

We discuss some problems of definability and decidability over rings of integers of algebraic extensions of ℚ. In particular, we show that for a large class of fields K there is a simple formula defining rational integers over OK. Below, UK is the group of units of OK.

RK= { x | ∀ ε ∈ UK \ {1} ∃ δ ∈ UK : x ≡ (δ-1)/(ε-1) mod (ε-1) }.