Danny Neftin: Reducible fibres and monodromy of polynomial maps

28th October, 2021

For a polynomial f∈ℚ[x], Hilbert's irreducibility theorem asserts that the fibre f-1(a) is irreducible over ℚ for all values a∈ℚ outside a "thin" set of exceptions Rf. The problem of describing Rf is closely related to determining the monodromy group of f, and has consequences to arithmetic dynamics, the Davenport-Lewis-Schinzel problem, and to the polynomial version of the question: "can you hear the shape of the drum?". We shall discuss recent progress on describing Rf and its consequences to the above topics.

Based on joint work with Joachim König.

Congling Qiu: Modularity and Heights of CM cycles on Kuga-Sato varieties

14th October, 2021

We study CM cycles on Kuga-Sato varieties over X(N) via theta lifting and relative trace formula. Our first result is the modularity of CM cycles, in the sense that the Hecke modules they generate are semisimple whose irreducible components are associated to higher-weight holomorphic cuspidal automorphic representations of GL2(ℚ). This is proved via theta lifting. Our second result is a higher weight analogue of the general Gross-Zagier formula of Yuan, S. Zhang and W. Zhang.

This is proved via relative trace formula, provided the modularity of CM cycles.

Xinyi Yuan: A uniform Bogomolov type of theorem for curves over global fields

15th September, 2021

In the recent breakthrough on the uniform Mordell-Lang problem by Dimitrov-Gao-Habegger and Kuhne, their key result is a uniform Bogomolov type of theorem for curves over number fields. In this talk, we introduce a refinement and generalization of the uniform Bogomolov conjecture over global fields, as a consequence of bigness of some adelic line bundles in the setting of Arakelov geometry. The treatment is based on the new theory of adelic line bundles of Yuan-Zhang and the admissible pairing over curves of Zhang.

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.

Andrei Rapinchuk: Groups with bounded generation: old and new

6th May, 2021

A group is said to have bounded generation (BG) if it is a finite product of cyclic subgroups. We will survey the known examples of groups with (BG) and their properties. We will then report on a recent result (joint with P. Corvaja, J. Ren and U. Zannier) that non-virtually abelian anisotropic linear groups (i. e. those consisting entirely of semi-simple elements) are not boundedly generated. The proofs rely on number-theoretic techniques.

Salim Tayou: Exceptional Jumps of Picard Rank of K3 Surfaces over Number Fields

18th February, 2021

Given a K3 surface X over a number field K, we prove that the set of primes of K where the geometric Picard rank jumps is infinite, assuming that X has everywhere potentially good reduction. This result is formulated in the general framework of GSpin Shimura varieties and I will explain other applications to abelian surfaces. I will also discuss applications to the existence of rational curves on K3 surfaces.

The results in this talk are joint work with Ananth Shankar, Arul Shankar and Yunqing Tang.

Julie Tzu-Yueh Wang: Pisot’s dth root conjecture for function fields and its complex analogue

29th September, 2020

Pisot's dth root conjecture, proved by Zannier in 2000, can be stated as follows. Let b be a linear recurrence over a number field k, and d ≥ 2 be an integer. Suppose that b(n) is the dth power of some element in k for all but finitely many n. Then there exists a linear recurrence a over k such that a(n)d = b(n) for all n.

In this talk, we propose a function-field analogue of this result and prove it under some 'non-triviality' assumption. We relate the problem to a result of Pasten-Wang on Büchi's dth power problem and develop a function-field GCD estimate for multivariable polynomials with 'small coefficients' evaluating at S-units arguments. We will also discuss its complex analogue in the notion of (generalized Ritt’s) exponential polynomials.

Jordan Ellenberg: What’s up in arithmetic statistics?

23rd July, 2020

If not for a global pandemic, a bunch of mathematicians would have gathered in Germany to talk about what's going on in the geometry of arithmetic statistics, which I would roughly describe as "methods from arithmetic geometry brought to bear on probabilistic questions about arithmetic objects". What does the maximal unramified extension of a random number field look like? What is the probability that a random elliptic curve has a 2-Selmer group of rank 100? How do you count points on a stack? I’ll give a survey of what’s happening in questions in this area, trying to emphasize open questions.

Vesselin Dimitrov: New constraints on the Galois configurations of algebraic integers in the complex plane

11th June, 2020

Fekete (1923) discovered the notion of transfinite diameter while studying the possible configurations of Galois orbits of algebraic integers in the complex plane. Based purely on the fact that the discriminants of monic integer irreducible polynomials P(X)∈ℤ[X] are at least 1 in magnitude (since they are non-zero integers), he found that the incidences (𝒦,P) between these polynomials P(X) and compacts 𝒦⊂ℂ of transfinite diameter d(𝒦)<1 have finite fibres over the argument 𝒦. Here we say that 𝒦 and P are in incidence if all the roots of P belong to 𝒦. The descendants of Fekete's theorem are vast and powerful, notably including the equidistribution theorems of Bilu, Rumely and Szpiro-Ullmo-Zhang or - in a different line of development - the root separation bound of Mahler. But the input on the discriminant is sometimes too coarse to be useful: in reality one expects, but cannot prove, that discriminants of polynomials are large, and dropping them by integrality is too crude in certain finer questions such as Lehmer's.

Breusch (1951) solved the non-reciprocal case of the Lehmer problem by taking up a lossless arithmetic input from resultants rather than discriminants. In this talk, I will present some further lossless constraints that derive from certain whole infinite sequences of Hankel determinants attached to the polynomial P(X) by algebraic operations. This will allow us to update on Fekete's theorem on the incidences (𝒦,P), by focusing this time on the fibres over the argument P for compacts 𝒦 that are made of finite unions of Jordan arcs continua covering the roots of P with certain congruence conditions on P and on the connected components of 𝒦. The ensuing taming on Galois orbits turn out to be sufficiently severe to resolve the conjecture of Schinzel and Zassenhaus (I will explain this case in detail), amidst certain other cases of the Lehmer problem that are far off from Salem's extreme. In a geometric formulation for 𝒜g with its Kobayashi metric, the root spacing constraints are likewise sufficiently severe to furthermore yield the exact ℳg analogues of the well-known polynomial counting theorems of Penner and Leininger-Margalit on the "L-short" geodesics of moduli space ℳg.

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.