University of Exeter: Monday 6th January 2025
Train station: travel to Exeter St. David's train station, not Exeter central.
Lecture room: Room 004, Harrison Building, North Park Road, Exeter, EX4 4QF. Harrison building is number 23 on this map.
Registration has now closed.
Schedule:
10:00-11:00am: Arrival and coffee
11:00am-12:00pm: Jimmy Tseng
12:10-1:10pm: Alex Torzewski
1:10-2.30pm: Lunch
2.30-3.30pm: Jessica Alessandri
3.30-4:00pm: Coffee
4:00pm-5:00pm: Andrei Seymour-Howell
6pm onwards: Dinner at Franco Manca Exeter, Queen St Dining, Guildhall Shopping Centre, Exeter, EX4 3EB
Speakers
Jimmy Tseng (Exeter)
Title: Interactions between homogeneous dynamics and number theory
Abstract: There have been a number of fruitful interactions between dynamics and number theory. In this talk, I will discuss one of these interactions, the notion of shrinking target horospherical equidistribution (STHE). STHE can be defined generally, but, in this talk, we will mostly discuss the case of targets (given in Grenier coordinates) shrinking into the cusp of and expanding horospheres (and related objects) on $\Gamma \backslash G$, the space of unimodular lattices. Here $G := \SL(d, \RR)$ and $\Gamma := \SL(d, \ZZ)$. There are two connections to number theory. One is that this gives a variant of Dani correspondence, which is the well-known connection between the dynamics on the homogenous space $\Gamma \backslash G$ and Diophantine approximation in number theory. Two is that the main technique requires (along with renormalization) the equidistribution of Farey sequences, which are objects in number theory.
If there is time, I will briefly discuss STHE for $G := \PSL(2, \RR)$ and $\Gamma$ a cofinite non-cocompact Fuchsian group. Here, there is a connection to analytic number theory, especially the double coset decomposition, Fourier techniques, and the spectral theory of the Laplace operator.
Alex Torzewski (King's College London)
Title: How common is formal complex multiplication?
Abstract: An elliptic curve over a characteristic zero field is said to have complex multiplication when its endomorphism ring is larger than Z ("E has extra endomorphisms"). Generic elliptic curves don't have complex multiplication. Similarly, when the Tate module of E has extra endomorphisms we say E has formal complex multiplication. Over a number field, E has formal complex multiplication if and only if it has complex multiplication. Over a local field this need not be the case. We investigate how often this happens via basic computations in p-adic Hodge theory.
Jessica Alessandri (Bath)
Title: A local-global problem for divisibility in algebraic groups
Abstract: In 2001, Dvornicich and Zannier, motivated by a particular case of the Hasse principle for quadratic forms and by the Grunwald-Wang Theorem, introduced the so-called “local-global divisibility problem” for commutative algebraic groups. During the last twenty years, several results have been produced for different algebraic groups. In this talk, I will give an overview on some recent developments, in particular for the case of algebraic tori and for elliptic curves. This is based on a joint work with Rocco Chirivì and Laura Paladino.
Andrei Seymour-Howell (Bristol)
Title: Unconditional computation of the class groups of real quadratic fields
Abstract: In the 19th century, Gauß famously conjectured that there are infinitely many real quadratic fields of class number one, in contrast to the finite number in the imaginary case. To inform his conjecture Gauß computed the class numbers of fields up to around discriminant 1000. Since the invention of the computer, this bound has been increased dramatically, however the current state of the art algorithms unfortunately all rely on GRH. In this talk, I will describe an algorithm to batch verify the output of the conditional algorithms, without assuming any unproven conjectures. Rather surprisingly, the main tool used will be the Selberg trace formula and explicit numerical computations of Maaß cusp forms. We have implemented this algorithm to compute class groups with discriminants up to X=10^11 and used the output to test various implications of the Cohen-Lenstra heuristics. This is joint work with Ce Bian, Andrew Booker, Austin Docherty and Michael Jacobson.