### Basel Number Theory Seminar: Spring 2023

Time: 14.15 on 23 Feb, 2023

Place: SR 05.002, Spiegelgasse 5

**Claire Burrin** (University of Zurich)

*Discrete lattice orbits in the plane*

I will discuss the following variant of the Gauss circle problem. Consider an orbit Gv, where G is a discrete group of invertible matrices and v is a vector in the plane. Under certain conditions (which can be made explicit) this orbit is discrete. How is this discrete set distributed in the plane? I will describe interesting features of such sets, and explain what makes them amenable to a mix of tools from number theory and harmonic analysis. My main goal will be to discuss a recent result with Samantha Fairchild on the 'weak uniform discreteness' of these sets.

Time: 14.15 on 9 Mar, 2023

Place: SR 05.002, Spiegelgasse 5

**Chris Daw** (University of Reading)

*Large Galois orbits for unlikely intersections*

I will touch upon the large Galois orbits philosophy from the realm of unlikely intersections. I will discuss recent work for Y(1)^n and work in progress for \mathcal{A}_g, all in collaboration with Martin Orr (Manchester, UK). The overarching theme is that of extending an old idea of André involving G-functions.

Time: 14.15 on 13 Apr, 2023

Place: SR 05.002, Spiegelgasse 5

**Riccardo Pengo** (Max Planck Institut for Mathematics Bonn)

*Limits of Mahler measures and successively exact polynomials*

The Mahler measure of a multivariate Laurent polynomial P is a real number which measures the arithmetic complexity of P, and appears in many areas of mathematics, ranging from the Iwasawa theory of knots to ergodic theory. The set of real numbers which can be expressed as the Mahler measure of a polynomial with integer coefficients has some very interesting topological properties, as observed by Boyd. In particular, one expects it to be closed. In this talk, based on joint work with François Brunault, Antonin Guilloux and Mahya Mehrabdollahei, I will show how one can produce many interesting Cauchy sequences of Mahler measures, which converge to a Mahler measure, therefore respecting Boyd's conjecture. We are able moreover to give an explicit bound for the error term in the convergence of these sequences, and a full asymptotic expansion for one explicit family of polynomials, whose Mahler measure can be expressed in terms of the Bloch-Wigner dilogarithm evaluated at certain roots of unity. This is due to the fact that these polynomials share the property of being exact, which was introduced in the work of Maillot and Lalin. In the second part of my talk, based on work in progress with François Brunault, I will introduce this notion briefly, and a generalization of it, called "successive exactness", which are particularly useful in predicting links between Mahler measures and special values of L-functions.

Time: 14.15 on 4 May, 2023

Place: SR 05.002, Spiegelgasse 5

**Jérôme Poineau** (University of Caen)

*Torsion points of elliptic curves via Berkovich spaces over Z*

Berkovich spaces over Z may be seen as fibrations containing complex analytic spaces as well as p-adic analytic spaces, for every prime number p. We will give an introduction to those spaces and explain how they may be used in an arithmetic context to prove height inequalities. As an application, following a strategy by DeMarco-Krieger-Ye, we will give a proof of a conjecture of Bogomolov-Fu-Tschinkel on uniform bounds on the number of common images on P^1 of torsion points of two elliptic curves.

Time: 14.15 on 8 Jun, 2023

Place: SR 05.001, Spiegelgasse 5

**Nuno Hultberg** (University of Copenhagen)

*Fields with few small points*

Using the Northcott property of number fields it is possible to show the finiteness of preperiodic rational points over number fields for endomorphisms f:P^n \to P^n of degree at least 2. We will explore generalizations of this approach to certain infinite dimensional algebraic extensions of the rational numbers. As a crucial tool, we introduce a refinement of Northcott's theorem.

Date | Speaker | Title |
---|---|---|

23 Feb, 2023 14.15 | Claire Burrin(University of Zurich) | Discrete lattice orbits in the plane |

9 Mar, 2023 14.15 | Chris Daw(University of Reading) | Large Galois orbits for unlikely intersections |

13 Apr, 2023 14.15 | Riccardo Pengo(Max Planck Institut for Mathematics Bonn) | Limits of Mahler measures and successively exact polynomials |

4 May, 2023 14.15 | Jérôme Poineau(University of Caen) | Torsion points of elliptic curves via Berkovich spaces over Z |

18 May, 2023 14.15 | Nick Rome | |

1 Jun, 2023 14.15 | Jakob Glas | |

8 Jun, 2023 14.15 | Nuno Hultberg(University of Copenhagen) | Fields with few small points |