### Basel Number Theory Seminar: Fall 2023

Time: 14.15 on 28 Sep, 2023

Place: SR 00.003, Spiegelgasse 1

**Gabriele Bogo** (TU Darmstadt)

*Supersingular abelian surfaces and orthogonal polynomials*

An abelian surface in characteristic p>0 is called supersingular if it is isogenous to the product of two elliptic curves with trivial p-torsion. I will show that the supersingular locus of the mod p reduction of genus zero curves in Hilbert modular surfaces can be described in terms of zeros of orthogonal polynomials. The proof is based on the theory of twisted modular forms. This is joint work with Yingkun Li.

Time: 14.15 on 19 Oct, 2023

Place: SR 00.003, Spiegelgasse 1

**Martín Sombra** (ICREA and Universitat de Barcelona)

*Equidistribution of small points in projective varieties*

Let $X$ be a projective variety over a number field equipped with a metrized line bundle $\overline{L}$. A generic sequence of algebraic points of $X$ is small if their heights with respect to $\overline{L}$ converge to the smallest possible value, namely the essential minimum of the height function. Yuan's equidistribution theorem (2008) describes the asymptotic distribution of the Galois orbits of the points in a small generic sequence, under the assumption that the essential minimum coincides with the normalized height of $X$. This hypothesis holds in important cases such as dynamical heights on projective varieties, but it fails for most choices of $(X,\overline{L})$.

In this talk I will present a generalization of this theorem, extending to the general projective setting a result by Burgos, Philippon, Rivera Letelier and the speaker for the toric case. In particular, it applies to the canonical height on a semiabelian variety, and thus permits to recover Kühne's equidistribution theorem (2019).

Joint work with François Ballaÿ (Caen).

Time: 14.15 on 23 Nov, 2023

Place: SR 00.003, Spiegelgasse 1

**Yuri Bilu** (Université de Bordeaux)

*Skolem meets Schanuel*

A *linear recurrence* of order~$r$ over a number field~$K$ is a map ${U:\mathbb{Z}\to K}$ satisfying a relation of the form $$U(n+r)=a_{r-1}U(n+r-1)+ \cdots+ a_0U(n) \qquad (n\in \mathbb{Z}), $$where ${a_0, \ldots, a_{r-1}\in K}$ and ${a_0\ne 0}$. A linear recurrence is called *simple* if the characteristic polynomial ${X^r-a_{r-1}X^{r-1}-\ldots- a_0}$ has only simple roots, and *non-degenerate* if ${\lambda/\lambda'}$ is not a root of unity for any two distinct roots $\lambda, \lambda'$ of the characteristic polynomial. The classical *Theorem of Skolem-Mahler-Lech* asserts that a non-degenerate linear recurrence may have at most finitely many zeros. However, all known proofs of this theorem are non-effective and do not produce any tool to determine the zeros.

In this talk I will describe a simple algorithm that, when terminates, produces the rigorously certified list of zeros of a given simple linear recurrence. This algorithm always terminates subject to two celebrated conjectures: the *$p$-adic Schanuel Conjecture*, and the *Exponential Local-Global Principle*. We do not give any complexity bound (even conditional to some conjectures), but the algorithm performs well in practice, and was implemented in the *Skolem tool*

https://skolem.mpi-sws.org/

that I will demonstrate.

A joint work with Florian Luca, Joris Nieuwveld, Joël Ouaknine, David Purser andJames Worrell.

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

28 Sep, 2023 14.15 | Gabriele Bogo(TU Darmstadt) | Supersingular abelian surfaces and orthogonal polynomials |

19 Oct, 2023 14.15 | Martín Sombra(ICREA and Universitat de Barcelona) | Equidistribution of small points in projective varieties |

23 Nov, 2023 14.15 | Yuri Bilu(Université de Bordeaux) | Skolem meets Schanuel |