# Number Theory Day 6.08.2024

We will have a number theory day with four talks:

**9:30-10:30**

**Speaker:**Adrien Morin (KU)

**Title:**Weil-étale cohomology and the ETNC for constructible sheaves.

**Abstract:**Let $X$ be a variety over a finite field. Given an order $R$ in a semisimple algebra $A$ over the rationals and a constructible étale sheaf $F$ of $R$-modules over $X$, one can consider a natural equivariant L-function associated with $F$. We will formulate and prove a special value conjecture at negative integers for this L-function, expressed in terms of Weil-étale cohomology, provided that the latter is “well-behaved”; this is a geometric analogue of, and implies, the equivariant Tamagawa conjecture for a Tate motive and its negative twists over a global function field. It also generalizes the results of Lichtenbaum and Geisser on special values at negative integers for zeta functions of varieties, and the work of Burns-Kakde in the case of the equivariant L-functions coming from a finite $G$-cover of varieties.

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**11:00-12:00**

**Speaker: **Richard Griffon (Univ Clermont Auvergne)**Title: **New cases of the generalized Brauer-Siegel theorem.**Abstract: **Given an infinite sequence $S$ of number fields one may wonder about the asymptotic behavior, as $K$ runs through $S$, of the product of the class number of $K$ by its regulator of units (in terms of the discriminant of $K$). The classical Brauer-Siegel theorem answers this question when the number fields in $S$ have bounded degree. In the early 2000's, Tsfasman and Vlăduţ suggested the Generalized Brauer-Siegel conjecture, GBS for short, which would answer this question for much more general sequences. Their conjecture, which would follow from GRH, is known to hold in a handful of situations : for instance, when the number fields in the sequence are almost normal over $\mathbb{Q}$ (Tsfasman, Vlăduţ, Zykin and Lebacque).In a very recent work with Philippe Lebacque and Gaël Rémond, we prove that GBS unconditionally holds in many more instances. As I will explain in the talk, the main ingredients in our proof are the introduction of the notion of Galois complexity of a number field, and a new "zero descent" principle. If time permits, I will exhibit a few concrete examples for which GBS is now proved.

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**13:30-14:30**

**Speaker: **Farbod Shokrieh (Univ Washington, Seattle)**Title: **Canonical local heights and Berkovich skeleta.**Abstract: **I will discuss canonical local heights on abelian varieties

over non-archimedean fields from the point of view of Berkovich

analytic spaces. Our main result is a refinement of Néron's classical

result relating canonical local heights with intersection

multiplicities on the Néron model. We also revisit Tate's explicit

formulas for Néron's canonical local heights on elliptic curves

(involving Bernoulli polynomials). Our results can be viewed as

extensions of Tate's formulas to higher dimensions.

(Based on joint work with Robin de Jong.)

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**15:00-16:00**

**Speaker: **Samuel Le Fourn (Univ Grenoble Alpes)**Title: **Rational points on $X_0(N)^*$ when $N$ is non-squarefree.**Abstract: **Let N be a non-squarefree number. The curve $X_0(N)^*$ is the quotient of $X_0(N)$ by the full group of Atkin--Lehner involutions. In ongoing work with Sachi Hashimoto and Timo Keller, we aim to show that the rational points on $X_0(N)^*$ are CM points or cusps for $N >> 0$ (and find precisely the exceptions to this principle when possible). Our strategy follows the work of Mazur, Momose, and Bilu--Parent for the families $X_0(p)$ and $X_0(p^r)^+$. In this talk, I will discuss how to prove the "first part" of such a result, i.e. the integrality of the associated $j$-invariants (and hopefully make clear why we need $N$ to be non squarefree), and go through some surprising exceptional cases.