# Seminar

#### Number Theory Seminar

Numerical equivalence of R-divisors and Shioda-Tate formula for arithmetic varieties

Speaker:   Paolo Dolce (University of Udine)
Date:  Tuesday, 02 March 2021 - 17:30
Place:  Online - Microsoft Teams. Please, contact daniel.macias@icmat.es to join the team

Abstract:
Let $X$ be an arithmetic variety over the ring of integers of a number field $K$, and let $X_K$ be its generic fiber. We give a formula that relates the dimension of the first Arakelov-Chow vector space of $X$ with the Mordell-Weil rank of the Albanese variety of $X_K$ and the rank of the Néron-Severi group of $X_K$. This is a higher dimensional and arithmetic version of the classical Shioda-Tate formula for elliptic surfaces. Such analogy is strengthened by the fact that we also show that the numerically trivial arithmetic $mathbb R$-divisors on $X$ are exactly the linear combinations of principal ones.