**seminarios Teoría de Números**

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

SPEAKER: Paolo Dolce (University of Udine)

DATE & TIME: Tuesday, March 02nd, 2021 - 17:30

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.

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