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arXiv · 2010.16134

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

Paolo Dolce·Roberto Gualdi
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Abstract

Let $X$ be an arithmetic variety over the ring of integers of a number field $K$, with smooth generic fiber $X_K$. 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 N\'eron-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 show that the numerically trivial arithmetic $\mathbb{R}$-divisors on $X$ are exactly the linear combinations of principal ones. This result is equivalent to the non-degeneracy of the arithmetic intersection pairing in the argument of divisors, partially confirming [GS94, Conjecture 1].

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Paolo Dolce, Roberto Gualdi. 2020-10-30. Numerical equivalence of $\mathbb R$-divisors and Shioda-Tate formula for arithmetic varieties. https://doi.org/10.1515/crelle-2021-0081

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