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

TL;DR

This paper extends the classical Shioda-Tate formula to higher-dimensional arithmetic varieties, relating Arakelov-Chow spaces, Mordell-Weil ranks, and Néron-Severi groups, and characterizes numerically trivial arithmetic divisors.

Contribution

It provides a higher-dimensional, arithmetic analogue of the Shioda-Tate formula and characterizes numerically trivial arithmetic divisors as linear combinations of principal divisors.

Findings

Derived a formula linking Arakelov-Chow space dimension with Mordell-Weil and Néron-Severi ranks.

Showed that numerically trivial arithmetic $ ext{R}$-divisors are exactly principal divisors.

Confirmed non-degeneracy of the arithmetic intersection pairing, supporting a conjecture.

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].