Let X be an arithmetic variety over the ring of integers of a number field K, with smooth generic fiber X K {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 {X_{K}} and the rank of the Néron-Severi group of X K {X_{K}}. This is a higher-dimensional and arithmetic version of the classical Shioda-Tate formula for elliptic surfaces. Such an analogy is strengthened by the fact that we show that the numerically trivial arithmetic ℝ {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 a conjecture by H. Gillet and C. Soulé.
Numerical equivalence of ℝ -divisors and Shioda-Tate formula for arithmetic varieties
Dolce P.;
2022-01-01
Abstract
Let X be an arithmetic variety over the ring of integers of a number field K, with smooth generic fiber X K {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 {X_{K}} and the rank of the Néron-Severi group of X K {X_{K}}. This is a higher-dimensional and arithmetic version of the classical Shioda-Tate formula for elliptic surfaces. Such an analogy is strengthened by the fact that we show that the numerically trivial arithmetic ℝ {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 a conjecture by H. Gillet and C. Soulé.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
