4.5 Article

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

Journal

JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
Volume 2022, Issue 784, Pages 131-154

Publisher

WALTER DE GRUYTER GMBH
DOI: 10.1515/crelle-2021-0081

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Funding

  1. Alexander von Humboldt Foundation
  2. MINECO [MTM2015-65361-P]

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The paper presents a formula relating the dimension of the first Arakelov-Chow vector space of an arithmetic variety X with the Mordell-Weil rank of the Albanese variety of X-K and the rank of the Neron-Seven group of X-K. Additionally, it proves that the numerically trivial arithmetic R-divisors on X are exactly the linear combinations of principal ones.
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 Neron-Seven group of 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-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. Soule.

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