Standard conjectures for abelian fourfolds
- PDF / 689,165 Bytes
- 64 Pages / 439.37 x 666.142 pts Page_size
- 95 Downloads / 192 Views
Standard conjectures for abelian fourfolds Giuseppe Ancona1
Received: 5 September 2018 / Accepted: 9 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Let A be an abelian fourfold in characteristic p. We prove the standard conjecture of Hodge type for A, namely that the intersection product 2 2 Znum (A)Q × Znum (A)Q −→ Q n (A) . (Equivalently, is of signature (ρ2 − ρ1 + 1; ρ1 − 1), with ρn = dim Znum Q it is positive definite when restricted to primitive classes for any choice of the polarization.) The approach consists in reformulating this question into a p-adic problem and then using p-adic Hodge theory to solve it. By combining this result with a theorem of Clozel we deduce that numerical equivalence on A coincides with -adic homological equivalence on A for infinitely many prime numbers . Hence, what is missing among the standard conjectures for abelian fourfolds is -independency of -adic homological equivalence.
Contents 1 2 3 4 5
Introduction . . . . . . . . . . . . Conventions . . . . . . . . . . . . Standard conjecture of Hodge type The motive of an abelian variety . . Lefschetz classes . . . . . . . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
B Giuseppe Ancona
[email protected]
1
Institut de Recherche Mathématique Avancée, Université de Strasbourg, Strasbourg, France
123
G. Ancona 6 Abelian varieties over finite fields 7 Exotic classes . . . . . . . . . . 8 Orthogonal motives of rank 2 . . 9 Quadratic forms . . . . . . . . . 10 The p-adic comparison theorem . 11 Characterization of p-adic periods 12 Computation of p-adic periods . 13 End of the proof . . . . . . . . . A Geometric examples . . . . . . . References . . . . . . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
1 Introduction In this paper we prove that the standard conjecture of Hodge type holds for abelian fourfolds. This is the first unconditional1 result on the conjecture since its formulation. Before giving the precise statement and a sketch of the proof, we briefly recall the history of the problem. In this introduction X will be a smooth, projective and geometrically connected variety over a base field k of characteristic p ≥
Data Loading...
