Related papers: On the Tate and Standard Conjectures over Finite F…

Given a K3 surface X over a field of characteristic p, Artin conjectured that if X is supersingular (meaning infinite height) then its Picard rank is 22. Along with work of Nygaard-Ogus, this conjecture implies the Tate conjecture for K3…

Using integral $p$-adic Hodge theory, Kato and Koshikawa define a generalization of the Faltings height of an abelian variety to motives defined over a number field. Assuming the adelic Mumford-Tate conjecture, we prove a finiteness…

We prove that Grothendieck's Hodge standard conjecture holds for abelian varieties in arbitrary characteristic if the Hodge conjecture holds for complex abelian varieties of CM-type. For abelian varieties with no exotic algebraic classes,…

We introduce a tensor decomposition of the $\ell$-adic Tate module of an abelian variety $A_0$ defined over a number field which is geometrically isotypic. If $A_0$ is potentially of $\GL_2$-type and defined over a totally real number…

We study Weil-etale cohomology, introduced by Lichtenbaum for varieties over finite fields. In the first half of the paper we give an explicit description of the base change from Weil-etale cohomology to etale cohomology. As a consequence,…

The main result of this paper concerns the positivity of the Hodge bundles of abelian varieties over global function fields. As applications, we obtain some partial results on the Tate--Shafarevich group and the Tate conjecture of surfaces…

Recently Engel et al. (2025) have shown that the integral Hodge conjecture fails for very general abelian varieties. Using Deligne's theory of absolute Hodge cycles, we deduce a similar statement for the integral Tate conjecture.

The relative proportionality principle of Hirzebruch and H\"ofer was discovered in the case of compactified ball quotient surfaces X when studying curves C in X. It can be expressed as an inequality which attains equality precisely when C…

We construct nontrivial L-equivalence between curves of genus one and degree five, and between elliptic surfaces of multisection index five. These results give the first examples of L-equivalence for curves (necessarily over…

We formulate an analogue of Tate conjecture on algebraic cycles, for the log geometry over a finite field. We show that the weight-monodromy conjecture follows from this conjecture and from the semi-simplicity of the Frobenius action. This…

The Hodge conjecture is shown to be equivalent to a question about the homology of very ample divisors with ordinary double point singularities. The infinitesimal version of the result is also discussed.

We prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky's theory of hyperholomorphic sheaves and a study of the cohomology…

Let $K$ be a field finitely generated over the field of rational numbers, $K(c)$ the extension of $K$ obtained by adjoining all roots of unity, $L$ an infinite Galois extension of $K$, $X$ an abelian variety defined over $K$. We prove that…

The Tate conjecture has two parts: i) Tate classes are linear combination of algebraic classes, ii) semisimplicity of Galois representations (for smooth projective varieties). B. Moonen proved that i) implies ii) in characteristic 0, using…

A classical theorem by K. Ribet asserts that an abelian variety defined over the maximal cyclotomic extension $K$ of a number field has only finitely many torsion points. We show that this statement can be viewed as a particular case of a…

We give a function field specific, algebraic proof of the main results of class field theory for abelian extensions of degree coprime to the characteristic. By adapting some methods known for number fields and combining them in a new way,…

As an application of P. Delgine's theorem (Esnault and Kerz in Acta Math. Vietnam. 37:531-562, 2012) on a finiteness of $l$-adic sheaves on a variety over a finite field, we show the finiteness of \'etale coverings of such a variety with…

We prove the abundance conjecture for projective slc surfaces over arbitrary fields of positive characteristic. The proof relies on abundance for lc surfaces over abritrary fields, proved by Tanaka, and on the technique of Hacon and Xu to…

We prove The Tate Thomason conjecture through Theorem 2.2. Fundamental is the work of R W Thomson and the proof also rests upon the theory of infinite abelian groups.

Let A be a CM abelian variety defined over a number field K. We compute congruence relations on units in fields generated by adjoining torsion points of A to K. For elliptic curves, congruence relations of the type we compute were used in…