The Tate conjecture for abelian fourfolds over finite fields

Abstract: The Tate conjecture describes the cohomology of algebraic varieties in terms of their geometry. In this talk, I will discuss the proof of the Tate conjecture for abelian fourfolds over finite fields. This is the first resolution of the conjecture for all abelian varieties of a fixed dimension over finite fields since the work of Tate in the 1960s. The argument relies on techniques from Ancona's proof of the standard conjecture of Hodge type for abelian fourfolds, and ultimately reduces to results of Markman on the Hodge conjecture for abelian varieties.

Combining the above case of the Tate conjecture with theorems of Ancona and Kahn, we deduce that the standard conjecture on homological versus numerical equivalence holds for abelian fourfolds over arbitrary fields. This completes the proof of the standard conjectures for abelian fourfolds.

number theory

Audience: researchers in the topic

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Five College Number Theory Seminar