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Let $A$ be an abelian variety over a field $k$ of dimension $g$, and $H$ be a Weil cohomology theory for smooth projective varieties over $k$ with characteristic $0$ coefficient field $E$.

Is it true that $\operatorname{dim} H^1(A)=2g$? I think this will follow from some standard conjectures, but do we know this unconditionally at present (at least for some low dimensional cases)?

For example, one can prove $\operatorname{dim} H^1(A) \leq 2g$ by some power tricks, see Chapter 3. Theorem 8.1 in https://pages.uoregon.edu/ddugger/wbook.pdf (it may contain some typos, but the idea works).

And if $k$ is algebraically closed and $A$ is superspecial, then $End_k(A)\otimes \mathbb Q \cong M_g(D_{p,\infty})$ is central simple algebra of dimension $4g^2$ over $\mathbb Q$, hence $End(A) \otimes E \hookrightarrow End(H^1(A))$, and $dim H^1(A)=2g$.

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    $\begingroup$ If $k$ is a finite field then a Weil cohomology theory applied to any smooth projective variety gives vector spaces of dimensions equal to that of l-adic etale cohomology. This was deduced by Katz and Messing from the Weil conjectures for etale cohomology and the fact that any Weil cohomology computes the same zeta function, see eudml.org/doc/142251 $\endgroup$
    – SashaP
    Commented Apr 18, 2019 at 23:40
  • $\begingroup$ @SashaP Thank you, this is a good application of gcd thm in Weil II, which can also be used to show integral etale cohomology is torsion free for large $\ell$. $\endgroup$
    – Zhiyu
    Commented Apr 19, 2019 at 16:49

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Yes, it's the same for all Weil cohomology theories --- see 2A.8 of Kleiman, S. L. Algebraic cycles and the Weil conjectures. Dix exposés sur la cohomologie des schémas, 359--386, Adv. Stud. Pure Math., 3, North-Holland, Amsterdam, 1968.

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  • $\begingroup$ Thank you! I have a simple question: how do we get $\beta : H^{2n-1}(X) \rightarrow H^1(Pic(X)^0_{red}) $ using the Poincare divisor $D$ as in 2A1 (ii)? $\endgroup$
    – Zhiyu
    Commented Apr 19, 2019 at 18:17

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