# References for period matrix of abelian variety

Hi, everyone.

I am looking for some references for period matrix of abelian variety over arbitrary field, if you know, could you please tell me?

For period matrix of abelian varieties, I means that if $A$ is an abelian variety over complex number field, $A \cong V/\Gamma$. If we chose a basis of $V$, and a basis of lattice $\Gamma$, write the basis of lattice in term by the basis of $V$, the matrix which is called period matrix.

I want to know how to define period matrix for abelian variety over arbitrary field, and some basic properties of this matrix.

Thank you very much!

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I am a little confused by your question. The period matrix arises when we compare different (co)homology theories for example Betti and De rham. See the papers of Faltings and Tsuji where they prove the comparison theorems. Also look at Tate's paper on p-divisible groups. I may be able to give you more concrete answers if you can kindly explain your question. Thanks. –  Arijit May 14 '13 at 10:07
@Arijit: Thanks Arijit, I modify the question. –  Ada Aimone May 14 '13 at 10:33
It's still not clear why you expect there to be a meaningful definition of "period matrix" in this wider context. –  David Loeffler May 14 '13 at 10:45
I agree with David as I dont see why do you want to compare $V$ and $\Gamma$. You can think of $\Gamma=H_{1,betti}(A(\mathbbC),\mathbbZ)$ –  Arijit May 14 '13 at 11:00

For an abelian variety $A$ over a subfield $k$ of the complex numbers, you have the first de Rham cohomology group of $A$ over $k$, which is a $k$-vector space, and the first singular cohomology group of $A/\mathbb{C}$, which is a $\mathbb{Q}$-vector space. When tensored up to $\mathbb{C}$ the two vector spaces are canonically isomorphic, and the matrix of periods of $A$ compares bases coming from the two vector spaces.
You won't get $A$ as a quotient of a vector space in general without some kind of strange transcendentality. For example, if $V$ is defined over $\mathbb{F}_p$, then any $S$-valued point of $V$ is $p$-torsion for any test object $S$. In particular, you can't possibly obtain any of the prime-to-$p$ torsion in $A$ without resorting to extraordinary means (whose existence I doubt).
There are some interesting treatments for certain complete topological fields other than $\mathbb{C}$, but "most" fields do not fit this description.