Martin's Blog

Motivic Galois groups and periods

Posted by Martin Orr on Tuesday, 03 November 2015 at 16:00

In my last post, I discussed how the existence of a polarisation implies an upper bound for the transcendence degree of the extended period matrix of an abelian variety, namely the dimension of the general symplectic group \operatorname{GSp}_{2g} (where g is the dimension of the abelian variety). In this post, I will discuss how this can be generalised to take into account all algebraic cycles on the abelian variety. The group \operatorname{GSp}_{2g} is replaced by the motivic Galois group of the abelian variety, which I will define. I will also mention how Deligne's theorem on absolute Hodge cycles allows us to replace the motivic Galois group by the Mumford-Tate group.

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