Mixed motives associated to elliptic curves
By Richard Hain
Also appears in collection : Moduli spaces in geometry / Espaces de modules en géométrie
The absolute Galois group of the rational numbers acts on the various flavours (profinite, prounipotent, pro-$\ell$) of the fundamental group of a smooth projective curve over the rationals. The image of the corresponding homomorphism normalizes the image of the profinite mapping class group in the automorphism group of the geometric fundamental group of the curve. The image of the Galois action modulo these “geometric automorphisms” is independent of the curve. A basic problem is to determine this image. This talk is a report on a joint project with Francis Brown whose goal is to understand the image mod geometric automorphisms in the prounipotent case. Standard arguments reduce the problem to one in genus 1, where one can approach the problem by studying the periods of iterated integrals of modular forms and their relation to multiple zeta values.
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