Cyclotomic and Elliptic Polylogarithms and Motivic Extensions of Q by Q(m)

Deligne and Goncharov constructed a neutral tannakian category of mixed Tate motives MTM_N unramified over Z[mu_N,1/N]. Brown and Hain--Matsumoto computed the first order action of its motivic Galois group on the unipotent fundamental group of a smoothing of the once punctured Tate elliptic curve when N = 1. In this thesis, we extend their work to N > 1 by computing the first order action of the motivic Galois group on the unipotent fundamental group of a smoothing of the Tate elliptic curve with a cyclic subgroup of order N removed.

One of our main tools is the elliptic KZB connection for the level N congruence subgroup of SL_2(Z). We adapt it to the arithmetic setting and prove that it underlies an admissible variation of mixed Hodge structure over the corresponding universal elliptic curve with its N-torsion removed. At the singular fiber above the distinguished cusp q = 0, we show that the KZB connection degenerates to the cyclotomic KZ connection and that the variation degenerates to a mixed Hodge structure that contains the Lie algebra of the motivic fundamental group pi_1^mot(P^1 - {0,mu_N,infty},v) studied by Deligne and Goncharov. This observation allows us to construct a higher level analogue of the Hain map. We further prove this map is a morphism in MTM_N by describing the Galois action on the motivic periods of the unipotent fundamental group of the punctured Tate curve.

The inclusion of pi_1^mot(P^1 - {0,mu_N,infty},v) into the limit mixed Hodge structure of the KZB variation allows us to relate the periods of the N-cyclotomic polylog variation of MHS to the periods of the level N elliptic polylog variation of MHS. This enables us to give a formula for the first order action of the Galois group of MTM_N on the elliptic polylogarithm. This is most explicit when N is a prime power.