Universal Mixed Elliptic Motives

TL;DR

This paper constructs a universal category of mixed elliptic motives over the moduli space of elliptic curves, analyzing its fundamental group and relations, and providing new proofs of known congruences.

Contribution

It introduces a new tannakian category of universal mixed elliptic motives and determines its fundamental group structure and relations, including arithmetic ones.

Findings

Determined the structure of the tannakian fundamental group of MEM_1.

Found lowest order relations and arithmetic relations describing the infinitesimal Galois action.

Provided a new proof of the Ihara-Takao congruences.

Abstract

In this paper we construct a Q-linear tannakian category MEM_1 of universal mixed elliptic motives over the moduli space M_{1,1} of elliptic curves. It contains MTM, the category of mixed Tate motives unramified over the integers. Each object of MEM_1 is an object of MTM endowed with an action of SL_2(Z) that is compatible with its structure. Universal mixed elliptic motives can be thought of as motivic local systems over M_{1,1} whose fiber over the tangential base point d/dq at the cusp is a mixed Tate motive. The basic structure of the tannakian fundamental group of MEM is determined and the lowest order terms of all relations are found (using computations of Francis Brown), including the arithmetic relations, which describe the "infinitesimal Galois action". We use the presentation to give a new and more conceptual proof of the Ihara-Takao congruences.