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UNIVERSAL MIXED ELLIPTIC MOTIVES

Part of: $K$-theory in geometry Curves Cycles and subschemes (Co)homology theory Arithmetic algebraic geometry

Published online by Cambridge University Press:  30 April 2018

Richard Hain  and
Makoto Matsumoto
Richard Hain
Affiliation:
Department of Mathematics, Duke University, Durham, NC 27708-0320, USA (hain@math.duke.edu)
Makoto Matsumoto
Affiliation:
Graduate School of Sciences, Hiroshima University, Hiroshima 739-8526, Japan (m-mat@math.sci.hiroshima-u.ac.jp)
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Abstract

In this paper we construct a $\mathbb{Q}$-linear tannakian category $\mathsf{MEM}_{1}$ of universal mixed elliptic motives over the moduli space ${\mathcal{M}}_{1,1}$ of elliptic curves. It contains $\mathsf{MTM}$, the category of mixed Tate motives unramified over the integers. Each object of $\mathsf{MEM}_{1}$ is an object of $\mathsf{MTM}$ endowed with an action of $\text{SL}_{2}(\mathbb{Z})$ that is compatible with its structure. Universal mixed elliptic motives can be thought of as motivic local systems over ${\mathcal{M}}_{1,1}$ whose fiber over the tangential base point $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}q$ at the cusp is a mixed Tate motive. The basic structure of the tannakian fundamental group of $\mathsf{MEM}$ is determined and the lowest order terms of a set (conjecturally, a minimal generating set) of relations are deduced from computations of Brown. This set of relations includes 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.

Keywords

mixed elliptic motivemixed Tate motivemodular curvemodular formelliptic polylogarithmvariation of mixed Hodge structure

MSC classification

Primary: 14F42: Motivic cohomology; motivic homotopy theory 14F35: Homotopy theory; fundamental groups 14H52: Elliptic curves
Secondary: 11G55: Polylogarithms and relations with $K$-theory 14C30: Transcendental methods, Hodge theory , Hodge conjecture 19E20: Relations with cohomology theories
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 3 , May 2020 , pp. 663 - 766
Copyright
© Cambridge University Press 2018

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Footnotes

The research was supported in part by the Scientific Grants-in-Aid 19204002, 23244002, 15K13460, JST-CREST 151001, and by the Core-to-Core grant 18005 from the Japan Society for the Promotion of Science; and grants DMS-0706955, DMS-1005675 and DMS-1406420 from the National Science Foundation.

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