[Paper Review] Motives for elliptic modular groups

Takashi Ichikawa|arXiv (Cornell University)|Sep 29, 2009

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper introduces a novel theory of Hecke operators on elliptic modular motives constructed from the fundamental group of compactified modular curves. By defining Hecke actions via quotients of the universal elliptic curve, the authors establish a congruence with Galois actions and prove that these motives decompose into pure motives over number fields, implying algebraicity of iterated Shimura integrals and periods of modular Ceresa cycles.

ABSTRACT

In the study of the arithmetic structure of elliptic modular groups which are the fundamental groups of compactified modular curves, these truncated group algebras and their direct sums are considered to construct elliptic modular motives. Our main result is a new theory of Hecke operators on these motives which gives a congruence relation to the Galois action, and their motivic decomposition. Using our Hecke theory, we show that elliptic modular motives are the direct sums of pure motives over certain number fields. This fact implies a kind of algebraicity on iterated Shimura integrals, i.e., multiple L-values of cusp forms of weight 2, and on the periods of modular Ceresa cycles.

Motivation & Objective

• To develop a motivic Hecke theory for elliptic modular groups that is compatible with Galois actions.
• To show that elliptic modular motives decompose into pure motives over number fields, enabling algebraicity results.
• To apply the Hecke theory to iterated Shimura integrals of cusp forms of weight 2 and to periods of modular Ceresa cycles.
• To provide a motivic framework for understanding the algebraicity of multiple L-values and divisor periods on modular curves.

Proposed method

• Construct elliptic modular motives as direct sums of truncated group algebras of the fundamental group of compactified modular curves at cusps.
• Define Hecke operators via quotients of the universal elliptic curve by finite subgroups, ensuring well-defined action on the motive.
• Establish a congruence relation between the Hecke action and the Galois action on the l-adic realization of the motive.
• Use the motivic decomposition into Hecke eigenspaces (scalar plus nilpotent) to analyze the structure of the motive.
• Apply results on algebraic fundamental groups and Malcev Lie algebras to verify the well-definedness of the Hecke action.
• Leverage mixed Hodge structures and Ext groups in the category of mixed Q-Hodge structures to analyze the Ceresa cycle and divisor periods.

Experimental results

Research questions

• RQ1 How can Hecke operators be defined on motivic fundamental groups of modular curves in a way compatible with Galois actions?
• RQ2 To what extent do elliptic modular motives decompose into pure motives over number fields?
• RQ3 What algebraicity properties do iterated Shimura integrals of cusp forms of weight 2 exhibit?
• RQ4 Are the periods of modular Ceresa cycles algebraic, and how can this be shown via motivic methods?
• RQ5 Can the nontriviality of the modular Ceresa cycle be detected through Hecke-invariant periods?

Key findings

• The Hecke operators on the elliptic modular motive satisfy a congruence relation with the Galois action, ensuring compatibility with the l-adic realization.
• The elliptic modular motive decomposes as a direct sum of pure motives over a number field, implying that its motivic structure is semisimple after base change.
• Iterated Shimura integrals of weight 2, which represent multiple L-values of cusp forms, are preserved under the Hecke action, providing a solution to a problem on their action.
• The periods of modular Ceresa cycles are algebraic, as shown by the triviality of associated Ext classes in the category of mixed Q-Hodge structures over Q-bar.
• The divisor $ D_0 $ associated to the Ceresa cycle has infinite order modulo torsion if any coefficient $ a_i $ in its period integral is irrational, which can be detected via Hecke computation.
• The motivic splitting of the exact sequence associated to the Ceresa cycle implies that the corresponding extension class is trivial in the category of mixed Q-Hodge structures over Q-bar.