# Mini-Workshop: Elliptic Multiple Zeta Values and Mixed Elliptic Motives

This is supported by JSPS KAKENHI JP18H01110.

## Date

28 (Sat). July. 2018.

## Organizer

Hidekazu Furusho (Nagoya University)

## Place

Rm A428, Science Building A (not Math Building!), Nagoya University (access).

## Program

28 (Sat). July. 2018.
13:30-14:30 Martin Gonzalez (IMJ, Paris), On the ellipsitomic KZB associator
15:00-16:00 Nils Matthes (Kyushu Univ), An algebraic characterization of the Kronecker function
16:30-17:30 Kenji Sakugawa (RIMS, Kyoto), On mixed elliptic motives over modular curves

## Abstracts

Martin Gonzalez (IMJ, Paris)
Title: On the ellipsitomic KZB associator
Abstract: The universal elliptic KZB connection has a twisted (or cyclotomic) counterpart. This is a flat connection defined on a G-principal bundle over the moduli space of elliptic curves with n marked points and a (M,N)-level structure. Here the Lie algebra associated to G is constructed from a twisted elliptic Kohno-Drinfeld Lie algebra, the Lie algebra $\frak{sl}_2$, and a twisted derivation algebra controlling the algebraic information of some modular forms. After presenting this connection I will retrieve an ellipsitomic (or twisted elliptic) KZB associator from its monodromy and elliptic multiple-zeta values at torsion points from the coefficients of this associator. Some parts of the results come from a joint work with Damien Calaque.

Nils Matthes (Kyushu Univ)
Title: An algebraic characterization of the Kronecker function
Abstract: Classical multiple zeta values can be written as iterated integrals of the differential forms $\frac{dz}{z}$ and $\frac{dz}{z-1}$. In a similar vein, elliptic multiple zeta values are defined as iterated integrals of (Laurent coefficients of) the form $F_{\tau}(\xi,\alpha)d\xi$ where $F_{\tau}$ is the Kronecker function (viewed as a Laurent series in $\alpha$).
In this talk, we characterize the Kronecker function using the Fay identity which is an elliptic analog of the partial fraction identities. If time permits, we give a similar characterization of the generating series of periods of modular forms studied by Zagier.

Kenji Sakugawa (RIMS, Kyoto)
Title: On mixed elliptic motives over modular curves
Abstract: Let Y be a modular curve. The category of elliptic motives over Y is a full subcategory of motivic local systems on Y in the sense of Arapura. In this talk, we give a structure theorem of the Tannakian fundamental group of this category. We also formulate a conjecture about the p-adic etale realization of the Tannakian fundamental group and see consequences of the conjecture. If time permits, I will explain a relation between our category and the category of universal mixed elliptic motives introduced by Hain and Matsumoto.

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