## Stockholm, May 2019

Instructors: Francis Brown (Oxford) and Richard Hain (Duke)

Mixed elliptic motives are motives associated to an elliptic curve $$E$$. They are iterated extensions of Tate twists of symmetric powers of $$H_1(E)$$. Mixed modular motives are iterated extensions of motives associated to (classical) modular forms. The unipotent completion of the fundamental group of a once punctured elliptic curve is a rich source of mixed elliptic motives. The relative completion of the fundamental group of a modular curve (a modular group) is a good source of mixed modular motives. The goal of these lecture series is to construct putative mixed elliptic motives and mixed modular motives in (relative) unipotent completions of fundamental groups of moduli spaces of elliptic curves and also to explain how mixed modular motives should control mixed elliptic motives.

Lectures:

1. (Brown) Overview: Projective line minus 3 points, I:
• iterated integrals and the KZ connection
• multiple zeta values (MZV)
• unipotent fundamental groups, Tate case
2. (Hain) Moduli of elliptic curves:
• the unipotent fundamental group of a punctured elliptic curve and its MHS
• moduli spaces of elliptic curves: $$\mathcal{M}_{1,1}$$, $$\mathcal{M}_{1,\vec{1}}$$, and $$\mathcal{M}_{1,2}$$
• the local system $$\mathbb H$$
3. (Brown) Projective line minus 3 points, II:
• Tannakian considerations: mixed Hodge structures, motives, and motivic periods
• Computation of action of motivic Galois group on $$\pi_1^{\mathrm{un}}(\mathbb{P}^1-\{0,1,\infty\},\vec{1})$$
4. (Hain) The elliptic KZB connection and its limit MHS:
• the elliptic KZB connection
• limit MHSs and regularized periods (notes)
• the limit MHS on $$\pi_1^{\mathrm{un}}(E_{\partial/\partial q}',\vec{v})$$
5. (Brown) Iterated Shimura integrals:
• Eichler-Shimura
• a universal connection
• the non-abelian cocycle
6. (Hain) Relative completion of $$\mathrm{SL}_2(\mathbb{Z})$$:
• definition and basic properties
• construction of the MHS (sketch)
• variations of MHS over $$\mathcal{M}_{1,1}$$
7. (Brown) Action of the motivic Galois group on $$G^{\mathrm{rel}}$$:
• review of the case of $$\mathbb{P}^1-\{0,1,\infty\}$$
• conjectural shape of the fundamental group of mixed modular motives
• the Galois action on $$G^{\mathrm{rel}}$$
8. (Hain) What are universal mixed elliptic motives?
• Eisenstein variations of MHS
• limit MHS at $$\partial/\partial q$$
• the Eisenstein quotient of $$G^{\mathrm{rel}}$$
9. (Brown) Multiple modular values:
• period computation using Rankin-Selberg
• transference
• the Pollack relations
10. (Hain) Select from:
• modular inverter(s)
• Goldman Turaev story

References:

1. F. Brown: Mixed Tate motives over $$\mathbb Z$$, Ann. of Math. (2) 175 (2012), 949-976, arXiv:1102.1312
2. F. Brown: Multiple Modular Values and the relative completion of the fundamental group of $$\mathcal{M}_{1,1}$$, arXiv:1407.5167
3. D. Calaque, B. Enriquez, P. Etingof: Universal KZB equations: the elliptic case, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, 165-266, Progr. Math. 269, Birkhäuser, 2009, arXiv:math/0702670
4. K.-T. Chen: Iterated path integrals Bull. Amer. Math. Soc. 83 (1977), 831-879.
5. B. Enriques: Elliptic associators, Selecta Math. (N.S.) 20 (2014), 491-584, arXiv:1003.1012
6. R. Hain: The geometry of the mixed Hodge structure on the fundamental group, Algebraic geometry, Bowdoin, 1985, 247-282, Proc. Sympos. Pure Math., 46, 1987, (pdf)
7. R. Hain: Lectures on moduli spaces of elliptic curves, Transformation groups and moduli spaces of curves, 95-166, Adv. Lect. Math. (ALM), 16, Int. Press, 2011, arXiv:0812.1803
8. R. Hain: Notes on the Universal Elliptic KZB Connection, PAMQ, to appear, arXiv:1309.0580
9. R. Hain: The Hodge-de Rham theory of modular groups, Recent advances in Hodge theory, 422-514, London Math. Soc. Lecture Note Ser., 427, Cambridge Univ. Press, 2016, arXiv:1403.6443
10. A. Levin, G. Racinet: Towards multiple elliptic polylogarithms, arXiv:math/0703237
11. Y. Manin: Iterated Shimura integrals, Mosc. Math. J. 5 (2005), 869-881, arXiv:math/0507438
12. Y. Manin: Iterated integrals of modular forms and noncommutative modular symbols, Algebraic geometry and number theory, 565-597, Progr. Math., 253, Birkhäuser Boston, Boston, MA, 2006, arXiv:math/0502576
13. S. Zucker: Hodge theory with degenerating coefficients. $$L_2$$ cohomology in the Poincaré metric, Ann. of Math. (2) 109 (1979), 415--476. (JSTOR)