Elliptic Motives

Stockholm, May 2019

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

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)