Elliptic Motives

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.


  1. (Brown) Overview: Projective line minus 3 points, I:
  2. (Hain) Moduli of elliptic curves:
  3. (Brown) Projective line minus 3 points, II:
  4. (Hain) The elliptic KZB connection and its limit MHS:
  5. (Brown) Iterated Shimura integrals:
  6. (Hain) Relative completion of \(\mathrm{SL}_2(\mathbb{Z})\):
  7. (Brown) Action of the motivic Galois group on \(G^{\mathrm{rel}}\):
  8. (Hain) What are universal mixed elliptic motives?
  9. (Brown) Multiple modular values:
  10. (Hain) Select from:


  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)

Return to: Master class: Mixed Elliptic Motives