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  • Richard Hain (a1) and Makoto Matsumoto (a2)

In this paper we construct a $\mathbb{Q}$ -linear tannakian category $\mathsf{MEM}_{1}$ of universal mixed elliptic motives over the moduli space ${\mathcal{M}}_{1,1}$ of elliptic curves. It contains $\mathsf{MTM}$ , the category of mixed Tate motives unramified over the integers. Each object of $\mathsf{MEM}_{1}$ is an object of $\mathsf{MTM}$ endowed with an action of $\text{SL}_{2}(\mathbb{Z})$ that is compatible with its structure. Universal mixed elliptic motives can be thought of as motivic local systems over ${\mathcal{M}}_{1,1}$ whose fiber over the tangential base point $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}q$ at the cusp is a mixed Tate motive. The basic structure of the tannakian fundamental group of $\mathsf{MEM}$ is determined and the lowest order terms of a set (conjecturally, a minimal generating set) of relations are deduced from computations of Brown. This set of relations includes the arithmetic relations, which describe the ‘infinitesimal Galois action’. We use the presentation to give a new and more conceptual proof of the Ihara–Takao congruences.

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1. Arapura, D., An abelian category of motivic sheaves, Adv. Math. 233 (2013), 135195.
2. Ayoub, J., A Guide to (Étale) Motivic Sheaves, Proceedings of the International Congress of Mathematicians, Seoul 2014, Volume II, pp. 11011124 (Kyung Moon Sa, Seoul).
3. Baumard, S. and Schneps, L., On the derivation representation of the fundamental Lie algebra of mixed elliptic motives, Ann. Math. Qué. 41 (2017), 4362.
4. Beilinson, A., Higher Regulators and Values of L-Functions, Current Problems in Mathematics, Volume 24, pp. 181238 (Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984).
5. Beilinson, A., Higher regulators of modular curves, in Applications of Algebraic K-Theory to Algebraic Geometry and Number theory, Parts I, II (Boulder, CO, 1983), Contemporary Mathematics, Volume 55, pp. 134 (American Mathematical Society, Providence, RI, 1986).
6. Beilinson, A., Notes on absolute Hodge cohomology, in Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Parts I, II (Boulder, CO, 1983), Contemporary Mathematics, Volume 55, pp. 3568 (American Mathematical Society, Providence, RI, 1986).
7. Beilinson, A. and Levin, A., The elliptic polylogarithm, in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., Volume 55, Part 2, pp. 123190 (American Mathematical Society, Providence, RI, 1994).
8. Borel, A., Cohomologie de SL n et valeurs de fonctions zeta aux points entiers, Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (1977), 613636.
9. Brown, F., Multiple zeta values and periods of moduli spaces M 0, n , Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 371489.
10. Brown, F., Mixed Tate motives over ℤ, Ann. of Math. (2) 175 (2012), 949976.
11. Brown, F., Multiple modular values for , Preprint, 2014, arXiv:1407.5167.
12. Brown, F., Zeta elements in depth 3 and the fundamental Lie algebra of the infinitesimal Tate curve, Forum Math. Sigma 5 (2017), e1, 56 pp.
13. Calaque, D., Enriquez, B. and Etingof, P., Universal KZB equations I: the elliptic case, in Algebra, Arithmetic, and Geometry: in Honor of Yu. I. Manin, Volume I, Progr. Math., Volume 269, pp. 165266 (Birkhäuser, Boston, 2009).
14. Deligne, P., La conjecture de Weil, II, Publ. Math. Inst. Hautes Études Sci. 52 (1980), 137252.
15. Deligne, P., Le groupe fondamental de la droite projective moins trois points, in Galois Groups Over ℚ (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., Volume 16, pp. 79297 (Springer, 1989).
16. Deligne, P., Le groupe fondamental unipotent motivique de G m -𝜇 N , pour N = 2, 3, 4, 6 ou 8, Publ. Math. Inst. Hautes Études Sci. 112 (2010), 101141.
17. Deligne, P. and Goncharov, A., Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), 156.
18. Drinfeld, V., On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(/ℚ), Algebra i Analiz 2 (1990), 149181; translation in Leningrad Math. J. 2 (1991), 829–860.
19. Enriquez, B., Elliptic associators, Selecta Math. (N.S.) 20 (2014), 491584.
20. Fontaine, J.-M. and Perrin-Riou, B., Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions L , in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., Volume 55, Part 1, pp. 599706 (American Mathematical Society, 1994).
21. Goncharov, A., Mixed elliptic motives, in Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996), London Mathematical Society Lecture Note Series, Volume 254, pp. 147221 (Cambridge University Press, 1998).
22. Goncharov, A., The dihedral Lie algebras and Galois symmetries of 𝜋1 ()1 - ({0, }∪𝜇 N )), Duke Math. J. 110 (2001), 397487.
23. Hain, R., Hodge–de Rham theory of relative Malcev completion, Ann. Sci. Éc. Norm. Supér. 31 (1998), 4792.
24. Hain, R., Relative weight filtrations on completions of mapping class groups, in Groups of Diffeomorphisms, Adv. Stud. Pure Mathematics, Volume 52, pp. 309368 (Math. Soc., Japan, Tokyo, 2008).
25. Hain, R., Letter to P. Deligne, December, 2009.
26. Hain, R., Lectures on moduli spaces of elliptic curves, in Transformation Groups and Moduli Spaces of Curves, Adv. Lect. Math. (ALM), Volume 16, pp. 95166 (International Press, 2011).
27. Hain, R., Notes on the Universal Elliptic KZB Equation, Pure and Applied Mathematics Quarterly, Volume 12, no. 2, (International Press, Somerville, MA, 2016).
28. Hain, R., The Hodge–de Rham theory of modular groups, in Recent Advances in Hodge Theory Period Domains, Algebraic Cycles, and Arithmetic (ed. Kerr, M. and Pearlstein, G.), LMS Lecture Notes Series, Volume 427, pp. 422514 (Cambridge University Press, Cambridge, 2016).
29. Hain, R., Deligne–Beilinson cohomology of affine groups, in Hodge Theory and L 2 Methods (ed. Ji, L. and Zucker, S.), pp. 377418 (International Press, Somerville, MA).
30. Hain, R., Unipotent path torsors of Ihara curves, in preparation.
31. Hain, R. and Matsumoto, M., Weighted completion of Galois groups and Galois actions on the fundamental group of ¶1 -{0, 1, }, Compositio Math. 139 (2003), 119167.
32. Hain, R. and Matsumoto, M., Tannakian fundamental groups associated to Galois groups, in Galois Groups and Fundamental Groups, Math. Sci. Res. Inst. Publ., Volume 41, pp. 183216 (Cambridge University Press, 2003).
33. Hain, R. and Matsumoto, M., Relative pro-l completions of mapping class groups, J. Algebra 321 (2009), 33353374.
34. Hain, R. and Zucker, S., Unipotent variations of mixed Hodge structure, Invent. Math. 88 (1987), 83124.
35. Hanamura, M., Mixed motives and algebraic cycles, I, Math. Res. Lett. 2 (1995), 811821.
36. Ihara, Y., Some arithmetic aspects of Galois actions in the pro-p fundamental group of ¶1 -{0, 1, }, in Arithmetic Fundamental Groups and Noncommutative Algebra (Berkeley, CA, 1999), Proceedings of Symposia in Pure Mathematics, Volume 70, pp. 247273 (American Mathematical Society, 2002).
37. Jantzen, J. C., Representations of Algebraic Groups, Pure and Applied Mathematics, Volume 131 (Academic Press, 1987).
38. Katz, N. and Mazur, B., Arithmetic Moduli of Elliptic Curves, Annals of Mathematics Studies, Volume 108 (Princeton University Press, 1985).
39. Knudsen, F., The projectivity of the moduli space of stable curves, III. The line bundles on M g, n , and a proof of the projectivity of M g, n in characteristic 0, Math. Scand. 52 (1983), 200212.
40. Lang, S., Introduction to Modular Forms, with appendixes by D. Zagier and Walter Feit, Grundlehren der Mathematischen Wissenschaften, Volume 222 (Springer, 1995). Corrected reprint of the 1976 original.
41. Levine, M., Tate motives and the vanishing conjectures for algebraic K-theory, in Algebraic K-Theory and Algebraic Topology (Lake Louise, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Volume 407, pp. 167188 (Kluwer, 1993).
42. Levine, M., Mixed Motives, Mathematical Surveys and Monographs, Volume 57 (American Mathematical Society, 1998).
43. Levin, A. and Racinet, G., Towards multiple elliptic polylogarithms, Preprint, 2007,arXiv:math/0703237.
44. Luo, M., The elliptic KZB connection and algebraic de Rham theory for unipotent fundamental groups of elliptic curves, Preprint, 2017, arXiv:1710.07691.
45. Nakamura, H., Tangential base points and Eisenstein power series, in Aspects of Galois Theory (Gainesville, FL, 1996), London Mathematical Society Lecture Note Series, Volume 256, pp. 202217 (Cambridge University Press, 1999).
46. May, P., Matric Massey products, J. Algebra 12 (1969), 533568.
47. Noohi, B., Fundamental groups of algebraic stacks, J. Inst. Math. Jussieu 3 (2004), 69103.
48. Olsson, M., Towards non-abelian p-adic Hodge theory in the good reduction case, Mem. Amer. Math. Soc. 210(990) (2011), vi+157 pp.
49. Pollack, A., Relations between derivations arising from modular forms, undergraduate thesis, Duke University, 2009. Available at:
50. Schmid, W., Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211319.
51. Schneps, L., On the Poisson bracket on the free Lie algebra in two generators, J. Lie Theory 16 (2006), 1937.
52. Scholl, A., Motives for modular forms, Invent. Math. 100 (1990), 419430.
53. Silverman, J., The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, Volume 106 (Springer, 1986).
54. Silverman, J., Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, Volume 151 (Springer, New York, 1994).
55. Soulé, C., On higher p-adic regulators, in Algebraic K-Theory, Evanston 1980 (Proc. Conf., Northwestern University, Evanston, IL, 1980), Lecture Notes in Mathematics, Volume 854, pp. 372401 (Springer, 1981).
56. Steenbrink, J. and Zucker, S., Variation of mixed Hodge structure, I, Invent. Math. 80 (1985), 489542.
57. Takao, N., Braid monodromies on proper curves and pro- Galois representations, J. Inst. Math. Jussieu 11 (2012), 161181.
58. Terasoma, T., Relative Deligne cohomologies and higher regulators for Kuga–Sato fiber spaces, Preprint, January 2011.
59. Tsunogai, H., On some derivations of Lie algebras related to Galois representations, Publ. Res. Inst. Math. Sci. 31 (1995), 113134.
60. Voevodsky, V., Suslin, A. and Friedlander, E., Cycles, Transfers, and Motivic Homology Theories, Annals of Mathematics Studies, Volume 143 (Princeton University Press, 2000).
61. Wasow, W., Asymptotic Expansions for Ordinary Differential Equations, Pure and Applied Mathematics, Volume XIV (Interscience Publishers, 1965).
62. Zucker, S., Hodge theory with degenerating coefficients, L 2 cohomology in the Poincaré metric, Ann. of Math. (2) 109 (1979), 415476.
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