## Algebra & Number Theory

### Cusp form motives and admissible $G$-covers

Dan Petersen

#### Abstract

There is a natural $Sn$-action on the moduli space $ℳ¯1,n(B(ℤ∕mℤ)2)$ of twisted stable maps into the stack $B(ℤ∕mℤ)2$, and so its cohomology may be decomposed into irreducible $Sn$-representations. Working over $Specℤ[1∕m‘]$ we show that the alternating part of the cohomology of one of its connected components is exactly the cohomology associated to cusp forms for $Γ(m‘)$. In particular this offers an alternative to Scholl’s construction of the Chow motive associated to such cusp forms. This answers in the affirmative a question of Manin on whether one can replace the Kuga–Sato varieties used by Scholl with some moduli space of pointed stable curves.

#### Article information

Source
Algebra Number Theory, Volume 6, Number 6 (2012), 1199-1221.

Dates
Accepted: 18 October 2011
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513729865

Digital Object Identifier
doi:10.2140/ant.2012.6.1199

Mathematical Reviews number (MathSciNet)
MR2968638

Zentralblatt MATH identifier
1303.14025

#### Citation

Petersen, Dan. Cusp form motives and admissible $G$-covers. Algebra Number Theory 6 (2012), no. 6, 1199--1221. doi:10.2140/ant.2012.6.1199. https://projecteuclid.org/euclid.ant/1513729865

#### References

• D. Abramovich, “Lectures on Gromov–Witten invariants of orbifolds”, pp. 1–48 in Enumerative invariants in algebraic geometry and string theory (Cetraro, 2005), edited by K. Behrend and M. Manetti, Lecture Notes in Math. 1947, Springer, Berlin, 2008.
• D. Abramovich and A. Vistoli, “Compactifying the space of stable maps”, J. Amer. Math. Soc. 15:1 (2002), 27–75.
• D. Abramovich, A. Corti, and A. Vistoli, “Twisted bundles and admissible covers”, Comm. Algebra 31:8 (2003), 3547–3618.
• D. Abramovich, T. Graber, M. Olsson, and H.-H. Tseng, “On the global quotient structure of the space of twisted stable maps to a quotient stack”, J. Algebraic Geom. 16:4 (2007), 731–751.
• D. Abramovich, M. Olsson, and A. Vistoli, “Twisted stable maps to tame Artin stacks”, J. Algebraic Geom. 20:3 (2011), 399–477.
• D. Blasius and J. D. Rogawski, “Motives for Hilbert modular forms”, Invent. Math. 114:1 (1993), 55–87.
• B. Conrad, “Arithmetic moduli of generalized elliptic curves”, J. Inst. Math. Jussieu 6:2 (2007), 209–278.
• C. Consani and C. Faber, “On the cusp form motives in genus 1 and level 1”, pp. 297–314 in Moduli spaces and arithmetic geometry (Kyoto, 2004), edited by S. Mukai et al., Adv. Stud. Pure Math. 45, Math. Soc. Japan, Tokyo, 2006.
• P. Deligne, “Formes modulaires et représentations $\ell$-adiques”, pp. 139–172 in Séminaire Bourbaki $1968/1969$ (Exposé 355), 1971.
• P. Deligne and M. Rapoport, “Les schémas de modules de courbes elliptiques”, pp. 143–316 in Modular functions of one variable, II (Antwerp, 1972), edited by P. Deligne and W. Kuyk, Lecture Notes in Math. 349, Springer, Berlin, 1973.
• F. Diamond and J. Shurman, A first course in modular forms, Graduate Texts in Mathematics 228, Springer, New York, 2005.
• G. Faltings, “Hodge–Tate structures and modular forms”, Math. Ann. 278 (1987), 133–149.
• E. Getzler, “Mixed Hodge structures of configuration spaces”, preprint 96-61, Max-Planck-Institut für Mathematik, Bonn, 1995.
• E. Getzler, “The semi-classical approximation for modular operads”, Comm. Math. Phys. 194:2 (1998), 481–492.
• E. Getzler and M. M. Kapranov, “Modular operads”, Compositio Math. 110:1 (1998), 65–126.
• A. Grothendieck (editor), Séminaire de Géométrie Algébrique du Bois Marie 1960–1961: Revêtements étales et groupe fondamental (SGA 1), Lecture Notes in Mathematics 224, Springer, Berlin, 1971. Reprinted Société Math. de France, Paris, 2003.
• T. J. Jarvis and T. Kimura, “Orbifold quantum cohomology of the classifying space of a finite group”, pp. 123–134 in Orbifolds in mathematics and physics (Madison, WI, 2001), edited by A. Adem et al., Contemp. Mathematics 310, Amer. Math. Soc., Providence, RI, 2002.
• N. M. Katz and B. Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies 108, Princeton University Press, 1985.
• S. L. Kleiman, “The standard conjectures”, pp. 3–20 in Motives (Seattle, WA, 1991), edited by U. Jannsen et al., Proc. Sympos. Pure Math. 55, Amer. Math. Soc., Providence, RI, 1994.
• A. Kresch and A. Vistoli, “On coverings of Deligne–Mumford stacks and surjectivity of the Brauer map”, Bull. London Math. Soc. 36:2 (2004), 188–192.
• I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, New York, 1995.
• Y. I. Manin, “Iterated Shimura integrals”, Mosc. Math. J. 5:4 (2005), 869–881, 973.
• Y. I. Manin, “Iterated integrals of modular forms and noncommutative modular symbols”, pp. 565–597 in Algebraic geometry and number theory, edited by V. Ginzburg, Progr. Math. 253, Birkhäuser, Boston, MA, 2006.
• A. Niles, “Moduli of elliptic curves via twisted stable maps”, preprint, 2012. http://www.arxiv.org/abs/1207.7280arXiv 1207.7280
• C. A. M. Peters and J. H. M. Steenbrink, Mixed Hodge structures, Ergeb. Math. Grenzgeb. (3) 52, Springer, Berlin, 2008.
• D. Petersen, “A remark on Getzler's semi-classical approximation”, pp. 309–316 in Geometry and arithmetic, edited by C. Faber et al., European Math. Soc., 2012. To appear.
• M. Romagny, “Group actions on stacks and applications”, Michigan Math. J. 53:1 (2005), 209–236.
• A. J. Scholl, “Motives for modular forms”, Invent. Math. 100:2 (1990), 419–430.
• B. Toën, “On motives for Deligne–Mumford stacks”, Internat. Math. Res. Notices 2000:17 (2000), 909–928.