This graduate mini-course was part of an initiative of the London Mathematical Society (LMS) and took place in Oxford, during the summer activities of the WORKing seminar in 2020. The lectures were live-streamed through Zoom to students in Oxford, Reading, and Warwick.

Schedule

  1. Friday 11/09/2020, 13:30 - 15:30
  2. Tuesday 15/09/2020, 13:30 -15:30
  3. Friday 18/09/2020, 13:30 - 15:30

The lectures will be given on Zoom. Let me know if you want a link and password. Update: Recordings of the lectures are now available on the LMS YouTube channel, check the links below.

Course description

This mini-course is part of a lecture series funded by the LMS and aimed at PhD students. It is an introduction to the relation between modular forms and cohomology (‘motives’) through the de Rham persective.

We will start with an introduction to some basic analytic aspects concerning modular forms and to their interpretation as sections of line bundles on modular curves. We will also explain how to algebraise this picture and how to work with the moduli stack of elliptic curves.

Our main goal is to attach certain 2-dimensional cohomology groups to Hecke eigenforms. This course will mainly deal with algebraic de Rham cohomology (which will be explained), but it can also serve to build geometric intuition to the l-adic setting, which gives rise to the famous l-adic representations attached to modular forms.

If time permits, we will also discuss the Eichler-Shimura isomorphism, periods of modular forms, and Manin’s theorem on the critical values of L-functions of modular forms.

Prerequisites: the basics of scheme theory and sheaf cohomology (e.g. Hartshorne’s chapters 2 and 3, or Liu’s chapters 2, 3 and 5).

Lecture notes

Please, beware that the following lecture notes are not polished and might contain mistakes.

  1. Modular forms and the moduli stack of complex tori
  2. Moduli stack of elliptic curves, algebraicity of modular forms, and de Rham cohomology
  3. De Rham bundle, twisted cohomology, and Eichler-Shimura

Videos

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3

References and further reading

Introduction to modular forms

A wonderful introduction to modular forms is the chapter 7 of [Serre73]. For an overview of the basic theory with many applications, see [Zagier08]. The book [Lang95] contains an exposition on periods of modular forms and Manin’s theorem. For a more complete discussion of modular curves and Hecke operators, [DS05] is a good modern reference.

  • [DS05] Diamond, Fred; Shurman, Jerry, A first course in modular forms. Graduate Texts in Mathematics, 228. Springer-Verlag, New York, 2005.
  • [Lang95] Lang, Serge, Introduction to modular forms. With appendices by D. Zagier and Walter Feit. Corrected reprint of the 1976 original. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 222. Springer-Verlag, Berlin, 1995.
  • [Serre73] Serre, J.-P., A course in arithmetic. Translated from the French. Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973.
  • [Zagier08] Zagier, Don, Elliptic modular forms and their applications. In The 1-2-3 of modular forms, 1-103, Universitext, Springer, Berlin, 2008.

Stacks

Although I will mention stacks, we will not use much of the general theory. For the reader wishing to know more, Vistoli’s article on [FGA05] is a great introduction to fibered categories and stacks. The appendix of [Vistoli89] is the fastest introduction to Deligne-Mumford stacks I know. For a more comprehensive exposition on algebraic stacks, see [Olsson16].

  • [FGA05] Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, Fundamental algebraic geometry. Grothendieck’s FGA explained. Mathematical Surveys and Monographs 123, Amer. Math. Soc. 2005.
  • [Olsson16] Olsson, Martin, Algebraic spaces and stacks. American Mathematical Society Colloquium Publications, 62. American Mathematical Society, Providence, RI, 2016.
  • [Vistoli89] Vistoli, Angelo, Intersection theory on algebraic stacks and on their moduli spaces. Invent. Math. 97 (1989), no. 3, 613-670.

Algebraic de Rham cohomology

The algebraic de Rham cohomology was first considered by Grothendieck in a letter to Atiyah [Grothendieck66]. The recent textbook [HMS17] contains an exposition on algebraic de Rham cohomology, the comparison isomorphism, and much more. For the Gauss-Manin connection, a classic reference is [KO68].

  • [Grothendieck66] Grothendieck, A., On the de Rham cohomology of algebraic varieties. Inst. Hautes Études Sci. Publ. Math. No. 29 (1966), 95-103.
  • [HMS17] Huber, Annette; Müller-Stach, Stefan, Periods and Nori motives. With contributions by Benjamin Friedrich and Jonas von Wangenheim. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 65. Springer, Cham, 2017.
  • [KO68] Katz, Nicholas M.; Oda, Tadao, On the differentiation of de Rham cohomology classes with respect to parameters. J. Math. Kyoto Univ. 8 (1968), 199-213.

Modular curves and geometric interpretation of modular forms

A good and classic introduction is [Katz73] (see Chapter 1 and Appendix A). Authoritative sources are [DP72] and [KM85]. For modular curves as Riemann surfaces, see [DS05] cited above.

  • [DP72] Deligne, P.; Rapoport, M., Les schémas de modules de courbes elliptiques. (French) Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 143-316. Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973.
  • [Katz73] Katz, Nicholas M., $p$-adic properties of modular schemes and modular forms. Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 69-190. Lecture Notes in Mathematics, Vol. 350, Springer, Berlin, 1973.
  • [KM85] Katz, Nicholas M.; Mazur, Barry, Arithmetic moduli of elliptic curves. Annals of Mathematics Studies, 108. Princeton University Press, Princeton, NJ, 1985.

Further references

  • Brown, Francis; Hain, Richard, Algebraic de Rham theory for weakly holomorphic modular forms of level one. Algebra Number Theory 12 (2018), no. 3, 723-750.
  • Coleman, R. F., Classical and overconvergent modular forms. Invent. Math. 124 (1996), no. 1-3, 215-241.
  • Guerzhoy, P., Hecke operators for weakly holomorphic modular forms and supersingular congruences. Proc. Amer. Math. Soc. 136 (2008), no. 9, 3051-3059.
  • M. Kazalicki, A. J. Scholl, Modular forms, de Rham cohomology and congruences. Trans. Amer. Math. Soc. 368 (2016), no. 10, 7097-7117.
  • A. J. Scholl, Motives for modular forms. Invent. Math. 100 (1990), no. 2, 419-430.