11:30 to 13:00 
Jan Vonk (Leiden University, Netherlands) 
Rational points on modular curves In this course, we will study the arithmetic properties of special points on modular curves. The main focus will be on rational points on modular Jacobians arising from the theory of complex multiplication. After a brief discussion of the basics of the theory, we give an overview of some of the most important results of the latter half of the 20th century concerning the question of when such points are nontorsion. Some of the main ideas of two results in particular will be our primary focus: Mazur’s work on torsion on elliptic curves, and the work of GrossZagier on heights of Heegner points.
References:
[DI95] Diamond, Fred; Im, John. Modular forms and modular curves. Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994), 39–133, CMS Conf. Proc., 17, Amer. Math. Soc., Providence, RI, 1995.
[Gro85] Gross, Benedict H.; Zagier, Don B. On singular moduli. J. Reine Angew. Math. 355 (1985), 191–220.
[Gro87] Gross, B.; Kohnen, W.; Zagier, D. Heegner points and derivatives of Lseries. II. Math. Ann. 278 (1987), no. 14, 497–562.
[Kol91] Kolyvagin, V. A. On the structure of ShafarevichTate groups. Algebraic geometry (Chicago, IL, 1989), 94–121, Lecture Notes in Math., 1479, Springer, Berlin, 1991.
[Lan87] Lang, Serge. Elliptic functions. With an appendix by J. Tate. Second edition. Graduate Texts in Mathematics, 112. SpringerVerlag, New York, 1987.
[Maz78] Mazur, B. Modular curves and the Eisenstein ideal. With an appendix by Mazur and M. Rapoport. Inst. Hautes Etudes Sci. Publ. Math. No. 47 (1977), 33–186 (1978). ́
[MazD78] Mazur, B. Rational isogenies of prime degree (with an appendix by D. Goldfeld). Invent. Math. 44 (1978), no. 2, 129–162.
[Wei76] Weil, Andr ́e. Elliptic functions according to Eisenstein and Kronecker. Reprint of the 1976 original. Classics in Mathematics. SpringerVerlag, Berlin, 1999.
Though the instructor will recall some of the necessary material during his course, the familiarity with the following topics may be useful.
• The basic knowledge on Elliptic Curves (Silverman’s book on Arithmetic of Elliptic curves), including the method of descent, and the theory of complex multiplication.
• The basic theory of modular curves as algebraic moduli spaces, the basic theory of modular forms, Wiles’ modularity theorem.



14:30 to 16:00 
Netan Dogra (King's College London, UK) 
The ChabautyColemanKim method: from theory to practice In the first part of this course I will discuss unipotent flat connections and unipotent isocrystals on varieties, from a theoretical and computational perspective. This will lead to a notion of iterated padic integrals as introduced by Coleman and Besser. In the second part, I will describe the Chabauty–Coleman–Kim method can sometimes be used to prove that these padic integrals satisfy ‘special’ identities when evaluated on rational points. I will also give some examples of how this can lead to a method for determining the set of rational points.
References:
[Ber] Berthelot, Pierre. Cohomologie rigide et cohomologie rigide `a support propre, `aparaˆıtre dans Ast ́erisque.
[Bes12] Besser, Amnon. Heidelberg lectures on Coleman integration. The arithmetic of fundamental groups–PIA 2010, 3–52, Contrib. Math. Comput. Sci., 2, Springer, Heidelberg, 2012.
[Bes02] Besser, Amnon. Coleman integration using the Tannakian formalism. Math. Ann. 322 (2002), no. 1, 19–48.
[Ked01] Kedlaya, Kiran S. Counting points on hyperelliptic curves using MonskyWashnitzer cohomology. J. Ramanujan Math. Soc. 16 (2001), no. 4, 323–338.
[Kim05] Kim, Minhyong. The motivic fundamental group of P 1 \ {0, 1, ∞} and the theorem of Siegel. Invent. Math. 161 (2005), no. 3, 629–656.
[Kim09] Kim, Minhyong. The unipotent Albanese map and Selmer varieties for curves. Publ. Res. Inst. Math. Sci. 45 (2009), no. 1, 89–133.



16:30 to 18:00 
Eknath Ghate (TIFR, Mumbai, India) 
Introduction to Modular Forms, Elliptic Curves, and Modular Curves. In this minicourse, we shall give a basic introduction to the topics listed in the title. In the first lecture, we shall introduce the main players and state the modularity theorem for elliptic curves. In the second lecture, we shall introduce Hecke operators on modular forms and modular curves. In the third lecture, we shall introduce the Jacobians of modular curves. In the last lecture, we shall describe how integral models of modular curves can be used in conjunction with the $q$expansion principle to give an integral structure to the space of modular forms.
References:
1. Diamond, Fred; Shurman, Jerry. A first course in modular forms. Graduate Texts in Mathematics, 228. SpringerVerlag, New York, 2005.
2. Diamond, Fred; Im, John. Modular forms and modular curves. Seminar on Fermat's Last Theorem (Toronto, ON, 1993–1994), 39133, CMS Conf. Proc., 17, Amer. Math. Soc., Providence, RI, 1995.



18:45 to 19:45 
Jennifer Balakrishnan (Boston University, USA) 
Computational aspects of nonabelian Chabauty (workshop lecture course) We give an overview of the computational tools that are useful for studying various aspects of the ChabautyColeman and nonabelian Chabauty methods. We will illustrate carrying out the method on various modular curves.


