# Program

Program of the school (28th august - 2nd september).

Mon Tue Wed Thu Fri
9:30-10:30 Vigni Vigni Venerucci Brasca Barrera
11:00-12:00 Vigni Vigni Venerucci Barrera Seveso
14:15-15:15 Longo Longo Brasca Barrera Seveso
15:30-16:30 Longo Venerucci Brasca Seveso

1. Modular curves and modular forms. (S. Vigni)
We plan to begin the summer school with a self contained introduction to modular curves, viewed as Riemann surfaces and as solutions of moduli problems. We will also give an self contained introduction to modular forms and Hecke algebras. We will finally state the main results concerning the Galois representations attached to modular forms, and sketch the construction in the case of modular forms of weight two. No prerequisites for this part, except basic knowledge of complex analysis, including Riemann surfaces.
2. Serre p-adic modular forms. (M. Longo)
Congruences between Fourier expansions of modular forms were first considered by Swinnerton-Dyer, in [3]. Developing his ideas, Serre introduced in [4] p-adic modular forms as p-adic limits of Fourier expansions of classical modular forms, deriving important results on Kubota-Leopoldt p-adic L-functions. Prerequisites: knowledge of basic properties of p-adic numbers.
3. Hida families (R. Venerucci)
The approach of Serre to p-adic modular forms, whose main example was that of p-adic families of Eisenstein series, was extended and generalized by Hida (see for example [5]) in the case of ordinary forms. This led to the notion of p-adic families of cusp forms, and families of Galois representations attached to them.
4. Katz p-adic modular forms. (R. Brasca)
Katz point of view on p-adic modular forms is more geometric that that the point of view adopted by Serre, since it is heavily based on the interpretation of modular curves as moduli spaces for elliptic curves equipped with a level structure. This geometric point of view makes use of powerful tools from arithmetic and algebraic geometry.
5. Overconvergent modular forms and Coleman families. (D. Barrera)
Using the language of Katz, and making use of rigid geometry, Coleman introduced the notion of overconvergent modular forms, studying their main properties and proving important results concerning the relation between classical and overconvergent modular forms. Building on this, Coleman also introduced families of modular forms, which can be seen as a natural extension of Hida families in non-ordinary situations.
6. The Eigencurve. (M. Seveso)
Coleman and Mazur introduced and studied, using rigid geometry and deformation theory, a geometric object (called ''the eigencurve'') which classifies Coleman families of modular forms.

References:
[1] Shimura, Goro. Introduction to the arithmetic theory of automorphic functions. Publications of the Mathematical Society of Japan, 11. Kanô Memorial Lectures, 1.
Princeton University Press, Princeton, NJ, 1994.
[2] F. Diamond, J. Shurman. A first course in modular forms. Graduate Texts in Mathematics, 228. Springer-Verlag, New York, 2005. xvi+436 pp.
[3] H.P.F. Swinnerton-Dyer. On l-adic representations and congruences for coefficients of modular forms. Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), pp. 1-55. Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973.
[4] J.-P. Serre. Formes modulaires et fonctions zêta p-adiques. Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), pp. 191-268. Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973.
[5] H. Hida. Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms. Invent. Math. 85 (1986), no. 3, 545-613.
[6] N. Katz. 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.
[7] R. Coleman. Classical and overconvergent modular forms. Invent. Math. 124 (1996), no. 1-3, 215-24.
[8] Robert Coleman, P-adic Banach spaces and families of modular forms, Invent. Math. 127, 417-479 (1997).
[9] R. Coleman, B. Mazur. The eigencurve. Galois representations in arithmetic algebraic geometry (Durham, 1996), 1-113, London Math. Soc. Lecture Note Ser., 254, Cambridge Univ. Press, Cambridge, 1998.

Program of the workshop (4- 6 september).

Mon Tue Wed
9:30-11:00 Andreatta Chojecki Barrera
11:30-13:00 Hernandez Johansson Brasca
14:30-16:00 Birkbeck Newton

Fabrizio Andreatta
Title Nearly overconvergent modular forms and applications.
Abstract (joint work with A. Iovita):I will start by reviewing the construction of $p$-adic families of modular forms due to myself, A. Iovita and V. Pilloni, I will then explain how to generalize this approach to get the $p$-adic interpolation of symmetric powers of the relative de Rham cohomology of the universal ellitpic curve over the relevant modular curve. I will provide applications to the definition of nearly overconvergent modular forms and to the construction of triple product $L$-functions for finite slope families of modular forms.

Daniel Barrera
Title: On the exceptional zeros of p-adic L-functions of Hilbert modular forms
Abstract: The use of modular symbols to attach p-adic L-functions to Hecke eigenforms goes back to the work of Manin et al in the 70s. In the 90s, Stevens developed his theory of overconvergent modular symbols, which was successfully used to construct p-adic L-functions on the eigenvariety. In this talk we will present a work in collaboration with Mladen Dimitrov and Andrei Jorza in which we generalize this approach to the Hilbert modular setting and prove new instances of the exceptional zero conjecture.

Riccardo Brasca
Title : Eigenvarieties for non-cuspidal Siegel modular forms.
Abstract : In a recent work Andreata, Iovita, and Pilloni constructed the eigenvariety for cuspidal Siegel modular forms. This eigenvariety has the expected dimension, namely the genus of the Siegel variety, but it parametrizes only cuspidal forms. We explain how to generalize the construction to the non-cuspidal case. To be precise, we introduce the notion of "degree of cuspidability" and we construct an eigenvariety that parametrizes forms of a given degree of cuspidability. The dimension of these eigenvarieties depends on the degree of cuspidability we want to consider: the more non-cuspidal the forms, the smaller the dimension. We also explain how to glue the (reduced) eigenvarieties we have constructed. This is a joint work with Giovanni Rosso.

Christopher Birkbeck.
Title: Slopes of Hilbert modular forms.
Abstract: Work of Buzzard and Kilford (among others) on slopes (the p-adic valuation of the Up eigenvalues) of overconvergent modular forms gave us great insights into the geometry of the associated eigenvarieties and are the basis of many conjectures. This is an active area of research and in many cases these conjectures are now proven, yet not much is known in the case of Hilbert modular forms. In my talk I will discuss how one computes slopes in the Hilbert case and what they suggest about the geometry of the associated eigenvarieties.

Przemyslaw Chojecki
Title: p-adic Jacquet-Langlands correspondence and patching
Abstract: We construct a natural candidate for the p-adic Jacquet-Langlands correspondence in the two-dimensional case, which we cut out from the cohomology of the Drinfeld tower. We show that it is isomorphic to a representation obtained from the functor appearing in Scholze’s work on the Lubin-Tate tower and moreover that it satisfies a local-global compatibility. Our proof uses global methods, in particular patching. This is a joint work with Erick Knight.

Valentin Hernandez
Title : An example of an arithmetic application of p-adic families of automorphic forms : constructing Galois extensions.
Abstract : In this talk I will try to explain a method discovered by Skinner-Urban and Bellaïche-Chenevier to construct Galois extensions in Selmer groups as predicted by a conjecture of Bloch and Kato. More precisely, in an article, Bellaïche and Chenevier construct for a quadratic imaginary field E and a polarized algebraic $p$-adic character $\chi$ of $G_E$ a non zero class in the Selmer group of $\chi$. This theorem relies on the construction of an eigenvariety for a (compact at infinity) unitary group in three variables and a careful analysis of deformations for a non tempered automorphic form associated to $\chi$. This relies heavily on the sign of $\chi$ being negative. In this talk, I will try to explain the lines of their method and more precisely what is the information given by the eigenvariety, and explain how it can be generalised to the case of a positive sign for $\chi$.

James Newton
Title: Extended eigenvarieties, the characteristic power series of U_p and the boundary of the eigencurve.
Abstract: Liu, Wan and Xiao recently established (a version of) a folklore conjecture describing the geometry of the eigencurve near the boundary of weight space. I will explain how a key estimate on the characteristic power series of U_p which appears in their work can be obtained as a simple consequence of a theory of overconvergent p-adic automorphic forms with mixed characteristic analytic weights (inspired by ideas of Coleman, Andreatta, Iovita and Pilloni). I will also discuss consequences for the eigenvarieties arising from Hilbert modular forms for totally real fields F in which p splits completely. This is all joint work with Christian Johansson.

Christian Johansson
Title: Extended eigenvarieties using overconvergent cohomology
Abstract: In this talk, I will start by discussing how one can construct the Coleman--Mazur eigencurve using overconvergent cohomology, as developed by Ash and Stevens. I will then talk about joint work with James Newton in which we construct an "integral model" of this eigencurve (and of all other eigenvarieties constructed by overconvergent cohmology), which agrees with that constructed by Andreatta--Iovita--Pilloni. The key technical part is to generalise the overconvergent distribution modules of Ash--Stevens to mixed characteristic.