# p-adic methods for Galois representations and modular forms

# p-adic methods for Galois representations and modular forms

January 30, 2017

January 27th, 2017.

CRM, UB and UPC

The theme of p-adic modular forms was initiated in the 70’s by Serre and Katz, who also gave the first important applications to the construction of p-adic L-functions. Since then, the theory has been extended and refined by Dwork, Hida, Mazur, Wiles, Coleman and many others. At the same time, the range of applications has grown substantially, so as to become a key tool in the study of the arithmetic of algebraic varieties and automorphic forms.

We organize three advanced courses by leading experts in the subject, suitable for PhD students, postdoc researchers and senior number-theorists.

January 30, 2017

January 27th, 2017.

The theme of p-adic modular forms was initiated in the 70’s by Serre and Katz, who also gave the first important applications to the construction of p-adic L-functions. Since then, the theory has been extended and refined by Dwork, Hida, Mazur, Wiles, Coleman and many others. At the same time, the range of applications has grown substantially, so as to become a key tool in the study of the arithmetic of algebraic varieties and automorphic forms.

We organize three advanced courses by leading experts in the subject, suitable for PhD students, postdoc researchers and senior number-theorists.

Adrian Iovita (Concordia University & Università degli Studi di Padova).

Vincent Pilloni (Ecole Normal Supérieure de Lyon).

Payman Kassaei (King’s College London).

Detailed Location

From Monday January 30 to Thursday February 2, 2017:

Centre de Recerca Matemàtica

at room A2 (first floor).

From Monday February 6 to Thursday February 9, 2017:

Facultat de Matemàtiques de la Universitat de Barcelona

at room B3.

From Monday February 20 to Wednesday February 22, 2017:

Facultat de Matemàtiques i Estadística de la Universitat Politècnica de Catalunya

(Room 103).

##### Schedule

**Integral overconvergent modular forms and nearly overconvergent modular forms of arbitrary weights**(Adrian Iovita)

Four lectures of two hours from Monday January 30 to Thursday February 2, 2017.

January 30, Monday: 9:30-10:30 and 11:30-12:30

January 31, Tuesday: 9:30-10:30 and 11:30-12:30

February 1, Wednesday: 9:30-10:30 and 11:30-12:30

February 2, Thursday: 9:00-10:00 and 10:30-11:30

**Siegel modular forms and abelian surfaces**(Vincent Pilloni)

Four lectures of two hours from Monday February 6 to Thursday February 9, 2017.

February 6, Monday: 10:30-11:30 and 11:45-12:45

February 7, Tuesday: 10:30-11:30 and 11:45-12:45

February 8, Wednesday: 10:30-11:30 and 11:45-12:45

February 9, Thursday: 10:30-11:30 and 11:45-12:45

##### Contents

- Integral overconvergent modular forms and nearly overconvergent modular forms of arbitrary weights by Adrian Iovita (Concordia University & Università degli Studi di Padova).
- Siegel modular forms and abelian surfaces by Vincent Pilloni (Ecole Normal Supérieure de Lyon).
- Hilbert modular forms: p-adic and mod p aspects by Payman Kassaei (King’s College).

##### Bibliography

- F. Andreatta, A. Iovita, V. Pilloni, The adic, cuspidal, Hilbert eigenvarieties, preprint.
- F. Andreatta, A. Iovita, V. Pilloni, Le halo spectral, preprint.
- F. Andreatta, A. Iovita, V. Pilloni, p-adic families of Siegel modular forms, Annals of Maths (2015).
- F. Andreatta, A. Iovita, V. Pilloni, On overconvergent Hilbert modular cusp forms to appear in Astérisque.
- F. Andreatta, A. Iovita, G. Stevens, Overconvergent Eichler-Shimura Isomorphisms, preprint.
- F. Andreatta, A. Iovita, G. Stevens, Overconvergent modular sheaves and modular forms for GL(2,F), to appear in Israel J. Math.
- S. Bijakowski, V. Pilloni, B. Stroh, Classicité de formes modulaires surconvergentes, to appear in Annals of Maths.
- R. Coleman, Classical and oveconvergent modular forms, Invent. Math., 124 (1996), 215-241.
- R. Coleman, p-adic Banach spaces and families of modular forms, Invent. Math., 127 (1997), 417-479.
- B. Conrad and K. Rubin, Arithmetic Algebraic Geometry, American Mathematical Society, IAS/Park City Mathematics Institute (2008).
- F. Diamond and J. Shurman, A First Course in Modular Forms, Springer-Verlag, Graduate Texts in Mathematics, No. 228, New York, 2005.
- T. Gee, P. Kassaei, Companion forms in parallel weight one, to appear in Compositio Mathematica.
- F. Gouvêa, p-adic Numbers, Springer-Verlag Berlin Heidelberg, Universitext, 1997.
- F. Gouvêa, Arithmetic of p-adic Modular Forms, Springer-Verlag, Lecture Notes in Mathematics, Vol 1304 (1988).
- F. Gouvêa and B. Mazur, Families of modular eigenforms, Math. Comp., Vol. 58, No. 198 (1992), 793-805.
- P. Kassaei, Modularity lifting in parallel one, Journal Amer. Math. Soc. 26 (2013), no. 1, 199–225.
- P. Kassaei, A gluing lemma and overconvergent modular forms, Duke Math. J. 132 (2006), no. 3, 509–529.
- N. Katz, p-adic properties of modular schemes and modular forms, in Modular Forms in One Variable III (SLN 350), Springer-Verlag, 1973, 69-190.
- L. Kilford, Modular Forms: A Classical and Computational Introduction, Imperial College Press, 2nd edition.
- N. Koblitz, p-adic numbers, p-adic Analysis, and Zeta-Functions, Springer-Verlag, Graduate Texts in Mathematics, No. 58, New York, 1984.
- V. Pilloni, B. Stroh, Cohomologie cohérente et représentations galoisiennes, preprint.
- V. Pilloni, B. Stroh, Surconvergence, ramification et modularité to apear in Astérisque.
- V. Pilloni, Formes modulaires p-adiques de Hilbert de poids 1, Invent. Math.
- J-P. Serre, Formes modulaires et fonctiones zeta p-adiques, in Modular Forms in One Variable III (SLN 350) 1973, Springer-Verlag, 191-268.
- J-P. Serre, Endomorphismes complètement continus des espaces de Banach padiques, in Publications Mathématiques de l’Institut des Hautes Études Scientifiques, December 1962, Volume 12, Issue 1, 69-85.