In this edition of the SNTB, the two learning courses of the seminar are interconnected. In the first course, the aim will be to present the foundations of Hida families. In the second one, we will give applications that this theory has found in the theory of elliptic curves. The first example of a p-adic family of modular forms was the Eisenstein family, considered by Serre in [Ser72]. The theory took off with the seminal works of Hida (see [Hid86a], [Hid86b]), where he constructed p-adic families of cuspforms (the so-called Hida families), varying continously with the weight, which are simultaneous eigenforms of the Hecke operators. Hida’s construction had a certain limitation: only p-ordinary Hecke eigenforms of level coprime to p belong to a Hida family. Coleman and Mazur constructed a p-adic rigid anlaytic curve containing any eigenform of level coprime to p. The goal of the first course is to review all these classical constructions. We will start the second course by presenting the p-adic Birch and Swinnerton- Dyer conjecture. The basic type of probelm that will concern us is the existence of an exceptional zero at the critical point of the p-adic L-function attached to an elliptic curve with split multiplicative reduction at p. This was experimen- tally observed by Mazur, Tate and Teitelbaum [MTT86], and proven by [GS93], using as a fundamental tool the theory of Hida families. The course will proceed by presenting results concerning related topics, whose proofs also make use of the theory of Hida families.

- $p$-adic analog of BSD conjecture. (Oscar Rivero)
- The exceptional zero conjecture. (Daniel Barrera)
- A second derivative result for $p$-adic $L$-functions (Carlos de Vera Piquero)
- The Mazur-Tate-Teitelbaum conjecture in the rank one setting. (Victor Rotger Cerda)
- The rationality of traces of Stark--Heegner points (Marc Masdeu)

[BC09] J. Bella ̈ıche, G. Chenevier, Families of Galois representations and Selmer groups, Ast\'erisque No. 324 (2009)

[BD07a] M. Bertolini, H. Darmon, Hida families and rational points on elliptic curves, Invent. Math. 168 (2007), no. 2, 371–431.

[BD07b] M. Bertolini, H. Darmon, The p-adic L-functions of modular elliptic curves, Mathematics Unlimited – 2001 and Beyond, Springer (2001) 109–170.

[BD09] M. Bertolini, H. Darmon, The rationality of Stark-Heegner points over genus fields of real quadratic fields, Annals of Mathematics 170 (2009) 343-369.

[CM96] R.F. Coleman, B. Mazur, The eigencurve, Galois representations in arithmetic algebraic grometry (Durham, 1996) (A.J. Scholl, R.L Taylor eds.), London Math. Soc. Lecture Notes Series 254, Cambridge Univ. Press, 1998, 1–113.

[DLR15] H. Darmon, A. Lauder, V. Rotger, Stark points and p-adic iterated integrals attached to modular forms of weight one, Forum of Mathematics, Pi (2015), Vol. 3, e8, 95 pages.

[DHHJPR16] E.P. Dummit, M. Hablicsek, R. Harron, L. Jain, R. Pollack, D. Ross, Explicit computations of Hida families via overconvergent modular symbols, to appear in Research in Number Theory, 2016.

[Eme09]M. Emerton, p-adic families of modular forms (after Hida, Coleman, and Mazur), S ́eminaire Bourbaki, 62`eme ann ́ee, 2009-2010, no. 1013.

[GS93] R. Greenberg, G. Stevens, p-adic L-functions and p-adic periods of modular forms, Invent. Math. 111(2), (1993), 407–447.

[Gui] X. Guitart, Mazur’s construction of the Kubota–Leopold p-adic L- function, available on Xevi Guitart’s web page.

[Hid86a] H. Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. Ecole Norm. Sup. 19 (1986), 231–273.

[Hid86b] H. Hida, Galois representations into GL 2 (Z p [[X]]) attached to ordi- nary cusp forms, invent. math. 85, 545–613 (1986).

[Laf] M.J. Lafferty, Hida Theory.

[MTT86] B. Mazur, J. Tate, J. Teitelbaum, On p-adic analogues of the con- jectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1) (1986), 1–48.

[Pil13] V. Pilloni, Overconvergent modular forms, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 1, 219–239.

[Rub92] K. Rubin, p-adic L-functions and rational points on elliptic curves with complex multiplication, Invent. Math. 84 (1) (1986), 323–350.

[Ser72] J.-P. Serre, Formes modulaires et fonctions zˆeta p-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Springer Lecture Notes 350, 191–268.

[Ven16] R. Venerucci, Exceptional zero formulae and a conjecture of Perrin– Riou, Invent. Math. 203 (3) (2016), 923-972.

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