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Where is the best place to learn about the p-adic L-functions of Elliptic Curves? Doing a bit of research I have found books like "An Introduction to Cyclotomic Fields" by Washington, but those only seem to discuss the p-adic analogues of Dirichlet L-functions.

As background, I am a student whose main relationship with algebraic Number Theory is Joe Silverman's "The Arithmetic of Elliptic Curves". I have also taken a course on Analytic Number Theory.

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    $\begingroup$ You could look at Mazur-Tate-Teitelbaum, On $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer. They give the definition for modular forms; the $p$-adic $L$-function of an elliptic curve $E$ is just the $p$-adic $L$-function of the modular form associated to $E$. $\endgroup$ – François Brunault Mar 4 at 22:09
  • $\begingroup$ @FrançoisBrunault thank you for your insight! If you want to put that same response as an answer, I could throw some reputation your way. $\endgroup$ – Milo Moses Mar 4 at 22:44
  • $\begingroup$ Even if you are not interested (maybe) in Dirichlet $p$-adic $L$-functions, I am pretty sure you will eventually have to learn the language of $p$-adic measures (if you have not studied it yet). Washington's book is an excelent place to start. $\endgroup$ – EFinat-S Mar 4 at 23:56
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We know how to attach $p$-adic $L$-function to elliptic curves over $\mathbb Q$ only because we know how to attach them to cuspidal modular eigenforms, and we know by Breuil-Conrad-Diamond-Taylor that every elliptic curves over $\mathbb Q$ are themselves attached to a cuspidal modular eigenform of weight $2$. Historically, the construction of $p$-adic $L$-functions attached to modular forms is the work of Mazur-Swinnerton-Dier, Manin, Amice-Velu, Shokurov. The article of Mazur-Tate-Teitelbaum quoted in comment contains a synthesis of all that.

But I think the best way to learn the theory now is to learn the version of the construction (not fundamentally different than the previously mentioned, but much simpler and clearer) given by Pollack and Stevens. For this, see the two papers by Pollack-Stevens: Overconvergent modular symbols and p-adic L-functions, Ann. Sci. Ec. Norm. Sup. (4) 44 (2011), no. 1, 1–42, and Critical slope $p$-adic $L$-functions, J. Lond. Math. Soc. (2) 87 (2013), no. 2, 428--452.

I have also written a book, an expanded version of a course I gave at Brandeis, on the subject of p-adic L-functions of modular forms (and families thereof). It exposes the Pollack-Stevens method, as well as their generalization and mine for the so-called "critical" case, that was missing at the time of Mazur-Tate-Teitelbaum, and many other things.

The book is called The Eigenbook. Eigenvarieties, families of Galois representations, p-adic L-functions and it will be published soon by Springer-Birkhauser, but you can also find a non-final version on my webpage.

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