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I have seen in some papers computation of $p$-adic periods of modular curves $X_0(N)$. Can somebody please explain to me what are the possible applications of such computations?

as a concrete example assume we have an algorithm to calculate the $p$ -adic period matrix of $X_0(71)$ what are the possible application of such a knowledge ?

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  • $\begingroup$ If you want to do math you need to get used to the idea that sometimes (not always!) the answer to this kind of question is "none". In this particular case, however, one might use these calculations to check the p-adic analogue of the Birch-Swinnerton-Dyer conjecture and maybe, assuming it, compute interesting things like the order of the Tate-Shafarevich group. $\endgroup$
    – Felipe Voloch
    Sep 16, 2018 at 6:37
  • $\begingroup$ Thanks! Can you please point me to the relevant papers that define the $p$ adic BSD conjecture ? I want to understand this application you're suggesting. I Googled them but I am not sure which papers are relevant to my question. Thanks again for the prompt answer ! $\endgroup$
    – natalie Kop
    Sep 16, 2018 at 13:16
  • $\begingroup$ Mazur, Tate, Teitelbaum, Inv. Math. 84 (1986) 1-48 and the papers that cite it. $\endgroup$
    – Felipe Voloch
    Sep 16, 2018 at 19:17

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