Arithmetic of Elliptic curves, one of fundamental research themes in modern number theory, is encoded in the special values of elliptic L-functions. For example, Mazur-Rubin heuristics on the distribution of the special L-values predict the behavior of ranks of elliptic curves. The average version of the heuristics was proved by several researchers including myself several years ago. In the talk, I will present how to use the dynamics of continued fractions to study the problem and introduce an approach to study the original problem, i.e., non-average version, after introducing a classical result, so-called Lochs' theorem, which compares entropies of two distinct dynamical systems.
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