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Heekyoung Hahn : Distribution of integer valued sequences associated to elliptic curves

Let $E$ be a non-CM elliptic curve defined over $\mathbb{Q}$. For each prime $p$ of good reduction, $E$ reduces to a curve $E_p$ over the finite field $\mathbb{F}_p$. In this talk, we are particularly interested in ssquare-free values of two sequences: $f_p(E) =p + 1 - a_p(E)$ and $f_p(E) = a_p(E)^2 - 4p$, where $a_p(E)=p+1-|E(\mathbb{F}_p)|$. More precisely for any fixed curve $E$, we first give an upper bound for the number of primes $p$ up to $X$ for which $f_p(E)$ is square-free. Second, we show that the average results on this prime counting function are compatible with the corresponding conjectures at the level of the constants, i.e., whether the average of the conjectured constants is equivalent to the constant obtained via the average conjecture. This is joint work with S. Akhtari, C. David and L. Thompson.

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