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- W4319988894 abstract "For an elliptic curve $E$ defined over a number field $K$, the heuristic density of the set of primes of $K$ for which $E$ has cyclic reduction is given by an inclusion-exclusion sum $delta_{E/K}$ involving the degrees of the $m$-division fields $K_m$ of $E$ over $K$. This density can be proved to be correct under assumption of GRH. For $E$ without complex multiplication (CM), we show that $delta_{E/K}$ is the product of an explicit non-negative rational number reflecting the finite entanglement of the division fields of $E$ and a universal infinite Artin-type product. For $E$ admitting CM over $K$ by a quadratic order ${mathcal{O}}$, we show that $delta_{E/K}$ admits a similar `factorization' in which the Artin type product also depends on ${mathcal{O}}$. For $E$ admitting CM over $bar K$ by an order ${mathcal{O}}notsubset K$, which occurs for $K={bf Q}$, the entanglement of division fields over $K$ is non-finite. In this case we write $delta_{E/K}$ as the sum of two contributions coming from the primes of $K$ that are split and inert in ${mathcal{O}}$. The split contribution can be dealt with by the previous methods, the inert contribution is of a different nature. We determine the ways in which the density can vanish, and provide numerical examples of the different kinds of densities." @default.
- W4319988894 created "2023-02-11" @default.
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- W4319988894 creator A5087168657 @default.
- W4319988894 date "2019-12-31" @default.
- W4319988894 modified "2023-09-28" @default.
- W4319988894 title "Cyclic reduction densities for elliptic curves" @default.
- W4319988894 doi "https://doi.org/10.48550/arxiv.2001.00028" @default.
- W4319988894 hasPublicationYear "2019" @default.
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