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Search: MSC category 14G10 ( Zeta-functions and related questions [See also 11G40] (Birch-Swinnerton-Dyer conjecture) )

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1. CJM Online first

Fité, Francesc; González, Josep; Lario, Joan Carles
Frobenius distribution for quotients of Fermat curves of prime exponent
Let $\mathcal{C}$ denote the Fermat curve over $\mathbb{Q}$ of prime exponent $\ell$. The Jacobian $\operatorname{Jac}(\mathcal{C})$ of~$\mathcal{C}$ splits over $\mathbb{Q}$ as the product of Jacobians $\operatorname{Jac}(\mathcal{C}_k)$, $1\leq k\leq \ell-2$, where $\mathcal{C}_k$ are curves obtained as quotients of $\mathcal{C}$ by certain subgroups of automorphisms of $\mathcal{C}$. It is well known that $\operatorname{Jac}(\mathcal{C}_k)$ is the power of an absolutely simple abelian variety $B_k$ with complex multiplication. We call degenerate those pairs $(\ell,k)$ for which $B_k$ has degenerate CM type. For a non-degenerate pair $(\ell,k)$, we compute the Sato-Tate group of $\operatorname{Jac}(\mathcal{C}_k)$, prove the generalized Sato-Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of $(\ell,k)$ being degenerate or not, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the $\ell$-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

Keywords:Sato-Tate group, Fermat curve, Frobenius distribution
Categories:11D41, 11M50, 11G10, 14G10

2. CJM 1997 (vol 49 pp. 749)

Howe, Lawrence
Twisted Hasse-Weil $L$-functions and the rank of Mordell-Weil groups
Following a method outlined by Greenberg, root number computations give a conjectural lower bound for the ranks of certain Mordell-Weil groups of elliptic curves. More specifically, for $\PQ_{n}$ a \pgl{{\bf Z}/p^{n}{\bf Z}}-extension of ${\bf Q}$ and $E$ an elliptic curve over {\bf Q}, define the motive $E \otimes \rho$, where $\rho$ is any irreducible representation of $\Gal (\PQ_{n}/{\bf Q})$. Under some restrictions, the root number in the conjectural functional equation for the $L$-function of $E \otimes \rho$ is easily computible, and a `$-1$' implies, by the Birch and Swinnerton-Dyer conjecture, that $\rho$ is found in $E(\PQ_{n}) \otimes {\bf C}$. Summing the dimensions of such $\rho$ gives a conjectural lower bound of $$ p^{2n} - p^{2n - 1} - p - 1 $$ for the rank of $E(\PQ_{n})$.

Categories:11G05, 14G10

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