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Register a new userOn the distribution of traces of Frobenius for families of elliptic curves and the Lang-Trotter conjecture on average
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We obtain distribution results for traces of Frobenius for various families of elliptic curves with respect to the Lang-Trotter conjecture, extremal primes, and the central limit theorem. This includes some generalisations and bounds related to the work of Sha-Shparlinski on the average Lang-Trotter conjecture for single-parametric families of elliptic curves and the work of various authors on the trace of Frobenius for primes in congruence classes. Some results are also obtained for modular forms.
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Let $C$ be a smooth projective curve over $mathbb{F}_q$ with function field $K$, $E/K$ a nonconstant elliptic curve and $phi:mathcal{E}to C$ its minimal regular model. For each $Pin C$ such that $E$ has good reduction at $P$, i.e., the fiber $mathcal{E}_P=phi^{-1}(P)$ is smooth, the eigenvalues of the zeta-function of $mathcal{E}_P$ over the residue field $kappa_P$ of $P$ are of the form $q_P^{1/2}e^{itheta_P},q_{P}e^{-itheta_P}$, where $q_P=q^{deg(P)}$ and $0letheta_Plepi$. The goal of this note is to determine given an integer $Bge 1$, $alpha,betain[0,pi]$ the number of $Pin C$ where the reduction of $E$ is good and such that $deg(P)le B$ and $alphaletheta_Plebeta$.
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