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Search: MSC category 11R45 ( Density theorems )

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1. CJM Online first

Cojocaru, Alina Carmen; Shulman, Andrew Michael
The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank
Let $\psi$ be a generic Drinfeld module of rank $r \geq 2$. We study the first elementary divisor $d_{1, \wp}(\psi)$ of the reduction of $\psi$ modulo a prime $\wp$, as $\wp$ varies. In particular, we prove the existence of the density of the primes $\wp$ for which $d_{1, \wp} (\psi)$ is fixed. For $r = 2$, we also study the second elementary divisor (the exponent) of the reduction of $\psi$ modulo $\wp$ and prove that, on average, it has a large norm. Our work is motivated by the study of J.-P. Serre of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M.R. Murty.

Keywords:Drinfeld modules, density theorems
Categories:11R45, 11G09, 11R58

2. CJM 2005 (vol 57 pp. 1155)

Cojocaru, Alina Carmen; Fouvry, Etienne; Murty, M. Ram
The Square Sieve and the Lang--Trotter Conjecture
Let $E$ be an elliptic curve defined over $\Q$ and without complex multiplication. Let $K$ be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes $p \leq x$ for which $\Q(\pi_p) = K$, where $\pi_p$ denotes the Frobenius endomorphism of $E$ at $p$. More precisely, under a generalized Riemann hypothesis we show that this number is $O_{E}(x^{\slfrac{17}{18}}\log x)$, and unconditionally we show that this number is $O_{E, K}\bigl(\frac{x(\log \log x)^{\slfrac{13}{12}}} {(\log x)^{\slfrac{25}{24}}}\bigr)$. We also prove that the number of imaginary quadratic fields $K$, with $-\disc K \leq x$ and of the form $K = \Q(\pi_p)$, is $\gg_E\log\log\log x$ for $x\geq x_0(E)$. These results represent progress towards a 1976 Lang--Trotter conjecture.

Keywords:Elliptic curves modulo $p$; Lang--Trotter conjecture;, applications of sieve methods
Categories:11G05, 11N36, 11R45

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