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Search: MSC category 14H52 ( Elliptic curves [See also 11G05, 11G07, 14Kxx] )

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1. CJM 2011 (vol 64 pp. 81)

David, C.; Wu, J.
Pseudoprime Reductions of Elliptic Curves
Let $E$ be an elliptic curve over $\mathbb Q$ without complex multiplication, and for each prime $p$ of good reduction, let $n_E(p) = | E(\mathbb F_p) |$. For any integer $b$, we consider elliptic pseudoprimes to the base $b$. More precisely, let $Q_{E,b}(x)$ be the number of primes $p \leq x$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$, and let $\pi_{E, b}^{\operatorname{pseu}}(x)$ be the number of compositive $n_E(p)$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$ (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address the problem of finding upper bounds for $Q_{E,b}(x)$ and $\pi_{E, b}^{\operatorname{pseu}}(x)$, generalising some of the literature for the classical pseudoprimes to this new setting.

Keywords:Rosser-Iwaniec sieve, group order of elliptic curves over finite fields, pseudoprimes
Categories:11N36, 14H52

2. CJM 2005 (vol 57 pp. 338)

Lange, Tanja; Shparlinski, Igor E.
Certain Exponential Sums and Random Walks on Elliptic Curves
For a given elliptic curve $\E$, we obtain an upper bound on the discrepancy of sets of multiples $z_sG$ where $z_s$ runs through a sequence $\cZ=\(z_1, \dots, z_T\)$ such that $k z_1,\dots, kz_T $ is a permutation of $z_1, \dots, z_T$, both sequences taken modulo $t$, for sufficiently many distinct values of $k$ modulo $t$. We apply this result to studying an analogue of the power generator over an elliptic curve. These results are elliptic curve analogues of those obtained for multiplicative groups of finite fields and residue rings.

Categories:11L07, 11T23, 11T71, 14H52, 94A60

3. CJM 1999 (vol 51 pp. 936)

David, Chantal; Kisilevsky, Hershy; Pappalardi, Francesco
Galois Representations with Non-Surjective Traces
Let $E$ be an elliptic curve over $\q$, and let $r$ be an integer. According to the Lang-Trotter conjecture, the number of primes $p$ such that $a_p(E) = r$ is either finite, or is asymptotic to $C_{E,r} {\sqrt{x}} / {\log{x}}$ where $C_{E,r}$ is a non-zero constant. A typical example of the former is the case of rational $\ell$-torsion, where $a_p(E) = r$ is impossible if $r \equiv 1 \pmod{\ell}$. We prove in this paper that, when $E$ has a rational $\ell$-isogeny and $\ell \neq 11$, the number of primes $p$ such that $a_p(E) \equiv r \pmod{\ell}$ is finite (for some $r$ modulo $\ell$) if and only if $E$ has rational $\ell$-torsion over the cyclotomic field $\q(\zeta_\ell)$. The case $\ell=11$ is special, and is also treated in the paper. We also classify all those occurences.

Category:14H52

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