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Search: MSC category 11N45 ( Asymptotic results on counting functions for algebraic and topological structures )

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1. CJM Online first

Chandee, Vorrapan; David, Chantal; Koukoulopoulos, Dimitris; Smith, Ethan
The frequency of elliptic curve groups over prime finite fields
Letting $p$ vary over all primes and $E$ vary over all elliptic curves over the finite field $\mathbb{F}_p$, we study the frequency to which a given group $G$ arises as a group of points $E(\mathbb{F}_p)$. It is well-known that the only permissible groups are of the form $G_{m,k}:=\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/mk\mathbb{Z}$. Given such a candidate group, we let $M(G_{m,k})$ be the frequency to which the group $G_{m,k}$ arises in this way. Previously, the second and fourth named authors determined an asymptotic formula for $M(G_{m,k})$ assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for $M(G_{m,k})$, pointwise and on average. In particular, we show that $M(G_{m,k})$ is bounded above by a constant multiple of the expected quantity when $m\le k^A$ and that the conjectured asymptotic for $M(G_{m,k})$ holds for almost all groups $G_{m,k}$ when $m\le k^{1/4-\epsilon}$. We also apply our methods to study the frequency to which a given integer $N$ arises as the group order $\#E(\mathbb{F}_p)$.

Keywords:average order, elliptic curves, primes in short intervals
Categories:11G07, 11N45, 11N13, 11N36

2. CJM 1997 (vol 49 pp. 641)

Burris, Stanley; Compton, Kevin; Odlyzko, Andrew; Richmond, Bruce
Fine spectra and limit laws II First-order 0--1 laws.
Using Feferman-Vaught techniques a condition on the fine spectrum of an admissible class of structures is found which leads to a first-order 0--1 law. The condition presented is best possible in the sense that if it is violated then one can find an admissible class with the same fine spectrum which does not have a first-order 0--1 law. If the condition is satisfied (and hence we have a first-order %% 0--1 law)

Categories:03N45, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81

3. CJM 1997 (vol 49 pp. 468)

Burris, Stanley; Sárközy, András
Fine spectra and limit laws I. First-order laws
Using Feferman-Vaught techniques we show a certain property of the fine spectrum of an admissible class of structures leads to a first-order law. The condition presented is best possible in the sense that if it is violated then one can find an admissible class with the same fine spectrum which does not have a first-order law. We present three conditions for verifying that the above property actually holds. The first condition is that the count function of an admissible class has regular variation with a certain uniformity of convergence. This applies to a wide range of admissible classes, including those satisfying Knopfmacher's Axiom A, and those satisfying Bateman and Diamond's condition. The second condition is similar to the first condition, but designed to handle the discrete case, {\it i.e.}, when the sizes of the structures in an admissible class $K$ are all powers of a single integer. It applies when either the class of indecomposables or the whole class satisfies Knopfmacher's Axiom A$^\#$. The third condition is also for the discrete case, when there is a uniform bound on the number of $K$-indecomposables of any given size.

Keywords:First order limit laws, generalized number theory
Categories:O3C13, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81

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