Local torsion on elliptic curves and the deformation theory of Galois representations

We prove that, on average, elliptic curves over Q have finitely many primes p for which they possess a p-adic point of order p. We include a discussion of applications to companion forms and the deformation theory of Galois representations

Discerning the Impact of Powder Feedstock Variability on Structure, Property, and Performance of Selective Laser Melted Alloy 718: A Principal Component Analysis (PCA) of Feedstock Variability

Extensive mechanical, chemical and microstructural analyses were conducted on additively manufactured Alloy 718 to characterize powders from multiple vendors to determine the effects of variations observed in the powders had on the consolidated material. With over 190 variables examined, it was necessary to reduce the number of variables and identify the variables and classes of variables that had the greatest effect. Principle Component Analysis (PCA) was used to reduce the number of variable to effectively 12 while identifying several classes of variables as most important

Elliptic curves with a given number of points over finite fields

Given an elliptic curve $E$ and a positive integer $N$, we consider the problem of counting the number of primes $p$ for which the reduction of $E$ modulo $p$ possesses exactly $N$ points over $\mathbb F_p$. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average.Comment: A mistake was discovered in the derivation of the product formula for K(N). The included corrigendum corrects this mistake. All page numbers in the corrigendum refer to the journal version of the manuscrip

Distribution of squarefree values of sequences associated with elliptic curves

Let E be a non-CM elliptic curve defined over Q. For each prime p of good reduction, E reduces to a curve E_p over the finite field F_p. For a given squarefree polynomial f(x,y), we examine the sequences f_p(E) := f(a_p(E), p), whose values are associated with the reduction of E over F_p. We are particularly interested in two sequences: f_p(E) =p + 1 - a_p(E) and f_p(E) = a_p(E)^2 - 4p. We present two results towards the goal of determining how often the values in a given sequence are squarefree. First, for any fixed curve E, we give an upper bound for the number of primes p up to X for which f_p(E) is squarefree. Moreover, we show that the conjectural asymptotic for the prime counting function \pi_{E,f}^{SF}(X) := #{p \leq X: f_p(E) is squarefree} is consistent with the asymptotic for the average over curves E in a suitable box

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 $\#E(\mathbb{F}_p)$.Comment: 40 pages, with an appendix by Chantal David, Greg Martin and Ethan Smith. Final version, to appear in the Canad. J. Math. Major reorganization of the paper, with the addition of a new section, where the main results are summarized and explaine

Institutions and Entrepreneurship: The Role of The Rule of Law

This paper examines variations in entrepreneurship across twenty developed countries, using three measures of entrepreneurship which we broadly describe as prestart, early-stage and established enterprises. It then links these measures to the economic institutional framework, holding constant a range of other factors. Two groups of conclusions emerge. The first is that the factors that influence pre-start, early-stage and established enterprises differ often quite sharply. Second, our results broadly confirm earlier work suggesting that social security entitlements, taxes, and employment protection legislation are negatively associated with (different forms of) entrepreneurial activity. However, our novel finding is that countries with a "better" rule of law have lower entrepreneurship. We explain this apparently counter-intuitive finding by arguing that in developed economies the benefits of the rule of law accrue primarily to large enterprises. �