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

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

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

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

"Konzept einer Berufswahlwoche" : Sehen Lehrpersonen der Sekundarstufe 1 Bedarf für eine unserem Konzept entsprechende Berufswahlwoche?

Diese Arbeit befasst sich mit dem Thema Berufswahlbereitschaft von Jugendlichen. In diesem Zusammenhang konzipierten die Autoren eine Berufswahlwoche, welche zum Ziel hat die Berufswahlbereitschaft der Jugendlichen zu steigern. Diese geplante Berufswahlwoche sollte im Rahmen des schon bestehenden institutionellen Angebots durch die Schulen und die Berufsberatung durchgeführt werden und ist als Ergänzung zu den bestehenden Angeboten gedacht. Um zu klären, ob von Seiten der Sekundarlehrpersonen Bedarf nach einer entsprechenden Ergänzung im Rahmen unserer Woche vorliegt, haben wir 42 Sekundarlehrpersonen mittels quantitativer elektronischer Umfrage zum fertigen Konzept befragt. Die konkrete Fragestellung lautete: Sehen Lehrpersonen der Sekundarstufe 1 Bedarf für eine unserem Konzept entsprechende Berufswahlwoche