A note on the Fourier coefficients of a Cohen-Eisenstein series

We prove a formula for the coefficients of a weight $3/2$ Cohen-Eisenstein series of square-free level $N$. This formula generalizes a result of Gross and in particular, it proves a conjecture of Quattrini. Let $l$ be an odd prime number. For any elliptic curve $E$ defined over $\mathbb{Q}$ of rank zero and square-free conductor $N$, if $l \mid |E(\mathbb{Q})|$, under certain conditions on the Shafarevich-Tate group $III_D$, we show that $l$ divides $|III_D|$ if and only if $l$ divides the class number $h(-D)$ of $\mathbb{Q}(\sqrt{-D}).$Comment: To appear in International Journal of Number Theor

Rank 2 Local Systems, Barsotti-Tate Groups, and Shimura Curves

We develop a descent criterion for $K$-linear abelian categories. Using recent advances in the Langlands correspondence due to Abe, we build a correspondence between certain rank 2 local systems and certain Barsotti-Tate groups on complete curves over a finite field. We conjecture that such Barsotti-Tate groups "come from" a family of fake elliptic curves. As an application of these ideas, we provide a criterion for being a Shimura curve over $\mathbb{F}_q$. Along the way, we formulate a conjecture on the field-of-coefficients of certain compatible systems.Comment: 30 pages. Part of author's PhD thesis. Comments welcome

An analysis of correlating parameters relating to hot gas ingestion characteristics of jet VTOL aircraft

Jet VTOL fighter-type model inlet-air temperature rise analysis with various exhaust pressure ratios and gas temperatures and surface wind velocities for correlating parameter

Store capacity optimisation

The problem is one of increasing the efficiency of distributing paper rolls from the manufacturing plants to the customers. A related problem is one of utilising the available capacity at the customer stores in an effective manner. During the MISG, several approaches to the above problems were proposed. In this report we describe the problem and several methods for solving it. Preliminary results are provided for some of these