The strong Birch--Swinnerton-Dyer conjecture and in particular the exact order of the Shafarevich--Tate group for abelian varieties over the rationals has only been known for elliptic curves (dimension 1) or in higher dimension where the conjecture could be reduced to dimension 1. We give the first absolutely simple examples of dimension 2 where the conjecture can be verified:
Let $X$ be...
In this talk, we'll demonstrate how to implement categories on a computer using our software project CAP - Categories, Algorithms, and Programming. We will discuss how to construct the bounded homotopy category of an additive or Abelian category, as well as the obstacles that arise and their practical categorical solutions.
We use these homotopy categories to demonstrate some applications...
As can be seen in the preceding talk ("Constructive Category Theory and Tilting equivalences via Strong Exceptional Sequences" by Kamal Saleh), building constructive towers of categories allows us to reach advanced and complex applications while retaining (relative) simplicity of both the mathematical descriptions and the organization of the software. Our software project CAP (Categories,...
Power series remain the best tool for effective evaluations of combinations of elementary functions. We overview a recent result concerning the symbolic computation of univariate formal power series.
Given a field K, of characteristic zero, a term a(n) is called m-fold hypergeometric for a positive integer m (mostly m>1), if the ratio a(n+m)/a(n) is rational in K(n). When the value of m is...
We develop an algorithm based on a greedy algorithm of Cuntz from his 2021 paper "A Greedy Algorithm to Compute Arrangements of Lines in the Projective Plane," to find arrangements of lines for which a "Mod 2-net coloring" is possible. In such a coloring we color the lines of the arrangement with one of two colors so, that for all lines going through a given point, we have an even number of...
Given a 0-dimensional projective scheme X in the projective n-space over a field K, we are interested in studying the Cayley-Bacharach property (CB-property) of X. Classically, a finite set of points X has the CB-property if every hypersurface of certain degree which contains all points of X but one automatically contains the last point. In this talk we shows how to check the CB-property of a...
An infinite sequence $\mathcal{ F } = (F_0, F_1, \dots)$ of function fields $F_n / \mathbb{F}_q $ of transcendence degree one with full constant field $ \mathbb{F}_q $ is called a tower of function fields if all extensions $F_{n + 1}/F_n$ are finite separable and the genus $g(F_n)$ tends to infinity as $n \to \infty$.
A tower is called asymptotically good if its limit $\lambda(\mathcal{F})...
