Algebra, Geometry, and Number Theory Seminar

Abstracts

- Fall Break

Tugba Yesin - Divisibility by 2 on quartic models of elliptic curves and rational Diophantine $D(q)$-quintuples

Let $C$ be a smooth genus one curve described by a quartic polynomial equation over the rational field $Q$ with $P \in C(Q)$. We give an explicit criterion for the divisibility-by-$2$ of a rational point on the elliptic curve $(C,P)$. This generalizes the classical criterion of the divisibility-by-$2$ on elliptic curves described by Weierstrass equations. We employ this criterion to investigate the question of extending a rational $D(q)$-quadruple to a quintuple. We give concrete examples to which we can give an affirmative answer. One of these results implies that although the rational $D(16t+9)$-quadruple $\{t,16t+8,225t+14,36t+20\}$ can not be extended to a polynomial $D(16t+9)$-quintuple using a linear polynomial, there are infinitely many rational values of $t$ for which the aforementioned rational $D(16t + 9)$-quadruple can be extended to a rational $D(16t + 9)$-quintuple. Moreover, these infinitely many values of $t$ are parametrized by the rational points on a certain elliptic curve of positive Mordell-Weil rank.

Charlotte Ure - A moduli interpretation of binary cubic forms

The subject of classifying homogeneous forms has fascinated mathematicians for centuries. In this talk, I will discuss joint work with Rajesh Kulkarni, giving a moduli interpretation to the quotient of binary cubic forms with respect to linear transformation of the variables. In particular, I will explain how these orbits correspond to certain coverings, and Brauer classes, on elliptic curves.

Brooke Ullery - Secants, Gorenstein ideals, and stable complexes

As Bertram describes in his thesis, certain rank two vector bundles on curves can be parametrized by a projective space into which the curve itself embeds. The idea is that the least so-called "stable" bundles correspond to points on the embedded curve, the next most unstable lie on the first secant variety, then on the second secant variety, and so on. In this talk, I'll describe joint work with Bertram in which we generalize this to $\mathbb{P}^2$ by replacing vector bundles with complexes of vector bundles. This leads to a surprising connection between height three Gorenstein ideals, secant varieties (and their generalizations), and stability conditions on the bounded derived category of coherent sheaves on $\mathbb{P}^2$.

Ken Ono - Variants of Lehmer’s Conjecture on Ramunajan’s tau-function

The theory of modular forms enjoys some of the most significant recent advances in number theory. This includes the discovery of ideas that underly the proof of Fermat’s Last Theorem, provide the framework for much of the Langlands Program, and gives glimpses of deep connections between geometry, number theory and physics. Despite these deep advances, the innocent (in appearance) Conjecture of D. H. Lehmer on the non-vanishing of Ramanujan's tau-function remains open. In this talk the speaker will tell the story of this important function, and describe recent results on variants of Lehmer’s Conjecture, where much progress has been made in the last few years. ​ (USC Colloquium)

- Thanksgiving

- Reserved (Palmetto Number Theory Series)

Last semester's seminar.