UCSD Number Theory Seminar (Math 209)

Thursday 2-3pm, AP&M 7421

Winter Quarter 2014

For previous quarters' schedule, click here.

January 9


January 16


January 23

Sorina Ionica (ENS Paris)
Isogeny graphs with maximal real multiplication

An isogeny graph is a graph whose vertices are principally polarized abelian varieties and whose edges are isogenies between these varieties. In his thesis, Kohel described the structure of isogeny graphs for elliptic curves and showed that one may compute the endomorphism ring of an elliptic curve defined over a finite field by using a depth first search algorithm in the graph. In dimension 2, the structure of isogeny graphs is less understood and existing algorithms for computing endomorphism rings are very expensive. We fully describe the isogeny graphs between genus 2 jacobians with complex multiplication, with the assumptions that the real multiplication subring is maximal and has class number one. We derive a depth first search algorithm for computing endomorphism rings locally at prime numbers, if the real multiplication is maximal. To the best of our knowledge, this is the first DFS-based algorithm in genus 2. (Joint work with Emmanuel Thomé).

January 30


February 6

Daniel Kane (Stanford)
Ranks of 2-Selmer Groups of Twists of an Elliptic Curve

Let E/Q be an elliptic curve with full 2-torsion over Q. We wish to study the distribution of the ranks of the 2-Selmer groups of twists of E as we vary the twist parameter. A recent result of Swinnerton-Dyer shows that if E has no cyclic 4-isogeny defined over Q, then the density of twists with given rank approaches a particular distribution. Unfortunately Swinnerton-Dyer used an unusual notion of density essentially given as the number of primes dividing the twist parameter goes to infinity. We extend this result to cover density in the natural sense.

February 13

Ronen Mukamel (Stanford)
Billiards, Hilbert modular forms and algebraic models for Teichmüller curves

For each real quadratic order O, there is a Weierstrass curve W in the Hilbert modular surface parametrizing Jacobians with real multiplication by O. The curve W emerges from the study of billiards in polygons and is important in Teichmüller theory because its natural immersion into the moduli space of curves is isometric. Such an immersion is called a Teichmüller curve. We will present explicit algebraic models for Weierstrass curves obtained by studying Hilbert modular forms. We will also present evidence from our examples that suggest a rich arithmetic associated to Teichmüller curves.
This work is joint with A. Kumar.

February 20


February 27


March 6

David Krumm (Claremont McKenna)
Squarefree parts of polynomial values

Let C be a hyperelliptic curve defined over the rational numbers, and consider the set S of all squarefree integers d such that the quadratic twist of C by d has a rational point. In this talk we will discuss the question of whether, given a prime number p, the set S contains representatives from all congruence classes modulo p. When C has genus 0 this question can be answered using elementary number theory, but for higher genera it seems to require the use of big conjectures in arithmetic geometry.

March 13

Gunther Cornelissen (Utrecht)
Graph spectra and diophantine equations

I will show how to find uniform finiteness results for certain diophantine equations in terms of the Laplace spectrum of an associated graph. The method is to bound the "gonality" of a curve (minimal degree of a map onto a line) by the "stable gonality" of an associated stable reduction graph, and then to bound this stable gonality of the graph (some kind of minimal degree of a map to a tree) in terms of spectral data. The latter bound is a graph theoretical analogue of a famous inequality of Li and Yau in differential geometry. An example of an application is to bound the degree of the modular parametrisation of elliptic curves over function fields. (Joint work with Fumiharu Kato and Janne Kool.)

March 20