Abstracts for the talks at Sage Days 36.
Stable p-adic recursions (Joe Buhler)
Michael Somos and Lewis Carroll found, respectively, one (circa 1990) and two dimensional (circa 1866) recursions that, unexpectedly, generate integers and satisfy the so-called Laurent phenomenon. David Robbins observed that the Carroll recursion seems to have an unexpected stability over DVRs when computed with finite precision. This conjecture seems to apply to the Somos recursions and to the much more general context of cluster algebras. I'll explain how some special cases of these conjectures can be proved, discuss some techniques for p-adic experiments on these conjectures, and explain how the conjectures can be reinterpreted in an entirely algebraic context that generalizes the Laurent phenomenon. This is joint work with Kiran Kedlaya.
Monday 2:00-3:00
Arithmetic aspects of triangle groups (John Voight)
Triangle groups, the symmetry groups of tessellations of the hyperbolic plane by triangles, have been studied since early work of Hecke and of Klein--the most famous triangle group being SL_2(ZZ). We present a construction of congruence subgroups of triangle groups (joint with Pete L. Clark) that gives rise to curves analogous to the modular curves, and provide some applications to arithmetic. We conclude with some computations that highlight the interesting features of these curves.
Tuesday 2:00-3:00
The divisibility of the Tate-Shafarevich group of an elliptic curve in the Weil-Chatelet group (Mirela Ciperiani)
In this talk I will report on progress on the following two questions, the first posed by Cassels in 1961 and the second considered by Bashmakov in 1974. The first question is whether the elements of the Tate-Shafarevich group are infinitely divisible when considered as elements of the Weil-Chatelet group. The second question concerns the intersection of the Tate-Shafarevich group with the maximal divisible subgroup of the Weil-Chatelet group. This is joint work with Jakob Stix.
Wednesday 4:30-5:30