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| == Definite quaternion algebras and triple product p-adic L-functions == | == Definite quaternion algebras and triple product p-adic L-functions (Matt Greenberg) == |
Abstracts for the talks at Sage Days 36.
The state of p-adics in Sage (David Roe)
Sunday 13:00-14:00
p-adics in FLINT (Sebastian Pancratz)
Monday 11:00-12:00
Single factor lifting for polynomials over local fields (Sebastian Pauli)
Monday 13:00-14:00
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 14:00-15:00
Computing p-adic L-functions of elliptic curves (William Stein)
Tuesday 11:00-12:00
Definite quaternion algebras and triple product p-adic L-functions (Matt Greenberg)
An extremely useful formula of Ichino expresses central values of triple product L-functions in terms of trilinear forms evaluated on specific test vectors. I will discuss joint work with Marco Seveso in which we show that, in the "definite case," these trilinear forms can be p-adically interpolated, giving rise to triple product p-adic L-functions.
Tuesday 13:00-14:00
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 14:00-15:00
Computations with Witt vectors (Luis Finotti)
Wednesday 11:00-12:00
Explicit p-adic Hodge theory (Xavier Caruso)
Wednesday 13:00-14:00
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 16:30-17:30
Combinatorics and Geometry (Fernando Rodriguez-Villegas)
In this talk I will discuss a combinatorial calculation of the polynomial that counts the number of indecomposable representations of a certain quiver and dimension vector. I will start by introducing quivers, their representations and Kac's results and conjectures on such counting polynomials in general. The combinatorial calculation involves the reliability polynomial of alternating graphs. I will end with the main motivation for the calculation: its relation to the geometry of character varieties.
Thursday 16:00-17:00