Abstracts for the talks at Sage Days 36.
The state of p-adics in Sage (David Roe)
p-adics in FLINT (Sebastian Pancratz)
Slides as of Monday 20 Feb, 12:50pm.
Single factor lifting for polynomials over local fields (Sebastian Pauli)
Slides as of Monday 20 Feb, 5:30pm.
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.
Computing p-adic L-functions attached to elliptic curves (William Stein)
I will describe what Sage currently includes for computing p-adic L-functions of elliptic curves and what is available in PSAGE. Then I will discuss Trac #12545, which is about polishing up the PSAGE code for inclusion in Sage and adding some additional functionality. All the code mentioned above uses Riemann sums, so is straightforward and fast for low precision, but slow as molasses if one wants high precision. I'll end my talk by briefly discuss Trac #812, which involves rewriting Robert Pollack's overconvergent modular symbols code for inclusion in Sage.
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.
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.
Computations with Witt vectors (Luis Finotti)
Computations with Witt vectors, when there is no canonical isomorphism onto a well known ring, can be quite demanding. Even computing the polynomials which define sums and products can be impractical (or even impossible) for large lengths. In this talk we will discuss alternative methods that avoid the computation of such polynomials and that can speed up the computations considerably in many cases. (The ideas involved are entirely elementary.)
Explicit p-adic Hodge theory (Xavier Caruso)
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.
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.