January 11 |
Nathan Green (Texas A&M) no pre-talk We study tensor powers of rank 1 Drinfeld A-modules, where A is the affine coordinate ring of an elliptic curve. Using the theory of A-motives, we find explicit formulas for the A-action of these modules. Then, by developing the theory of vector valued Anderson generating functions, we give formulas for the coefficients of the logarithm and exponential functions associated to these A-modules, as well as formulas for the fun- damental period. This allows us to relate function field zeta values to evaluations of the logarithm function and prove transcendence facts about these zeta values. |
January 12 |
Joseph Ferrara (UC Santa Cruz) no pre-talk; APM 6402 In the 1970’s Stark made precise conjectures about the leading term of the Taylor series expansion at s = 0 of Artin L-functions, refining Dirichlet’s class number formula. Around the same time Barsky, Cassou-Nogu`es, and Deligne and Ribet for totally real fields, along with Katz for CM fields defined p-adic L-functions of ray class characters. Since then Stark-type conjectures for these p-adic L-functions have been formulated, and progress has been made in some cases. The goal of this talk is to discuss a new definition of a p-adic L-function and Stark conjecture for a mixed signature character of a real quadratic field. After stating the definition and conjecture, theoretical and numerical evidence will be discussed. |
January 18 |
Florian Sprung (Arizona State) + pre-talk Iwasawa theory is a bridge between algebraic and analytic invariants attached to an arithmetic object, for a given prime p. When this arithmetic object is an elliptic curve or a modular form, the primes come in two flavors -- ordinary and supersingular. When p is ordinary, the theory has historically been relatively well behaved. When p is supersingular, there are several difficulties, and we explain how to address the difficulties involved in the case of elliptic curves, culminating in the proof of the Main Conjecture. If time permits, we will sketch joint work in progress with Castella, Ciperiani, and Skinner concerning main conjecture for weight-two modular forms. |
January 25 |
Daxin Xu (Caltech) + pre-talk Deninger and Werner developed an analogue for p-adic curves of the classical correspondence of Narasimhan and Seshadri between stable bundles of degree zero and unitary representations of the topological fundamental group for a complex smooth proper curve. Using parallel transport, they associated functorially to every vector bundle on a p-adic curve whose reduction is strongly semi-stable of degree 0 a p-adic representation of the etale fundamental group of the curve. They asked several questions: whether their functor is fully faithful and what is its essential image; whether the cohomology of the local systems produced by this functor admits a Hodge-Tate filtration; and whether their construction is compatible with the p-adic Simpson correspondence developed by Faltings. We will answer these questions in this talk. |
February 1 |
Koji Shimizu (Harvard) + pre-talk Sen attached to each p-adic Galois representation of a p-adic field a multiset of numbers called generalized Hodge-Tate weights. In this talk, we regard a p-adic local system on a rigid analytic variety as a geometric family of Galois representations and show that the multiset of generalized Hodge-Tate weights of the local system is constant. |
February 8 |
TBA TBA |
February 15 |
TBA TBA |
February 22 |
Shrenik Shah (Columbia) TBA |
March 1 |
Taylor Mcadam (UCSD) TBA |
March 8 |
Sean Howe (Stanford) TBA |
March 15 |
TBA TBA |