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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. |
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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. |
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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. |
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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. |
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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. |
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February 8 |
NO MEETING |
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February 15 |
Peter Wear (UCSD) |
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February 22 |
Shrenik Shah (Columbia) + pre-talk The class number formula connects the residue of the Dedekind zeta function at s=1 to the regulator, which measures the covolume of the lattice generated by logarithms of units. Beilinson defined a vast generalization of the regulator morphism and conjectured a class number formula associated to the cohomology of any smooth proper variety over a number field. His formula provides arithmetic meaning for the orders of the so-called trivial zeros of L-functions at integer points as well as the value of the first nonzero derivative at these points. We study this conjecture for the middle degree cohomology of compactified Shimura varieties associated to unitary groups of signature (2,1) and (2,2) over Q. We construct explicit Beilinson-Flach elements in the motivic cohomology of these varieties and compute their regulator. This is joint work with Aaron Pollack. |
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March 1 |
Taylor McAdam (UCSD) no pre-talk Equidistribution results play an important role in dynamical systems and their applications in number theory. Often in such applications it is necessary for equidistribution to be effective (i.e. the rate of convergence is known) so that number-theoretic methods such as sieving can be applied. In this talk, we will give a brief history of effective equidistribution results in homogeneous dynamics and their applications to number theory. We will then present an effective equidistribution result for certain flows on the space of lattices and discuss a number-theoretic application regarding almost-prime times for these flows. |
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March 8 |
Sean Howe (Stanford) + pre-talk In a 1987 letter to Tate, Serre showed that the Hecke eigensystems appearing in mod p modular forms are the same as those appearing in mod p functions on a finite double coset constructed from the quaternion algebra ramified at p and infinity. At the end of the letter, he asked whether there might be a similar relation between p-adic modular forms and p-adic functions on the quaternion algebra. We show the answer is yes: the completed Hecke algebra of p-adic modular forms is the same as the completed Hecke algebra of naive p-adic automorphic functions on the quaternion algebra. The resulting p-adic Jacquet-Langlands correspondence is richer than the classical Jacquet-Langlands correspondence -- for example, Ramanujan's delta function, which is invisible to the classical correspondence, appears. The proof is a lifting of Serre's geometric argument from characteristic p to characteristic zero; the quaternionic double coset is realized as a fiber of the Hodge-Tate period map, and eigensystems are extended off of the fiber using a variant of Scholze's fake Hasse invariants. |
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March 15 |
NO MEETING |