JHU Number Theory Seminar, Fall 2005Place: 304 Krieger |
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| Time | Speaker | Title | |
| September 21 4:30pm--5:30pm |
Kathrin Bringmann University of Wisconsin at Madison |
Arithmetic properties of coefficients of Maass Poincare series of half-integral weight We generalize a result of Zagier, describing the duality of weakly holomorphic modular forms of weight 1/2 and 3/2. We show that Zagier's result is a special case of a generic duality for general weight. For this we consider Maass Poincare series and conclude the duality by comparing Fourier coefficients. Moreover we write these coefficients as traces of singular moduli. As a plus we obtain exact formulas for these traces. |
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| September 27 4:30pm--5:30pm |
Critian Popescu UCSD |
Gross-Rubin-Stark-Tate We discuss a refinement of the Rubin-Stark Conjecture for abelian $L$--functions of arbitrary order of vanishing at $s=0$. This generalizes Gross's $v$-adic refinement of the abelian, order of vanishing $1$, integral Stark Conjecture and predicts a link between special values of derivatives of $p$--adic and global $L$-functions. Time permitting, we will also show how our refinement relates to a recent strengthening of Gross's Conjecture due to Tate. |
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| September 28 4:30pm--5:30pm |
Erez Lapid Hebrew University of Jerusalem and IAS |
The relative trace formula and its applications The relative trace formula is a variant of the Arthur-Selberg trace formula which is a tool to study periods of automorphic forms over suitable period subgroups. It was invented by Jacquet. I will explain its scope through some examples and tell about recent developments. |
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| Octorber 19 4:30pm--5:30pm |
Amir Jafari IAS |
Algebraic cycles, iterated integrals and trees This is a joint work with H. Furusho. Bloch and Kriz described a Hopf algebra, built out of algebraic cycles, which should be the coordinate ring of the Tannakian fundamnetal group of the category of mixed Tate motives. They explained how to define elements of this Hopf algebra that correspond to the polylogarithms. I will explain how to extend this work to include all generic iterated integrals of the punctured projective line. Gangle, Goncharov and Levin have also found similar results. |
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| Octorber 21 4:30pm--5:30pm |
Hidekazu Furusho Nagoya University |
Generalized double shuffle relations for p-adic multiple zeta values This is a joint work with A.Jafari. In my talk I will explain the regularization of p-adic multiple zeta values and prove double shuffle relations for them. |
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| November 9 4:00pm--5:00pm |
Alina Ioana Bucur Brown University |
Moments of quadratic Dirichlet L-functions In this talk I will outline the methods used in the proof of the fourth moment conjecture of quadratic Dirichlet L-functions over rational function fields. The challenge here comes from the fact that the multiple Dirichlet series has an infinite group of functional equations, as opposed to the first three moments where the group is finite. This has proved to be a major stumbling block in the past, but Adrian Diaconu and myself managed to make progress in dealing with this situation. |
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| November 16 4:00pm--5:00pm |
Jayce Getz University of Wisconsin at Madison |
Hilbert modular generating series with coefficients in intersection homology In a famous paper, Hirzebruch and Zagier considered families of homology classes $\{Z_m\}_{m \in \ZZ_{\geq 0}}$ on certain Hilbert modular surfaces and showed that the generating series $\sum_{n=0}^{\infty} Z_m \cdot Z_n q^n$ are elliptic modular forms with nebentypus. This work can be seen as giving a geometric interpretation of the Doi-Naganuma lifting. We prove the modularity of analogous generating series in the context of intersection homology classes on Hilbert modular varieties of arbitrary dimension. This a joint project with M. Goresky. The aim of this work-in-progress is to give a geometric/topological interpretation of abelian base change. |
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| November 30 4:00pm--5:00pm |
Alexandru Popa Princeton University |
Special values of Rankin L-functions over real quadratic fields and applications We discuss two applications of a central value formula for the Rankin L-function of a cuspidal modular form f of even weight and square-free level N, twisted by narrow class characters of a real quadratic field K. In the first application, subconvexity bounds for the central value when $f$ is a weight 0 Maass Form imply the equidistribution of _individual_ long closed geodesics on the modular curve X_0(N), generalizing a result of Duke on the equidistribution of _all_ geodesics attached to K. The second application regards the base change of an elliptic curve E defined over the rationals, to K. Assuming the Birch and Swinnerton-Dyer conjecture, when E(K) is finite we give an interpretation for the order of the Tate-Shafarevich group of E over K in terms of the homology class of the sum of geodesic cycles attached to the quadratic field K. |
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