Joint IAS/PU Number Theory

Complex analytic vanishing cycles for formal schemes

Vladimir Berkovich
Weizmann Institute of Sciences; Member, School of Mathematics
December 12, 2013
Let \(R={\cal O}_{{\bf C},0}\) be the ring of power series convergent in a neighborhood of zero in the complex plane. Every scheme \(\cal X\) of finite type over \(R\) defines a complex analytic space \({\cal X}^h\) over an open disc \(D\) of small radius with center at zero. The preimage of the punctured disc \(D^\ast=D\backslash\{0\}\) is denoted by \({\cal X}^h_\eta\), and the preimage of zero coincides with the analytification \({\cal X}_s^h\) of the closed fiber \({\cal X}_s\) of \(\cal X\).

Genus of abstract modular curves with level \(\ell\) structure

Ana Cadoret
Ecole Polytechnique; Member, School of Mathematics
November 21, 2013
To any bounded family of \(\mathbb F_\ell\)-linear representations of the etale fundamental of a curve \(X\) one can associate families of abstract modular curves which, in this setting, generalize the `usual' modular curves with level \(\ell\) structure (\(Y_0(\ell), Y_1(\ell), Y(\ell)\) etc.). Under mild hypotheses, it is expected that the genus (and even the geometric gonality) of these curves goes to \(\infty\) with \(\ell\). I will sketch a purely algebraic proof of the growth of the genus - working in particular in positive characteristic.

Independence of \(\ell\) and local terms

Martin Olsson
University of California, Berkeley
November 14, 2013
Let \(k\) be an algebraically closed field and let \(c:C\rightarrow X\times X\) be a correspondence. Let \(\ell \) be a prime invertible in \(k\) and let \(K\in D^b_c(X, \overline {\mathbb Q}_\ell )\) be a complex. An action of \(c\) on \(K\) is by definition a map \(u:c_1^*K\rightarrow c_2^!K\). For such an action one can define for each proper component \(Z\) of the fixed point scheme of \(c\) a local term \(\text{lt}_Z(K, u)\in \overline {\mathbb Q}_\ell \).

A Converse to a Theorem of Gross-Zaqier-Kolyvagin

Christopher Skinner
Princeton University; Member, School of Mathematics
April 4, 2013

The theorem of the title is that if the L-function L(E,s) of an elliptic curve E over the rationals vanishes to order r=0 or 1 at s=1 then the rank of the group of rational rational points of E equals r and the Tate-Shafarevich group of E is finite. This talk will describe an approach to the converse. The methods are mostly p-adic.

An Analogue of the Ichino-Ikeda Conjecture for Whittaker Coefficients of the Metaplectic Group

Erez Lapid
Hebrew University of Jerusalem and Weizmann Institute of Science
March 14, 2013

A few years ago Ichino-Ikeda formulated a quantitative version of the Gross-Prasad conjecture, modeled after the classical work of Waldspurger. This is a powerful local-to-global principle which is very suitable for analytic and arithmetic applications. One can formulate a Whittaker analogue of the Ichino-Ikeda conjecture. We use the descent method of Ginzburg-Rallis-Soudry to reduce the Whittaker version to a purely local identity which we prove in the p-adic case under some mild hypotheses. Joint work with Zhengyu Mao