Joint IAS/PU Number Theory

The standard \(L\)-function for \(G_2\): a "new way"

Nadya Gurevich
Ben-Gurion University of the Negev
October 2, 2014
We consider a Rankin-Selberg integral representation of a cuspidal (not necessarily generic) representation of the exceptional group \(G_2\). Although the integral unfolds with a non-unique model, it turns out to be Eulerian and represents the standard \(L\)-function of degree 7. We discuss a general approach to the integrals with non-unique models. The integral can be used to describe the representations of \(G_2\) for which the (twisted) \(L\)-function has a pole as functorial lifts. This is a joint work with Avner Segal.

Recovering elliptic curves from their \(p\)-torsion

Benjamin Bakker
New York University
May 2, 2014
Given an elliptic curve \(E\) over a field \(k\), its \(p\)-torsion \(E[p]\) gives a 2-dimensional representation of the Galois group \(G_k\) over \(\mathbb F_p\). The Frey-Mazur conjecture asserts that for \(k= \mathbb Q\) and \(p > 13\), \(E\) is in fact determined up to isogeny by the representation \(E[p]\). In joint work with J.

Applications of additive combinatorics to Diophantine equations

Alexei Skorobogatov
Imperial College London
April 10, 2014
The work of Green, Tao and Ziegler can be used to prove existence and approximation properties for rational solutions of the Diophantine equations that describe representations of a product of norm forms by a product of linear polynomials. One can also prove that the Brauer-Manin obstruction precisely describes the closure of rational points in the adelic points for pencils of conics and quadrics over \(\mathbb Q\) when the degenerate fibres are all defined over \(\mathbb Q\).

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 \(\text{lt}_Z(K, u)\in \overline {\mathbb Q}_\ell \).