# NTS/Abstracts/Fall2010

## Contents

- 1 Jordan Ellenberg, UW Madison
- 2 Shuichiro Takeda, Purdue
- 3 Xinyi Yuan
- 4 Jared Weinstein, IAS
- 5 David Zywina, U Penn
- 6 Soroosh Yazdani, UBC and SFU
- 7 Zhiwei Yun, MIT
- 8 David Brown, UW Madison
- 9 Bryden Cais, UW Madison
- 10 Tom Hales, University of Pittsburg
- 11 Jay Pottharst, Boston University
- 12 Melanie Matchett Wood, Stanford and AIM
- 13 Samit Dasgupta, UC Santa Cruz
- 14 David Geraghty, Princeton and IAS
- 15 Toby Gee, Northwestern
- 16 Organizer contact information

## Jordan Ellenberg, UW Madison

Title: Expander graphs, gonality, and Galois representations |

Abstract: TBA |

## Shuichiro Takeda, Purdue

Title: On the regularized Siegel-Weil formula for the second terms and
non-vanishing of theta lifts from orthogonal groups |

Abstract: In this talk, we will discuss (a certain form of) the Siegel-Weil formula for the second terms (the weak second term identity). If time permits, we will give an application of the Siegel-Weil formula to non-vanishing problems of theta lifts. (This is a joint with W. Gan.) |

## Xinyi Yuan

Volumes of arithmetic line bundles and equidistribution |

In this talk, I will introduce equidistribution of small points in algebraic dynamical systems. The result is a corollary of the differentiability of volumes of arithmetic line bundles in Arakelov geometry. For example, the equidistribution theorem on abelian varieties by Szpiro-Ullmo-Zhang is a consequence of the arithmetic Hilbert-Samuel formula by Gillet-Soule. |

## Jared Weinstein, IAS

Title: Resolution of singularities on the tower of modular curves |

Abstract: The family of modular curves X(p^n) provides the geometric link between two types of objects: On the one hand, 2-dimensional representations of the absolute Galois group of Q_p, and on the other, admissible representations of the group GL_2(Q_p). This relationship, known as the local Langlands correspondence, is realized in the cohomology of the modular curves. Unfortunately, the Galois-module structure of the cohomology of X(p^n) is obscured by the fact that integral models have very bad reduction. In this talk we present a new combinatorial picture of the resolution of singularities of the tower of modular curves, and demonstrate how this picture encodes some features of the local Langlands correspondence. |

## David Zywina, U Penn

Title: Bounds for Serre's open image theorem |

Abstract. |

## Soroosh Yazdani, UBC and SFU

Title: Local Szpiro Conjecture |

For any elliptic curve E over Q, let N(E) and Delta(E) denote it's conductor and minimal discriminant. Szpiro conjecture states that for any epsilon>0, there exists a constant C such that Abs(Delta(E)) < C (N(E))^{6+\epsilon} for any elliptic curve E. This conjecture, if true, will have applications to many Diophantine equations. Assuming Szpiro conjecture, one expects that there are only finitely many semistable elliptic curves E such that min_{p|N(E)} v_p(\Delta(E)) >6. We conjecture that, in fact, there are none. In this talk we study this conjecture in some special cases, and provide some evidence towards this conjecture. |

## Zhiwei Yun, MIT

Title: From automorphic forms to Kloosterman sheaves (joint work with J.Heinloth and B-C.Ngo) |

Abstract: Classical Kloosterman sheaves are rank n local systems on the punctured line (over a finite field) which incarnate Kloosterman sums in a geometric way. The arithmetic properties of the Kloosterman sums (such as estimate of absolute values and distribution of angles) can be deduced from geometric properties of these sheaves. In this talk, we will construct generalized Kloosterman local systems with an arbitrary reductive structure group using the geometric Langlands correspondence. They provide new examples of exponential sums with nice arithmetic properties. In particular, we will see exponential sums whose equidistribution laws are controlled by exceptional groups E_7,E_8,F_4 and G_2. |

## David Brown, UW Madison

Title |

Abstract. |

## Bryden Cais, UW Madison

Title: On the restriction of crystalline Galois representations |

Abstract: We formulate a generalization of a conjecture of Breuil (now a theorem of Kisin) on the restriction of crystalline p-adic Galois
representations to a general class of infinite index subgroups of the Galois group. Following arguments of Breuil, we will explain the proof of our generalization in the Barsotti-Tate case. |

## Tom Hales, University of Pittsburg

Title |

At the International Congress of Mathematicians in India in August, Ngo Bao Chau was awarded a Fields medal for his proof of the "Fundamental Lemma." This talk is particularly intended for students and mathematicians who are not specialists in the theory of Automorphic Representions. I will describe the significance and some of the applications of the "Fundamental Lemma." I will explain why this problem turned out to be so difficult to solve and will give some of the key ideas that go into the proof. |

## Jay Pottharst, Boston University

Title: Iwasawa theory at nonordinary primes |

Abstract. |

## Melanie Matchett Wood, Stanford and AIM

Title: Geometric parametrizations of ideal classes |

In a ring of algebraic integers, the ideal class group measures the failure of unique factorization. A classical correspondence due to Dirichlet and Dedekind allows us to work with ideal classes of quadratic rings concretely in terms of binary quadratic forms with integer coefficients. A recent result of Bhargava gives an analogous correspondence between ideal classes of cubic rings and certain trilinear forms. From another point of view, the ideal class group is the group of invertible modules of a ring, whose geometric analog is the Picard group of line bundles on a space. We discuss how we can view these correspondences between ideal classes and forms geometrically, and give new results on parametrizations of ideal classes of certain rank n rings (e.g. orders in degree n number fields) by trilinear forms. |

## Samit Dasgupta, UC Santa Cruz

On Greenberg's conjecture on derivatives of p-adic L-functions with
trivial zeroes |

In 1991, Ralph Greenberg stated a conjecture about p-adic L-functions that have a trivial zero at s=1. Here "trivial" means that the zero arises from the vanishing of an Euler factor that must be removed in order to state the interpolation property of the p-adic L-function. Greenberg's conjecture concerns the value of the derivative of the p-adic L-function at s=1. An example of this conjecture is the case of the p-adic L-function of an elliptic curve E/Q with split multiplicative reduction at p. In this case, Greenberg's conjecture reduces to an earlier conjecture by Mazur, Tate, and Teitelbaum, and was proven by Greenberg himself in joint work with Glenn Stevens. In this talk, we will describe a strategy to prove new cases of Greenberg's conjecture. We will concentrate on the case of the symmetric square of an elliptic curve with good reduction at p. The strategy is a generalization of my previous work with Darmon and Pollack proving certain cases of the Gross--Stark conjecture (which can also be viewed as a special case of Greenberg's conjecture). The method involves studying explicit p-adic families of modular forms on GSp_4 and their associated Galois representations. |

## David Geraghty, Princeton and IAS

Title: Potential automorphy for compatible systems |

Abstract: I will describe a joint work with Barnet-Lamb, Gee and Taylor where we establish a potential automorphy result for compatible systems of Galois representations over totally real and CM fields. This is deduced from a potential automorphy result for single l-adic Galois representations satisfying a `diagonalizability' condition at the places dividing l. |

## Toby Gee, Northwestern

Title: Potential automorphy for compatible systems |

Abstract: I will continue David Geraghty's talk, and discuss a number of applications. |

## Organizer contact information

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