# Difference between revisions of "NTS/Abstracts/Fall2010"

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− | + | 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 | ||

+ | |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. | ||

|} | |} | ||

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## Revision as of 21:27, 7 October 2010

## 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 Bryden Cais, UW Madison
- 9 David Brown, UW Madison
- 10 Jay Pottharst, Boston University
- 11 Alex Paulin, Berkeley
- 12 Samit Dasgupta, UC Santa Cruz
- 13 David Geraghty, Princeton and IAS
- 14 Toby Gee, Northwestern
- 15 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 |
< 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. |

## Bryden Cais, UW Madison

Title |

Abstract. |

## David Brown, UW Madison

Title |

Abstract. |

## Jay Pottharst, Boston University

Title: Iwasawa theory at nonordinary primes |

Abstract. |

## Alex Paulin, Berkeley

Title |

Abstract. |

## Samit Dasgupta, UC Santa Cruz

Title |

Abstract. |

## David Geraghty, Princeton and IAS

Title |

Abstract. |

## Toby Gee, Northwestern

Title |

Abstract. |

## Organizer contact information

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