Saturday, January 28, 2017

Stanford University

Location: Mathematics Department (Building 380)

Talks in room 380-C (in basement)

Chantal David, Concordia

Brian Lawrence, Stanford

Philippe Michel, EPFL

Dinakar Ramakrishnan, Caltech

Zev Rosengarten, Stanford

9:30-10:00 | Coffee/Bagels |

10:00-11:00 | Brian Lawrence |

11:00-11:15 | Coffee Break |

11:15-12:15 |
Chantal David |

12:15-1:45 |
Lunch |

1:45-2:45 | Dinakar Ramakrishnan |

2:45-3:10 |
Coffee Break |

3:10-4:10 |
Zev Rosengarten |

4:10-4:30 |
Break |

4:30-5:30 |
Philippe Michel |

6:00 |
Dinner, Three Seasons Please RSVP to sdasgup2 (at) ucsc (dot) edu |

We investigate the average root number (i.e., sign of the functional equation) of 1-parameter families of elliptic curves over Q (i.e., elliptic curves over Q(t), or equivalently elliptic surfaces over Q). For most 1-parameter families, the average root number is predicted to be 0. Helfgott showed that under Chowlaâ€™s conjecture and the square-free conjecture, the average root number is 0 unless the curve has no place of multiplicative reduction over Q(t). We build families with no place of multiplicative reduction and compute their average root number; some have periodic root number, giving a rational average, and others have an average expressed as an infinite Euler product.

We also prove several density results for the average root number of 1-parameter families, over Z and over Q, and exhibit some surprising examples. For instance, we make a non-isotrivial family with rank r over Q(t) for which the root number averaged across integral values of t is -(-1)

This is joint work with S. Bettin and C. Delaunay.

We explore the comparison isomorphism of p-adic Hodge theory in the case of elliptic curves, and discuss some ideas which may be used to prove the S-unit theorem and the finiteness of rational points on higher-genus curves (Faltings' theorem).

In a series of recent works Blomer, Fouvry, Kowalski, Milicevic, Sawin and myself have been able to solve the vexing problem of evaluating asymptotically the second moment of the central L-values of character twists (of large prime conductor) of a fixed modular form; the solution combine the spectral theory of modular forms, bounds for bilinear sums of Kloosterman sums and advanced methods in l-adic cohomology. We will describe this and some recent applications (by the same authors) to the non-vanishing of these central values.

After briefly recalling the basic questions on the points of smooth projective surfaces of general type over number fields, we will discuss a bit of progress for those covered by the unit ball. The talk will end with a discussion of a program (joint with M. Dimitrov) to establish a weak analogue of a result of Mazur for modular curves.

The results of Poitou and Tate on the cohomology of finite discrete Galois modules G over local and global fields (i.e., local duality, the 9-term exact sequence, etc.) are the fundamental results on the Galois cohomology for such fields. These results classically required the assumption that the order of G not be divisible by the characteristic of the field. Cesnavicius recently removed this assumption on the order, and thereby generalized the results of Poitou and Tate to arbitrary finite commutative group schemes (in modern language, the classical results only applied to finite etale group schemes with etale Cartier dual); for this, one must replace etale cohomology with fppf cohomology.

We generalize these results to arbitrary affine commutative group schemes of finite type. This involves several interesting new ideas, especially in the function field case. Time permitting, we will discuss an application to computing Tamagawa numbers, which was the initial motivation for this work.

Parking is free and plentiful in the Oval and surrounding lots (on Roth Way and Lausen St) on weekends. Here is a campus map, with the math building (380) labelled "Math Corner."

There is no formal registration, but if you plan to attend, we would appreciate an email to sdasgup2 at ucsc dot edu to
help plan the event, especially if you plan to attend the dinner afterwards.

There will be a dinner following the conference at 6:00pm, at Three Seasons in Palo Alto. Graduate students will be partially subsidized for the dinner. Thanks to the Stanford MRC and the math departments at UC Berkeley and UC Santa Cruz for support for the conference and the dinner.