Saturday, December 5, 2015

University of California, Santa Cruz

Room: McHenry Library, room 1240

Jeremy Booher, Stanford University

Jayce Getz, Duke University

Matthew Greenberg, University of Calgary

Rachel Pries, Colorado State University

Zhiwei Yun, Stanford University

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

10:00-11:00 | Rachel Pries |

11:00-11:15 | Coffee Break |

11:15-12:15 |
Jayce Getz |

12:15-1:45 |
Lunch |

1:45-2:45 | Zhiwei Yun |

2:45-3:10 |
Coffee Break |

3:10-4:10 |
Jeremy Booher |

4:10-4:30 |
Break |

4:30-5:30 |
Matthew Greenberg |

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

For a representation of the absolute Galois group of the rationals over a finite field of characteristic p, we would like to know if there exists a lift to characteristic zero with nice properties. In particular, we would like it to be geometric in the sense of the Fontaine-Mazur conjecture: ramified at finitely many primes and potentially semistable at p. For two-dimensional representations, Ramakrishna proved that under technical assumptions odd representations admit geometric lifts. We generalize this to higher dimensional orthogonal and symplectic representations. The key ingredient is a smooth local deformation condition obtained by analysing unipotent orbits and their centralizers in the relative situation, not just over fields.

We introduce the Langlands' beyond endoscopy proposal from the perspective of the Kuznetsov formula (following Sarnak and Venkatesh). Special attention will be paid to the analytic difficulties of the proposal and how certain modifications of the formula might provide avenues to surmount these difficulties.

In this talk, I will describe the construction of a p-adic period integral interpolating, over balanced subsets of weight space, the period integral functional appearing in Ichino's special value formula for the triple product L-function. Since Ichino's formula relates period integrals to special values of L-functions, these p-adic period integrals can be considered as p-adic L-functions. The key step in the construction is proving that the classical branching laws for representations of GL

In positive characteristic, an elliptic curve can be ordinary or supersingular. For abelian varieties of higher dimension, there are several invariants which can be used to generalize this distinction, including the p-rank, Newton polygon, or Ekedahl-Oort type. There are well-understood stratifications of the moduli space of principally polarized abelian varieties in positive characteristic by these invariants. In the first half of the talk, I will describe the current state of knowledge about which of these invariants occur for Jacobians of curves and how the Torelli locus intersects these strata. In the second half of the talk, I will discuss recent research with Ekin Ozman about p-ranks of Prym varieties.

In joint work with Wei Zhang, we prove a generalization of the Gross-Zagier formula in the function field setting. Our formula relates self-intersection number of certain cycles on the moduli of Drinfeld Shtukas for PGL(2) to higher derivatives of automorphic L-functions at the central point.

The closest parking is at the Hahn parking lot. Parking is free and no permit is required on Saturdays, as long as you park in a legal parking space and do not block emergency access, or park in spaces that are posted and reserved for special use. There is a foot bridge that connects the Hahn parking lot to McHenry Library. If you click on this map and zoom out one level, the bridge is indicated by a dotted line. Halfway through the bridge, there is a path that leads to the left, but you want to ignore this path and continue straight until the end of the bridge. After crossing the bridge, make a sharp left slightly downhill to enter the building at the first level. (The main entrance is actually on the second level, so if you enter at the main entrance you will have to take the elevator down one level.)

Here are printable maps: McHenry Library, and more specifically how to get to Hahn parking lot and walk to McHenry.

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 Laili Restaurant in Downtown Santa Cruz. Please send an email to sdasgup2 at ucsc dot edu if you plan to attend. Graduate students will be partially subsidized at the dinner. We thank the UC Santa Cruz Mathematics Department, the Math Research Center at Stanford University, and the UC Berkeley Mathematics Department for partial financial support.