Bay Area Algebraic Number Theory and Arithmetic Geometry Day 5
Saturday, November 10, 2012
University of California, Santa Cruz
Room: McHenry Library, room 1240
Michael Daub, University of California, Berkeley
Martin Luu, Stanford University
Cristian Popescu, University of California, San Diego
Michael Spiess, Universitat Bielefeld
Xinyi Yuan, University of California, Berkeley
All talks will be in the ground floor of McHenry Library at UCSC, room 1240. Refreshments (including the morning coffee)
will be in room 1257.
|Dinner, to be announced
Please RSVP to sdasgup2 (at) ucsc (dot) edu
Titles and Abstracts:
Michael Daub, "Computations of Chow-Heegner points via p-adic methods"
I will present a recent construction by Darmon and Rotger of rational points on modular elliptic curves via intersections of algebraic cycles on products of Kuga-Sato varieties. In the case of a triple product of modular curves, the conditions under which the points are nontrivial are well understood thanks to the generalized Gross-Zagier formula of Yuan-Zhang-Zhang. Upon replacing two of the curves in the product with two copies of a Kuga-Sato variety, a similar situation is expected to emerge, although it is still conjectural due to the lack of a Gross-Zagier formula in this setting. I will demonstrate how these points can be computed using the theory of p-adic modular forms, allowing for computational verification of certain parts of the conjecture.
Martin Luu, "Congruences and Monodromy"
I will describe how congruences of automorphic forms can be used to calculate monodromy operators of automorphic Galois representations. In particular, I will talk about Hilbert modular forms of partial weight one: For such forms the construction by Jarvis of the Galois representation can not tell non-triviality of monodromy at places of Steinberg ramification. I will also explain how gamma-factors from the doubling method can be used in this circle of ideas, for example for automorphic forms on symplectic groups. Along the way, I will talk about some new potential level-lowering results and how they fit into the above described relation between congruences and monodromy operators.
Cristian Popescu, "The Arithmetic of Special Values"
I will describe some of my recent joint work with
various coauthors on several conjectures on special values of
equivariant Artin L-functions.
Michael Spiess, "Shintani cocycle and p-adic L-series of totally real fields"
We give a cohomological construction of the p-adic
L-series Lp(χ, s) of a Hecke character χ of a totally real
number field F. As an application we explain a purely analytic
proof of the fact that Lp(χ,s) has a trivial zero at
s=0 of order at least equal to the number of places of F above
p where the local component of χ is trivial.
"Heegner points and congruent numbers"
In this talk, I will talk about the recent work of Ye Tian
on the congruent number problem. The main result claims a large class
of integers to be congruent. The proof uses Heegner points and the
Gross-Zagier formula proved by Yuan-Zhang-Zhang.
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 6:15pm at
Le Cigare Volant
in 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.