Bay Area Algebraic Number Theory and Arithmetic Geometry Day II

Saturday, April 2, 2011
University of California, Berkeley
Room: Talks will be in Evans Hall, Room 60. There will be two coffee/snack breaks in Room 1015 on the 10th floor (see schedule below).


Speakers:

Manjul Bhargava (Princeton University)
Benedict Gross (Harvard University)
Payman Kassaei (King's College London)
Fernando Rodriguez-Villegas (University of Texas, Austin)
Melanie Wood (Stanford University)


Schedule:

All talks are in room 60 on the ground floor of Evans Hall

9:15-9:45 Coffee (downstairs, room 70)
9:45-10:45 Manjul Bhargava
10:45-11:30 Coffee/Snack Break (upstairs, room 1015)
11:30-12:30
Benedict Gross
12:30-2:00
Lunch
2:00-3:00 Melanie Wood
3:00-3:15
Break
3:15-4:15
Payman Kassaei
4:15-5:00
Coffee/Snack Break (upstairs, room 1015)
5:00-6:00
Fernando Rodriguez-Villegas
6:30
Dinner, Ruen Pair Thai Cuisine
Please RSVP to sdasgup2 (at) ucsc (dot) edu





Titles and Abstracts:

Manjul Bhargava, "A positive proportion of plane cubics fail the Hasse principle"
In 1957, Selmer showed that the plane cubic curve cut out by 3x3+4y3+5z3 has no rational point, despite having a point locally at all places of Q. That is, this curve fails the Hasse principle. How rare are such counterexamples to the Hasse principle among plane cubic curves? In this talk, using recent work with Arul Shankar, we show that in fact a positive proportion of all plane cubics (when ordered by the heights of their defining equations) fail the Hasse principle.

Benedict Gross, "Stable orbits and Selmer groups"
In this talk, which is a report on joint work with Manjul Bhargava, I will consider the orbits in two representations of the split orthogonal group SO(2n+1) = SO(V) over a field k, with char(k) not equal to 2. The first is the standard representation V. The second is the representation W on the submodule of Sym2(V) which is the kernel of the invariant bilinear form on V. The analysis of the stable orbits in W involves the 2-torsion subgroup of the Jacobians of hyperelliptic curves of genus n over k, with a k-rational Weierstrass point. When k is a global field, these orbits can be used to study the 2-Selmer group.

Payman Kassaei, "Analytic continuation over Hilbert modular varieties"
I will discuss some results regarding the analytic continuation of overconvergent p-adic Hilbert modular eigenforms in the case where p is unramified in the totally real field.

Fernando Rodriguez-Villegas, "Hypergeometric motives"
This is work in progress joint with H. Cohen. In this talk I will discuss the motives of the title and their associated L-functions. These families of motives arise from the classical hypergeometric differential equations in one variable and are defined by very simple data. They have the virtue that their Euler factors can be generally computed without recourse to an automorphic input or the direct counting of points of varieties over finite fields. This results in a large number of explicitly computable motivic L-functions with Euler factors of higher degree (three or more). A representative example is a degree four L-function coming from a piece of the middle cohomology of the Dwork pencil of quintic threefolds.

Melanie Wood, "Quartic orders associated to binary quartic forms"
It is natural to associate to a quartic number field the minimal (quartic) polynomial of a generating element. Moreover, a binary quartic form with integral coefficients gives rise naturally to a quartic ring. We will see how this relationship between quartic rings/fields and quartic polynomials relates to Bhargava's parametrization of quartic rings with their cubic resolvents by pairs of ternary quadratic forms. This will allow us to see that the quartic rings associated to binary quartic forms are exactly those with a monogenic cubic resolvent. We will be able to view this phenomenon from both a concrete and geometric point of view.


Parking:

To park in a campus parking lot, you can purchase a permit from a ticket dispenser located in the lot. Permits should be placed on your dashboard. It costs $12 to park all day, and the dispensers take credit cards. If you have a parking permit of the correct "strength" from another UC campus, you can park for free by displaying your UC permit.

A list of parking lots is available here; clicking on the first link "Campus Parking Lots" sends you to a Google map displaying the lots. Scroll down on the left panel until you see the name of the desired parking lot; click on the name and you get a description of the lot and its capacities.

Two recommended parking locations are the Upper Hearst Parking Structure and the Lower Hearst Parking Structure. The easiest solution is to park on level 1 of the Lower Hearst Structure. This structure is on the north side of Hearst Avenue, just south of Euclid. Because Hearst is a divided road, you need to be heading west on Hearst in order to turn into the lot. Walking to Evans Hall from either structure takes less than 5 minutes.


Registration:

There is no official registration, but please send an email to sdasgup2 (at) ucsc (dot) edu by Monday, March 28 if you are planning to attend (especially if you will be attending the dinner, see below).


Dinner:

After the talks, there will be a dinner at Ruen Pair Thai Cuisine at 6:30pm. Please send me an email if you are planning to attend so that we can give the restaurant an accurate head count.