Bay Area Algebraic Number Theory and Arithmetic Geometry Day I

Saturday, November 6, 2010
University of California, Santa Cruz
Engineering 2 building, Room 194
Located at B4 in the Campus Map grid.
Room 194 is located on the ground floor and opens directly to the outside. (It is not inside the building.)


Burcu Baran (Stanford University)
Joel Bellaiche (Brandeis University)
Brian Conrad (Stanford University)
Kai-Wen Lan (Princeton University/IAS)
George Schaeffer (University of California, Berkeley)


9:30-10:00 Coffee/Snacks
10:00-11:00 Joel Bellaiche
11:00-11:20 Coffee/Snack Break
Burcu Baran
2:00-3:00 Brian Conrad
Coffee/Snack Break
George Schaeffer
Coffee/Snack Break
Kai-Wen Lan

Titles and Abstracts:

Burcu Baran, "Computing a level-13 modular curve over Q via representation theory"
For any n > 0, let Xns(n) denote the modular curve over Q associated to the normalizer of a non-split Cartan subgroup of level n. The integral points and rational points of Xns(n) are crucial in two interesting problems: the class number one problem and Serre's uniformity conjecture. In this talk we focus on the genus-3 curve Xns(13). It has no Q-rational cusp (as for any level n > 2), so to compute an equation for this curve as a quartic in P2Q we use representation theory. With the same methods, we see that there exists a surprising exceptional Q-isomorphism to another modular curve. The j-function of degree 78 on Xns(13) will also be explicitly given at the end of the talk.

Joel Bellaiche, "p-adic adjoint L-function and ramification points on the eigencurve (after Walter Kim)"
In his 2006 Berkeley PhD thesis (a remarkable work), Walter Kim constructed a function on the eigencurve interpolating the values at 1 of the adjoint L-function of modular forms, and related its zero locus to the ramification locus of the eigencurve over the weight space. Unfortunately, some arguments were incomplete and Kim left academia without publishing his thesis, which has been a little bit overlooked since then. In this talk, I will explain Kim's results and arguments, and show how one can precise and generalize them.

Brian Conrad, "A local-global principle for nth roots of characters"
The Grunwald-Wang theorem characterizes when the local-global principle holds for nth roots in a global field, allowing to ignore a specified finite set of places. This is important when constructing global finite-order Galois characters with finitely many specified local restrictions. A natural and useful variant has not been addressed before: does the local-global principle hold for extracting the nth root of a global Galois character (with values in C* or Ql*)? The answer is closely tied up with the original Grunwald-Wang theorem and Tate global duality. We explain the solution, address refinements with control on local ramification (including l-adic Hodge theory aspects), and give explicit examples to illustrate exceptional behavior that can arise.

Kai-Wen Lan, "Vanishing theorems for torsion automorphic sheaves"
In this talk, I will explain my joint work with Junecue Suh on when and why the cohomology of Shimura varieties (with nontrivial integral coefficients) has no torsion, based on certain vanishing theorems we proved recently. (All conditions involved will be explicit, independent of level, and effectively computable.)

George Schaeffer, "Computing weight $1$ modular forms mod p"
Weight 1 modular forms mod p and their associated Galois representations have many unusual arithmetic properties, but computational challenges have somewhat impeded their study. In this talk, we discuss an efficient heuristic/proto-algorithm for computing q-expansions of weight 1 modular forms mod p which builds on ideas of K. Buzzard.

Lunch options:

Subs and pizzas

Burritos, etc.

Dining Hall brunch

(It costs $10+tax for the dining hall, but if enough people are interested, let me know and I can organize a group rate.)


The campus parking map is here. Parking in the Core West parking structure, located at C3 in the campus parking map grid, is free with no permit required on weekends (avoid the pay stalls on the second level and park in the regular unmarked spaces).