Southern California Number Theory Day 2009
Saturday, February 28, 2009 on the Caltech Campus
Speakers Location Schedule Abstracts Parking Directions to Caltech Dinner Contact
Invited Speakers: Christopher Skinner (Princeton University), David Helm (University of Texas), Ben Howard (Boston College), Philippe Michel (Ecole Polytechnique Federale de Lausanne)
Location: All talks will be held in the Sloan Building (Department of Math) room 151 on the Caltech campus. Click on campus map to locate Sloan building (we are building number 37).
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Click here to see list of registered participants
Schedule:
9:30  10:00  Refreshments
10:00  11:00  Christopher Skinner (Princeton University),
"Families of automorphic forms and applications"
11:00  11:30  Break
11:30  12:30  David Helm (University of Texas),
"On ladic families of admissible representatons of GL_2(Q_p)"
12:30  2:30  Lunch Break
2:30  3:30  Ben Howard (Boston College),
"Intersection theory on Shimura surfaces"
3:30  4:00  Break
4:00  5:00  Philippe Michel (Ecole Polytechnique Federale de Lausanne),
"The subconvexity problem for GL_2"
5:30  Dinner  see below
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Abstracts:

Christopher Skinner (Princeton University)
Families of automorphic forms and applications
Families of automorphic forms, particularly padic families, have been a key ingredient in efforts to understand the arithmetic significance of Lvalues. I will discuss some recent results about the existence of such families and their connections to values of Lfunctions and order of Selmer groups.
Biography 

David Helm (University of Texas)
On ladic families of admissible representations of GL_{2}(Q_{p})
The “mod l” local Langlands correspondence of MarieFrance Vigneras establishes a bijection between admissible representations of GL_{n}(Q_{p}) and ndimensional Frobeniussemisimple representations of G_{{Qp}} over an algebraic closure of F. For n = 2, we compare the deformation theory of a given admissible representation with that of the corresponding galois representation; in almost all cases there is a canonical isomorphism between the deformation rings. The failure of this to occur in all cases is closely connected to the existence of congruences between modular forms; when such congruences exist the deformation theory of the Galois representation is “richer” than that of the corresponding admissible representation.
A result of Matthew Emerton provides a way of remedying this deficiency.
His ideas give rise to a notion of "family of admissible representations'' that is weaker than the deformation theoretic notion.
We show that using this weaker notion one can associate a family of admissible representations to a given ladic family of Galois representations in a way that gives a "local Langlands correspondence in families.''


Ben Howard (Boston College)
Intersection theory on Shimura surfaces
Kudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura varieties to Fourier coefficients of Eisenstein series. The lowest dimensional case, in which one intersects two codimension one cycles on the integral model of a Shimura curve, has been completed by KudlaRapoportYang. I will describe results in a higher dimensional setting: on the integral model of a Shimura surface one can consider the intersection of an embedded Shimura curve with a family of codimension two cycles of complex multiplication points. The intersection numbers of these cycles are related to Fourier coefficients of a Hilbert modular form of halfintegral weight.
Ben Howard was an undergraduate at the University of Chicago, and went on to complete his Ph.D. at Stanford University under the supervision of Karl Rubin. He is currently an associate professor at Boston College. 

Philippe Michel (Ecole Polytechnique Federale de Lausanne)
The subconvexity problem for GL_{2}
In this talk we will describe the general subconvexity problem for central value of Lfunctions.
We will also explain the resolution of this problem for GL_{1} and GL_{2} automorphic Lfunctions over a general number field, and this uniformly in all parameters (the spectral, level and saspects). The main ingredient of the proof are suitable representations of the central values in terms of automorphic periods which factor over local period integrals of matrix coefficients, the spectral decomposition of such periods the spectral gap property for GL_{2} matrix coefficients and the amplification method of Iwaniec If time permits we will also describe an application explained to us by Andre Reznikov of the uniformity of our bounds to the study of
the restriction of Maass forms of large Laplace eigenvalue along a fixed closed geodesic. This is joint work with Akshay Venkatesh. 
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Parking: Parking at Caltech is free on the weekends. Please park in the lot located on California Blvd which can be located on this map.
Directions: Directions to Caltech can be found by following this link. A map of the Pasadena area can be found here.
Dinner: The dinner following the talk will be held at Heidar Baba Restaurant. It is within walking distance of Caltech. Food will be served buffet style and there will be vegarian options. The charge is $20 per person.
Contact Organizers: Dinakar Ramakrishnan, Matthias Flach, Elena Mantovan
