Bay Area Algebraic Number Theory and Arithmetic Geometry Day 8

Saturday, April 26, 2014
Stanford University Mathematics Department (Building 380)
Morning coffee/bagels in 2nd floor common room
Talks in room 380-C (in basement)


Speakers:

Henri Darmon, McGill University
Michael Larsen, Indiana University
Michael Magee, University of California, Santa Cruz
Alice Silverberg, University of California, Irvine
Ander Steele, University of Calgary


Schedule:

9:30-10:00 Coffee/Bagels
10:00-11:00 Henri Darmon
11:00-11:15 Coffee Break
11:15-12:15
Ander Steele
12:15-1:45
Lunch
1:45-2:45 Alice Silverberg
2:45-3:10
Coffee Break
3:10-4:10
Michael Magee
4:10-4:30
Break
4:30-5:30
Michael Larsen
6:00
Dinner, Three Seasons





Titles and Abstracts:

Henri Darmon, “The Birch and Swinnerton Dyer conjecture for ring class characters of real quadratic fields”

Let E be an elliptic curve over Q and let χ be a ring class character of a real quadratic field K. I will explain the proof that the non-vanishing of the central critical value L(E/K,χ,1) of the Hasse-Weil L-series of E twisted by χ implies the triviality of the χ-component of the Mordell-Weil group of E, in line with a natural Galois-equivariant refinement of the Birch and Swinnerton-Dyer conjecture. The proof relies on Gross-Kudla-Schoen diagonal cycles and on their variation in p-adic families. Possible applications to the theory of “Stark-Heegner points” and to explicit class field theory for real quadratic fields will also be evoked. All of this is joint work with Victor Rotger.

Michael Magee, “Zero sets of Hecke polynomials on the sphere”

The eigenspaces of the Laplacian on the two dimensional sphere consist of homogeneous polynomials and occur with increasing dimension as the eigenvalue grows. I'll explain how one can remove this high multiplicity by using arithmetic Hecke operators which arise from the Hamilton quaternions. The resulting Hecke eigenfunctions are subject to predictions arising from random function theory and quantum chaos, in particular concerning the topology of their zero sets. I'll discuss what is known in this area and how one can try to study these 'nodal lines'.

Michael Larsen, “Diophantine properties of fields with finitely generated Galois group”

I will discuss a number of related conjectures concerning the rational points of varieties (especially curves and abelian varieties) over fields with finitely generated Galois group and present some evidence from algebraic number theory, Diophantine geometry, and additive combinatorics in support of these conjectures.

Alice Silverberg, “Deterministic elliptic curve primality proving for special sequences”

In joint work with Alexander Abatzoglou, Andrew Sutherland, and Angela Wong, we use elliptic curves with complex multiplication to obtain necessary and sufficient conditions for primality of integers in certain sequences. We give fast deterministic primality proving algorithms for such integers. We use these algorithms to efficiently search for very large primes, and prove the primality of several integers with more than 100,000 decimal digits, including some with over a million bits in their binary representations. We obtain the largest proven prime N for which no significant partial factorization of N-1 or N+1 is known. We also give a general framework, that builds on earlier work of Chudnovsky-Chudnovsky, Gross, and Denomme-Savin.

Ander Steele, “The Shintani modular symbol and evil Eisenstein series”

Let f be an eigenform on Γ1(N) ∩ Γ0(p), p not dividing N. If f is non-critical then the work of Amice and Velu provides us with a p-adic L-function Lp(f,s) interpolating the critical values of f. Thanks to the work of Pollack-Stevens and Bellaiche we now have a natural notion of Lp(f, s) even when f has critical slope. In this talk we will relate the Shintani modular symbol, which parameterizes special values of L-functions of totally real fields, to the p-adic L-functions of critical slope Eisenstein series. This gives a new computation, originally due to Bellaiche and Dasgupta, of the p-adic L-function of critical slope Eisenstein series.


Parking:

Parking is free and plentiful in the Oval and surrounding lots (on Roth Way and Lausen St) on weekends. Here is a campus map, with the math building (380) labelled "Math Corner."


Registration:

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.


Dinner:

There will be a dinner following the conference at 6pm at Three Seasons in downtown Palo Alto. Please send an email to sdasgup2 at ucsc dot edu if you plan to attend.