On the restriction of irreducible representations of the group U_n(k) to the subgroup U_{n−1}(k)

Speaker: 

Benedict Gross

Institution: 

Harvard University

Time: 

Tuesday, February 17, 2009 - 2:00pm

Location: 

RH 306

Let k be a local field, and let K be a separable quadratic field extension of k. It is known that an irreducible complex representation π_1 of the unitary group G_1 = U_n(k) has a multiplicity free restriction to the subgroup G_2 = U{n−1}(k) fixing a non-isotropic line in the corresponding Hermitian space over K. More precisely, if π_2 is an irreducible representation of G_2 , then π = π_1 ⊗ π_2 is an irreducible representation of the product G = G_1 G_2 which we can restrict to the subgroup H = G_2 , diagonally embedded in G. The space of H-invariant linear forms on π has dimension ≤ 1.

In this talk, I will use the local Langlands correspondence and some number theoretic invariants of the Langlands parameter of π to predict when the dimension of H-invariant forms is equal to 1, i.e. when the dual of π_2 occurs in the restriction of π_1 . I will also illustrate this prediction with several examples, including the classical branching formula for representations of compact unitary groups. This is joint work with Wee Teck Gan and Dipendra Prasad.

An upper bound for the dimension of q-ary trace codes

Speaker: 

Phong Le

Institution: 

UCI

Time: 

Tuesday, October 21, 2008 - 2:00pm

Location: 

RH 306

Extending results of Van Der Vlugt,
I shall derive a new non-trivial upper bound for the dimension of trace
codes connected to algebraic-geometric codes. Furthermore, I will deduce
their true dimension if certain conditions are satisfied. Finally,
potential areas of improvement and other related results will be outlined.

Local arithmetic constants of elliptic curves and applications

Speaker: 

Sunil Chetty

Institution: 

UCI

Time: 

Thursday, October 23, 2008 - 3:00pm

Location: 

RH 306

This talk will discuss developments in the theory of local
arithmetic constants associated to an elliptic curve E over a number field
k, as introduced and studied by Mazur and Rubin. I calculate the
arithmetic constant for places of k where E has bad reduction, giving a
more general setting in which one has a lower bound for the rank of the
p-power Selmer group of E over extensions of k. Also, by comparing the
local arithmetic constants with the local analytic root numbers of E, I
determine a setting in which one can verify a (relative) parity conjecture
for E.

Odd counts of partitions

Speaker: 

Dennis Eichhorn

Institution: 

UCI

Time: 

Thursday, May 29, 2008 - 3:00pm

Location: 

MSTB 254

How many ways can an integer n be expressed as a sum of positive integers? This question is the cornerstone of Partition Theory, and it is surprisingly difficult to answer. For example, if we let p(n) be the number of these expressions of n, even the parity of p(n) remains something of a mystery, despite the fact that it has been studied for over a century. In particular, although empirical evidence (the first several million values) seems to indicate that Po(N) = [the number of odd values of p(n) up to N] is asymptotic to N/2, no one has even been able to show that Po(N) is larger than the square root of N for N sufficiently large. Many advances in discovering the mod 2 behavior of p(n) have been made over the past several years, and most of them have required properties of l-adic Galois representations and the theory of modular forms. However, one lower bound for Po(N) (which was the state-of-the-art for a brief period) was proven using only elementary generating function techniques and results from classical analytic number theory. In this talk, we develop the history of the mystery, and we prove the latter lower bound. The talk will be aimed at the partition theoretically uninitiated, and a great deal of background will be provided.

Even sharper upper bounds on the number of points on curves

Speaker: 

Everett Howe

Institution: 

CCR - La Jolla

Time: 

Thursday, April 17, 2008 - 4:00pm

Location: 

MSTB 254

For every prime power q and positive integer g, we let N_q(g) denote the maximum value of #C(F_q), where C ranges over all genus-g curves over F_q. Several years ago Kristin Lauter and I used a number of techniques to improve the known upper bounds on N_q(g) for specific values of q and g. The key to many of our improvements was a numerical invariant attached to non-simple isogeny classes of abelian varieties over finite fields; when this invariant is small, any Jacobian in the given isogeny class must satisfy restrictive conditions. Now Lauter and I have come up with a better invariant, which allows us to make even stronger deductions about Jacobians in isogeny classes. In this talk, I will explain how we have been able to use this new invariant, together with arguments about short vectors in Hermitian lattices over imaginary quadratic fields, to pin down several more values of N_q(g) and to improve the best known upper bounds in a number of other cases.

Iteration Dynamics from Cryptology on Exceptional Covers

Speaker: 

Professor Mike Fried

Institution: 

Montana State U-Billings, Emeritus UCI

Time: 

Tuesday, May 20, 2008 - 2:00pm

Location: 

MSTB 254

Let Fq be the finite field and : XY an Fq cover of normal varieties. We call exceptional if it maps 1-1 on Fqt points for an infinity of t. We say over Q is exceptional if it is exceptional mod infinitely many p. When X=Y, and is over Q, we have a map: exceptional p period of mod p. RSA cryptography uses x xk (k odd) and Euler's Theorem gives us its periods.

We give a paragraph of history: Schur (1921) posed a list of all Q exceptional polynomials. This inspired Davenport and Lewis (1961) to propose that a geometric property C D-L C would imply a polynomial is exceptional. Both were right (1969). Serre's O(pen) I(mage) T(heorem) produces most remaining exceptional Q rational functions (1977).

We use the D-L generalization to show exceptional covers (of Y over Fq) form a category with fiber products: the (Y,Fq) exceptional tower. Using that we can generate subtowers that connect the tower to two famous results.

I. Denef-Loeser-Nicaise motives: They attach a "motivic Poincare series" to any problem over Q. A generalization of exceptional covers produces (we say Weil) relations among Poincare series over (Y,Fq). The easiest converse question is this: If the zeta functions of X and P1 have a special Weil relation, is X an exceptional cover?

II. Serre's O(pen) I(mage) T(heorem): Rational functions from the OIT generate two (P1,Fq) exceptional subtower. The C(omplex) M(ultiplication) part of the OIT produces exceptional covers. We see their periods from the CM analog of Euler's Theorem. Periods of the subtower from the G(eneral) L(inear) part of the OIT give our most serious challenge.

Truncated Euler systems

Speaker: 

Soogil Seo

Institution: 

Yonsei University

Time: 

Monday, March 10, 2008 - 3:00pm

Location: 

MSTB 254

Let K be an imaginary quadratic field and let F be an abelian extension of K. It is known that the order of the class group Cl_F of F is equal to the order of the quotient U_F/El_F of the group of global units U_F by the group of elliptic units El_F of F. We introduce a filtration on El_F made from the so-called truncated Euler systems and conjecture that the associated graded module is isomorphic, as a Galois module, to the class group.

On the Coates-Sinnott Conjectures

Speaker: 

Cristian Popescu

Institution: 

UCSD

Time: 

Thursday, April 10, 2008 - 3:00pm

Location: 

MSTB 254

The conjectures in the title were formulated in the late 1970's as vast generalizations of the classical theorem of Stickelberger. They make a subtle connection between the Z[G(L/k)]-module structure of the Quillen K-groups K*(OL) in an abelian extension L/k of number fields and the values at negative integers of the associated G(L/k)-equivariant L-functions.

These conjectures are known to hold true if the base field k is Q, due to work of Coates-Sinnott and Kurihara. In this talk, we will provide evidence in support of these conjectures over arbitrary totally real number fields k.

PEL moduli spaces without C-valued points

Speaker: 

Oliver Bueltel

Institution: 

University of Heidelberg

Time: 

Thursday, April 3, 2008 - 3:00pm

Location: 

MSTB 254

The moduli space A_g of principally polarized abelian
g-folds may be viewed as a prime motivation for the theory of
Shimura varieties. I will explain this, along with variants of
such moduli interpretations (of Hodge-type or PEL).

I will then discuss mod p reductions and some of their moduli
interpretations which are outside the Hodge or PEL class.

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