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Mathematics

The Distribution of J-invariants for CM Elliptic Curves defined over Zp

Speaker: 
Andrew Fiori
Date: 
Thu, Jan 28, 2016
Location: 
PIMS, University of Calgary
Conference: 
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract: 
Let K be a quadratic imaginary field, and p be a prime which is inert in K. It is known that the mod p reductions of the j-invariants of elliptic curves defined over the algebraic closure of Qp which admit CM by an order of K are equidistributed among the supersingular values in F{p2}. By contrast, if we replace this algebraically closed field by Qp, the j-invariants for many natural families of orders do not share this same distribution and are simply not uniformly distributed among all the supersingular values in Fp. In this talk I will explain why this occurs, and some of the computations which led me to consider this question.

Bi-cross-validation for factor analysis

Speaker: 
Art Owen
Date: 
Tue, Jan 19, 2016
Location: 
PIMS, University of British Columbia
Conference: 
Constance van Eeden Invited Speaker, UBC Statistics Department
Abstract: 

Factor analysis is a core technique in applied statistics with implications for biology, education, finance, psychology and engineering. It represents a large matrix of data through a small number k of latent variables or factors. Despite more than 100 years of use, it remains challenging to choose k from the data. Ad hoc and subjective methods are popular, but subject to confirmation bias and they do not scale to automatic uses. There are many recent tools in random matrix theory (RMT) that apply to the factor analysis setting, so long as the noise has constant variance. Real data usually involves heteroscedasticity foiling those techniques. There are also tools in the econometrics literature, but those apply mostly to the strong factor setting unlike RMT which handles weaker factors. The best published method is parallel analysis, but that is only justified by simulations. We propose a bi-cross-validation approach holding out some rows and some columns of the data matrix, predicting the held out data via a factor analysis on the held in data. We also use simulations to justify the method, though our simulations are designed using recent findings from RMT. The new approach outperforms previous methods that we found, as measured by recovery of a true underlying factor matrix.

 

This is joint work with Jingshu Wang of Stanford University.

 

Biosketch: Art Owen is a professor of statistics at Stanford University. He is best known for developing empirical likelihood and randomized quasi-Monte Carlo. Empirical likelihood is an inferential method that uses a data driven likelihood without requiring the user to specify a parametric family of distributions. It yields very powerful tests and is used in econometrics. Randomized quasi-Monte Carlo sampling, is a quadrature method that can attain nearly O(n**-3) mean squared errors on smooth enough functions. It is useful in valuation of options and in computer graphics. His present research interests focus on large scale data matrices. Professor Owen's teaching is focused on doctoral applied courses including linear modeling, categorical data, and stochastic simulation (Monte Carlo).

Abelian Varieties Multi-Site Seminar Series: Drew Sutherland

Speaker: 
Drew Sutherland
Date: 
Tue, Jan 12, 2016
Location: 
PIMS, University of Washington
Conference: 
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract: 
Let A be an abelian variety of dimension g over a number field K. The Sato-Tate group ST(A) is a compact subgroup of the unitary symplectic group USp(2g) that can be defined in terms of the l-adic Galois representation associated to A. Under the generalized Sato-Tate conjecture, the Haar measure of ST(A) governs the distribution of various arithmetic statistics associated to A, including the distribution of normalized Frobenius traces at primes of good reduction. The Sato-Tate groups that can and do arise for g=1 and g=2 have been completely determined, but the case g=3 remains open. I will give a brief overview of the classification for g=2 and then discuss the current state of progress for g=3.

OM representation of prime ideals and applications in function fields

Speaker: 
Jens Bauch
Date: 
Thu, Dec 10, 2015
Location: 
PIMS, Simon Fraser University
Conference: 
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract: 
Let $A$ be a Dedekind domain, $K$ the fraction field of $A$, and $f\in A[x]$ a monic irreducible separable polynomial. Denote by $\theta\in K^{\mathrm{sep}}$ a root of $f$ and let $F=K(\theta)$ be the finite separable extension of $K$ generated by $\theta$. We consider $\mathcal{O}$ the integral closure of $A$ in $L$. For a given non-zero prime ideal $\mathfrak{p}$ of $A$ the Montes algorithm determines a parametrization (OM representation) for every prime ideal $\mathfrak{P}$ of $\mathcal{O}$ lying over $\mathfrak{p}$. For a field $k$ and $f\in k[t,x]$ this yields a new representation of places of the function field $F/k$ determined by $f$. In this talk we summarize some applications which improve the arithmetic in the divisor class group of $F$ using this new representation.

An arithmetic intersection formula for denominators of Igusa class polynomials

Speaker: 
Bianca Viray
Date: 
Thu, Nov 12, 2015
Location: 
PIMS, University of Washington
Conference: 
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract: 
Igusa class polynomials are the genus 2 analogue of Hilbert class polynomials; their roots are invariants of genus 2 curves that have complex multiplication by a fixed order. The coefficients of Igusa class polynomials are rational, but, unlike in genus 1, are not integral. An exact formula, or tight upper bound, for these denominators is needed to compute Igusa class polynomials and has applications to cryptography. In this talk, we explain how to obtain a formula for the arithmetic intersection number G1.CM(K) and how this results in a bound for denominators of Igusa class polynomials. We also explain how the formula for G1.CM(K) leads us to a generalization of Gross and Zagier's formula for differences of CM j-invariants. This is joint work with Kristin Lauter.

Local-global principles for quadratic forms

Author: 
Raman Parimala
Date: 
Fri, Oct 30, 2015
Location: 
PIMS, University of British Columbia
Conference: 
PIMS/UBC Distinguished Colloquium
Abstract: 
The classical theorem of Hasse-Minkowski asserts that a quadratic form over a number field represents zero nontrivially provided it represents zero nontrivially over its completions at all its places. We discuss analogous local global principles over function fields of p-adic curves. Such local-global principles in the general setting for homogeneous spaces have implications to the understanding of the arithmetic of these fields.

Landing a Faculty Position

Author: 
Phillip Loewen
Date: 
Fri, Oct 23, 2015
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-Math Job Forum
Abstract: 
The PIMS-Math Job Forum is an annual Forum to help graduate students and postdoctoral fellows in the Mathematics Department with their job searches. The session is divided in two parts: short presentations from our panel followed by a discussion. Learn the secrets of writing an effective research statement, developing an outstanding CV, and giving a winning job talk. We will address questions like: Who do I ask for recommendation letters? What kind of jobs should I apply to? What can I do to maximize my chances of success?

Finding Your Place at a liberal arts college

Author: 
Kathryn Nyman
Date: 
Fri, Oct 23, 2015
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-Math Job Forum
Abstract: 
The PIMS-Math Job Forum is an annual Forum to help graduate students and postdoctoral fellows in the Mathematics Department with their job searches. The session is divided in two parts: short presentations from our panel followed by a discussion. Learn the secrets of writing an effective research statement, developing an outstanding CV, and giving a winning job talk. We will address questions like: Who do I ask for recommendation letters? What kind of jobs should I apply to? What can I do to maximize my chances of success?

My Life in "industry"

Author: 
Richard Liang
Date: 
Fri, Oct 23, 2015
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-Math Job Forum
Abstract: 
The PIMS-Math Job Forum is an annual Forum to help graduate students and postdoctoral fellows in the Mathematics Department with their job searches. The session is divided in two parts: short presentations from our panel followed by a discussion. Learn the secrets of writing an effective research statement, developing an outstanding CV, and giving a winning job talk. We will address questions like: Who do I ask for recommendation letters? What kind of jobs should I apply to? What can I do to maximize my chances of success?

Signs of abelian varieties and representations

Speaker: 
Matthew Greenberg
Date: 
Thu, Oct 15, 2015
Location: 
PIMS, University of Calgary
Conference: 
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract: 
The sign is a fundamental invariant of an abelian variety defined over a local (archimedian or p-adic) or global (number or function) field. The sign of an abelian varieties over a global field has arithmetic significance: it is the parity of Mordell-Weil group of the abelian variety. The sign also appears in the functional equation of the L-function of abelian variety, determining the parity of its order of vanishing at s=1. The modularity conjecture says that this L-function coincides with the L-function of an automorphic representation, and the sign can be expressed in terms of this representation. Although we know how to compute this sign using representation theory, this computation does not really shed any light on the representation theoretic significance of the sign. This representation theoretic significance was articulated first by Dipendra Prasad (in his thesis), where he relates the sign of a representation to branching laws — laws that govern how an irreducible group representation decomposes when restricted to a subgroup. The globalization of Prasad’s theory culminates in the conjectures of Gan, Gross and Prasad. These conjectures suggest non-torsion elements in Mordell-Weil groups of abelian varieties can be obstructions to the existence of branching laws. By exploiting p-adic variation, though, one can hope to actually produce the Mordell-Weil elements giving rise to these obstructions. Aspects of this last point are joint work with Marco Seveso.
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