Arizona State University

College of Liberal Arts and Sciences

 

 

 

 

 

 

Number Theory Talks

Spring, 2008


 

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Calendar

February 1

Number Theory Seminar

1:40-2:30 p.m.

PSA 308

Jing Long Hoelscher, University of Arizona

Galois extensions with prescribed ramification

Abstract. With given places ramifying, which groups can occur as Galois groups over a field K? For function fields K of curves over algebraically closed fields k of characteristic p, this is answered by Abyhankar's Conjecture, which was proved by M. Raynaud and D. Harbater. For number fields K, I will generalize some results in Harbater's papers to give some answers to the problem above, especially when there is just one ramified prime. For function fields K over finite fields, I will give some constraints on the groups that can occur with given ramification, and take steps to develop Iwasawa theory for cyclotomic function fields.

 

February 22

Colloquium

1:30-2:30 p.m.

PSF 101

Wayne Raskind, University of Southern California

Public Key Cryptography, Number Theory and Arithmetic Geometry

Abstract. Many public key cryptographic systems are based on the discrete logarithm problem for finite abelian groups. In this talk, we will briefly review how this works and present some of our research (joint with Ming-Deh Huang of the USC Computer Science Department) on a general method for treating this problem for abelian algebraic groups over finite fields. We will then describe in some detail the cases of the multiplicative group and elliptic curves over finite fields, which reveal some surprising connections between the discrete logarithm problem and some fundamental problems in number theory and arithmetic geometry.

 

February 27

Number Theory Seminar

1:40-2:30 p.m.

PSA 308

Helene Nehrebecki

Waring’s problem

Abstract. Waring’s problem states that every positive integer is the sum of 9 cubes, 19 fourth powers, and so on. In this talk, we will discuss results leading to the solution of Waring’s problem. Finally, we will explore recent advancements in this area. 

 

March 5

Number Theory Seminar

1:40-2:30 p.m.

PSA 308

Alejandra Alvarado

Arithmetic progressions on curves

Abstract. An arithmetic progression is a sequence of numbers such that the difference between any two consecutive numbers is constant.  When we talk about an arithmetic progression on a curve F(x,y)=0, we mean an arithmetic progression in the x coordinates.  In this talk we will discuss APs on curves of genus greater than zero.  In particular, we will look at APs on certain types of elliptic curves.

 

March 19

Number Theory Seminar

1:40-2:30 p.m.

PSA 308

David Roberts, University of Minnesota at Morris

Chebyshev covers and exceptional number fields

Abstract. Click here to view.

 

 

March 26

Number Theory Seminar

1:40-2:30 p.m.

PSA 308

Daniel Corral

A connected topology for the integers, I

Abstract. We discuss a paper of Solomon W. Golomb, where arithmetic progressions are used to define a basis of open sets, yielding a topology on Z. We show how this topology can be used to prove the infinitude of primes.

 

April 2

Number Theory Seminar

1:40-2:30 p.m.

PSA 308

Daniel Corral

A connected topology for the integers, II

Abstract. We resume our exposition of the paper "A Connected Topology for the Integers" by Solomon W. Golomb.  As in the previous talk, let D denote the Golomb topology generated by the empty set and a certain type of arithmetic progression in the integers.  In this talk we will discuss a theorem of Golomb which states that the Dirchlet theorem on arithmetic progressions is equivalent to the density of the primes in (Z, D).  It will also be shown that the set of primes has empty interior in (Z, D).  The final result in the Golomb paper is that the set of positive integers n, where 6n-1 and 6n+1 are a prime pairing, is closed in a stronger topology than D.  If time permits we will begin a discussion of a paper of A. M Kirch or present the related H. Furstenberg proof of the infinitude of primes.

 

April 9

Number Theory Seminar

1:40-2:30 p.m.

PSA 308

Keenan Kidwell

The theory and computation of finite extensions of the p-adic numbers, I

Abstract. The fields of real and p-adic numbers are examples of local fields, (locally compact topological fields), and are the only completions of Q, while number fields (finite extensions of Q) are examples of global fields. The number theoretic motivation for studying the properties of real and p-adic numbers is to obtain local information, which (hopefully) can be assembled into global information about number fields; an example of the success of this strategy is the Hasse-Minkowski Theorem, a local-global principle relating the existence of zeros of quadratic forms over the real and p-adic numbers to that of zeros over the rationals. One connection between number fields and p-adic fields is given by the so-called p-adic algebra of a number field F: F\otimes Qp \cong\prod_{i=1}^g Kp,i , where each Kp,i is a finite extension of the p-adic numbers. Within a fixed algebraic closure, there are only finitely many extensions of  the p-adic numbers of a given degree. In this two-part talk, we will discuss the theoretical aspects of p-adic fields, including residue degree, ramification index, unramified and totally ramified extensions, the continuity of roots of polynomial equations, and the aforementioned finiteness result, as well as some of the work of J. Jones and D. Roberts involving the determination and classification of the extensions of the p-adic numbers of a given degree.

 

April 16

Number Theory Seminar

1:40-2:30 p.m.

PSA 308

Keenan Kidwell

The theory and computation of finite extensions of the p-adic numbers, II

Abstract. The fields of real and p-adic numbers are examples of local fields, (locally compact topological fields), and the only completions of Q, while number fields (finite extensions of Q) are examples of global fields. The number theoretic motivation for studying the properties of real and p-adic numbers is to obtain local information, which (hopefully) can be assembled into global information about number fields; an example of the success of this strategy is the Hasse-Minkowski Theorem, a local-global principle relating the existence of zeros of quadratic forms over the real and p-adic numbers to that of zeros over the rationals. One connection between number fields and p-adic fields is given by the so-called p-adic algebra of a number field F: F\otimes Qp \cong\prod_{i=1}^g Kp,i , where each Kp,i is a finite extension of the p-adic numbers. Within a fixed algebraic closure, there are only finitely many extensions of  the p-adic numbers of a given degree. In this two-part talk, we will discuss the theoretical aspects of p-adic fields, including residue degree, ramification index, unramified and totally ramified extensions, the continuity of roots of polynomial equations, and the aforementioned finiteness result, as well as some of the work of J. Jones and D. Roberts involving the determination and classification of the extensions of the p-adic numbers of a given degree.

 

April 23

Number Theory Seminar

1:40-2:30 p.m.

PSA 308

Dong Quan Nguyen

On the Duffin-Schaeffer conjecture, part I

Abstract. We will prove the Duffin-Schaeffer theorem, which motivates the formulation of the Duffin-Schaeffer conjecture.

 

April 30

Number Theory Seminar

1:40-2:30 p.m.

PSA 230

Dong Quan Nguyen

On the Duffin-Schaeffer conjecture, part II

Abstract. We will prove the Duffin-Schaeffer theorem, which motivates the formulation of the Duffin-Schaeffer conjecture.

 

 

Autumn, 2007 Calendar

October 24

Number Theory Seminar

10:40-11:30 a.m. PSH 552

Keenan Kidwell

Quadratic Forms Over the Field of Rationals and its Completions and the Hasse-Minkowski Theorem

Abstract. We will define quadratic forms and quadratic modules, focusing on forms over the fields of rational, real, and p-adic numbers. The Hilbert symbol will be introduced, as well as the rank and discriminant of a quadratic form, and these tools will be used to give a necessary and sufficient condition for the equivalence of two forms over a field of characteristic different from 2. We will conclude with a statement and discussion of the proof of the Hasse-Minkowski Theorem, which relates the existence of zeros of a quadratic form over the rationals to that of zeros of the form over the real and p-adic fields.

 

October 31

Number Theory Seminar

10:40-11:30 a.m. PSH 552

Dong Quan Nguyen

Algebraic Points on Cubic Hypersurfaces I

Abstract. We prove the Cassels Swinnerton-Dyer conjecture for the case n=3 following a paper of Coray.

 

November 7

Number Theory Seminar

10:40-11:30 a.m. PSH 552

Dong Quan Nguyen

Algebraic Points on Cubic Hypersurfaces II

Abstract. We prove the Cassels Swinnerton-Dyer conjecture for the case n=3 following a paper of Coray.

 

November 14

Number Theory Seminar

10:40-11:30 a.m. PSH 552

Alejandra Alvarado

Integral Arithmetic Progressions on y2 = x3 + k

Abstract. Consider elliptic curves of the form y2 = x3 + k over the rationals. We will investigate arithmetic progressions on these curves, of length three, four and five.

 

November 28

Number Theory Seminar

10:40-11:30 a.m. PSH 552

Chase Franks

On the mu-invariant of the Gamma-transform of a rational function

Abstract. p-adic L-functions arise as Gamma-transforms of p-adic measures that are constructed from rational functions. Following Sinnott, we will compute the mu-invariant of the Gamma-transform of a rational function measure. This will lead to a proof of the Ferrero-Washington theorem, which says that mu = 0 for the basic Zp-extension of an abelian number field.

 

 

 

Number Theory Talks (2006-2007)

Number Theory Talks (2005-2006)