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# Number Theory

## Our Research

Number Theory is one of the oldest branches of modern mathematics. It is motivated by the study of properties of integers and solutions to equations in integers. Many of its problems can be stated easily, but often require sophisticated methods from a diverse spectrum of areas in order to study. Its modern formulations are wide reaching and have close ties to algebraic geometry, analysis, and group theory; together with computational aspects. Perhaps due to the fundamental and profound nature of the integers, Number Theory plays a special role in mathematics and applications: two of the Clay Millennium Prize Problems are in Number Theory, and many internet security protocols are based on number theoretic problems.

Number Theory is an active area of research for faculty at SFU, and together with faculty at UBC, we form one of the largest communities of Number Theory researchers in North America.

## People

### Faculty

**Nils Bruin**

Diophantine and Arithmetic Geometry

**Imin Chen**

Number Theory and Arithmetic Geometry

**Stephen Choi**

Analytic Number Theory

### Postdoctoral Fellows and Visitors

### PhD Graduate Students

### MSc Graduate Students

- Andrew Cha

- Eugene Filatov

## Current Theses

## Recent Courses

### Graduate Program

#### Regular

- MATH 724 - Applications of Complex Analysis
- MATH 725 - Real Analysis
- MATH 740 - Galois Theory
- MATH 741 - Commutative Algebra and Algebraic Geometry
- MATH 817 - Groups and Rings
- MATH 818 - Algebra and Geometry
- MATH 842 - Algebraic Number Theory
- MATH 843 - Analytic and Diophantine Number Theory

#### Applications

- MATH 447 - Coding Theory
- MACM 401 - Introduction to Computer Algebra
- MACM 442 - Cryptography

## Resources

#### Links

- PIMS Abelian Varieties Multi-Site Seminar
- Joint PIMS/SFU/UBC Number Theory Seminar

## Past Events & Talks

#### Events

- PIMS Workshop on Computational Arithmetic Geometry
- Pacific Northwest Number Theory Conference 2014, May 17-18, 2014 at IRMACS, SFU.
- PIMS Distinguished Visitor Colloquium: Frits Beukers, May 2, 2013.
- Riemann Zeta Function Workshop, IRMACS, November 2-3, 2012.
- BIRS Summer School on Contemporary Methods for Solving Diophantine Equations,

June 10-17th, 2012. - Pacific Northwest Number Theory Conference 2010.
- Special session in AMS Western Section meeting: West End Number Theory, October 4-5, 2008.
- CNTA IX, Vancouver, July 9-14, 2006.
- 10th Annual Pacific Northwest Number Theory Conference, Digipen/Microsoft Research, February 25-26, 2006.
- Pacific Northwest Number Theory Conference 9, April 23, 2005.
- Workshop on computational arithmetic geometry, July 5-9, 2004.
- Joint PIMS/SFU/UBC Number Theory Seminar: Past semesters.

#### Talks

**Speaker: Nils Bruin, SFU
Title: Invisible Sha[4]
Date: Thu 14 Nov 2013
**Mazur observed that for a lot of elliptic curves E with non-trivial elements in Sha(E/Q)[n], one can find another elliptic curve E' that is n-congruent to E and for which the corresponding element in H1(Q,E′[n])H1(Q,E′[n]) lies in the image of the Mordell-Weil group of E'. Such an element in Sha is said to be made visible by E'.

It was since proved that for n=2,3, one can always find such an E', both when the n-congruence preserves Weil-pairing and when it inverts it. For given E and n=4, the question boils down to deciding if a certain K3 surface has a rational point. In joint work with Tom Fisher, we have been able to finally find equations for these K3 surfaces, which allows us to determine visibility computationally in specific cases.

**Speaker: Imin Chen, SFU
Title: Darmon's program for x^p + y^p = z^r and first case solutions
Date: Thu 7 Nov 2013
**Darmon has developed a program to resolve the generalized Fermat equation xp + yp = zr using Galois representations and abelian varieties of GL2 type over a totally real field. I will survey some parts of his program and point out the key difficulties which remain. Recently, numerous irreducibility criteria for the mod p representations attached to elliptic curves over totally real fields have been developed (David, Billerey, Freitas-Dieulefait, Freitas-Siksek). These are based on a technique which first appeared in Serre's 1972 Inventiones paper. I will explain how this method can be adapted to Darmon's Frey abelian varieties of GL2 type over a totally real field and thereby show that the above equation does not have any non-trivial first case solutions for p large enough compared to r a regular prime ≥ 5.

**Speaker: Colin Weir, SFU
Title: Counting dihedral function fields
Date: Thu 24 Oct 2013
**In the early 70's Davenport and Heilbronn derived the leading term in the asymptotic formula for the number of cubic number fields with bounded discriminant. However, as algorithmic data became available, a large "gap" became evident between the actual number of cubic number fields of small discriminant and the asymptotic prediction. We will discuss this and the analogous situation in the function field setting. We will present methods for constructing and tabulating dihedral function fields (which includes non-Galois cubics) and prove the existence of a similar "gap" for cubic function fields of small discriminant and the leading term of the corresponding asymptotic.

**Speaker: Michael Coons, University of Newcastle, Australia
Title: Mahler's Method, digital expansions, and algebraic numbers (or not)
Date: Thu 26 Sep 2013, 3:30pm
**In this talk, we survey past, present, and possible future results concerning the arithmetic nature of low complexity sequences. For example, what properties can be exhibited by numbers whose base expansion can be determined by a finite automaton? In the current context, this line of questioning was unknowingly initiated by Mahler, and later championed by Loxton and van der Poorten following the work of Cobham and Mendes France. In addition to describing some historical work, this talk will describe some of the the current advancements and generalisations concerning Mahler's method.

**Series: PIMS Colloquium
Speaker: Frits Beukers, Utrecht University
Title: What are hypergeometric functions?
Date: Thursday, May 2, 2013
**Hypergeometric functions occur in many shapes and flavours throughout mathematics and mathematical physics. The first such functions were introduced by Euler and studied in depth by Gauss. Since the end of the 19th the concept of hypergeometric functions was extended in many directions, thus creating a veritable zoo of different functions both inone variable and several variables. By the end of the 1980's Gel'fand, Kapranov and Zelevinsky introduced the concept of A-hypergeometric functions,which created a remarkable amount of order through combinatorial ideas. In this lecture we give a first introduction to hypergeometric functions and explain the idea of

A-hypergeometric functions.

**Series: PIMS Colloquium
Speaker: Tom Archibald, SFU
Title: The hypergeometric series and the hypergeometric equation: highlights of their roles in classical mathematics
Date: Thursday, May 2, 2013
**Things hypergeometric reach out in various directions that may be a little surprising. In this talk we will look at some nineteenth-century developments. Beginning with some results of Gauss, we will sample from work by E. E. Kummer (who, in providing solutions for the hypergeometric equation, had noticed connections to Legendre's period relations for elliptic integrals); and by L. Fuchs (who characterized the hypergeometric equation among linear DEs of the "Fuchsian" class). These studies are linked to work by Fuchs, Hermite and others on modular equations, and the detailed history reveals some surprising connections in classical mathematics.

**PIMS Number Theory Seminar
Tuesday, May 7, 2013, SFU K9509
2:00 pm, Frits Beukers (Utrecht University)
Title: Analytic aspects of hypergeometric functions
**The hypergeometric functions of Gauss formed the perfect testing ground for Riemann's ideas on analytic continuation of complex analytic functions. Many properties of hypergeometric functions became evident through the use of the so-called monodromy group. We shall explain these ideas and show some applications. Time permitting, we discuss possibilities to extend these ideas to the several variable setting.

**3:00pm, Frits Beukers (Utrecht University)
Title: Arithmetic aspects of hypergeometric functions
**By the end of the 1980's several authors introduced the concept of hypergeometric function on a finite field. Although this is a purely number theoretical finite sum, it shares many properties with its analytic counterpart. The special values of these functions turn out to be related to point counting on algebraic varieties over finite fields or better, traces of Frobenius operators. In this lecture we introduce these finite hypergeometric functions and describe some of their properties.

**Number Theory Seminar
Anna Haensch, Wesleyan University
Thu 15 Mar 2012, 3:00pm
Title: A characterization of almost universal ternary inhomogeneous quadratic forms**

A fundamental question in the study of integral quadratic forms is the representation problem which asks for an effective determination of the set of integers represented by a given quadratic form. A slightly different, but equally interesting problem, is the representation problem for inhomogeneous quadratic forms. In this talk, we will discuss a characterization of positive definite almost universal ternary inhomogeneous quadratic forms which satisfy some mild arithmetic conditions. Using these general results, we will then characterize almost universal ternary sums of polygonal numbers.

**Adrian Belshaw, Capilano University
Thu 16 Feb 2012, 4:10pm
Strong normality**

At a previous seminar, we proposed a "strong normality" test, to exclude numbers like Champernowne's number. Now we give a sharp version of this test. Almost all numbers are strongly normal, and every strongly normal number is normal. We use a method of Sierpinski to construct an absolutely normal number satisfying the new criterion. (This is joint work with Peter Borwein.)

**Himadri Ganguli, SFU
Thu 16 Feb 2012, 3:00pm
On the correlation of completely multiplicative functions**

Let f(n) be an arithmetic function and x > 0, then we define the correlation function C(f, x) = P n≤x f(n)f(n + 1)f(n + 2). In this talk we present an asymptotic formula for C(f, x) in the case when f(n) is a completely multiplicative function and |f(n)| ≤ 1 for all n ∈ N. Let λy(n) denote the truncated Liouville function which equals +1 or −1 according n has odd or even number of prime divisors p ≤ y counted with multiplicity. It follows from the main theorem that C(λy, x) = o(x) whenever y = x o(1) and speaks in favour of the Chowla conjecture that C(λ, x) = o(x) where λ is the classical liouville function.

**Imin Chen, SFU
Thu 26 Jan 2012, 4:10pm
On the equation a^3 + b^{3n} = c^2**

I will explain how to apply the modular method to resolve cases of this family of generalized Fermat equations. (joint with M. Bennett, S. Dahmen, S. Yazdani).

**Stephen Choi, SFU
Thu 26 Jan 2012, 3:00pm
On small fractional parts**

This is joint work in progress with Alan Haynes and Jeffrey Vaaler. Let A be a finite, nonempty set of positive integers. For x in R/Z, we study Delta(A,x) := min { ||ax|| : a \in A }, where || y || = \min { |y-n| : n \in Z } is the distance from y to the nearest integer. If each element of A is odd, then it is obvious that Delta(A,1/2)=1/2. However, in this talk, we will show that for most points x in R/Z the value of Delta(A,x) is not much bigger than |A|-1/2.

**Paul Mezo, Carleton University
Thu 17 Nov 2011, 4:10pm
Character identities in real twisted endoscopy**

Part of the Langlands Program is to find a meaningful correspondence between representations of Galois groups and representations of reductive algebraic groups. I will attempt to motivate this through an example and then concentrate on what happens at a (real) Archimedean place of the global picture. In this context the idea of endoscopy arises in a natural fashion and suggests identities between representations of different Lie groups. These identities have been proven by Shelstad. I will sketch the theory of endoscopy under twisting by a group automorphism and describe character identities between discrete series representations.