Past Junior Number Theory Seminar

We describe various approaches to the problem of expressing a polynomial $f(x) = \sum_{i=0}^{m} a_i x^i$ in terms of a different radix $r(x)$ as $f(x) = \sum_{j=0}^{n} b_j(x) r(x)^j$ with $0 \leq \deg(b_j) < \deg(r)$. Two approaches, the naive repeated division by $r(x)$ and the divide and conquer strategy, are well known. We also describe an approach based on the use of precomputed Newton inverses, which appears to offer significant practical improvements. A potential application of interest to number theorists is the fibration method for point counting, in current implementations of which the runtime is typically dominated by radix conversions.

The theory of modular forms owes in many ways lots of its results to the existence of the Hecke operators and their nice properties. However, even acting on modular forms of level 1, lots of basic questions remain unresolved. We will describe and prove some known properties of the Hecke operators, and state Maeda's conjecture. This conjecture, if true, has many deep consequences in the theory. In particular, we will indicate how it implies the nonvanishing of certain L-functions.

We discuss conjectures and results concerning small gaps between primes. In particular, we consider the work of Goldston, Pintz and Yildrim which shows that infinitely often there are gaps which have size an arbitrarily small proportion of the average gap.

(Note change in time and location)

The purpose of this talk is to give an introduction to the theory and

practice of integer factorization. More precisely, I plan to talk about the

p-1 method, the elliptic curve method, the quadratic sieve, and if time

permits the number field sieve.

I will describe why e^{\pi\sqrt{163}} is almost an integer and how this is related to Q(\sqrt{-163}) having class number one and why n^2-n+41 is prime for n=0,...,39. Bits and pieces about Gauss's class number problem, Heegner numbers, the j-invariant and complex multiplication on elliptic curves will be discussed along the way.

The main aim of this talk will be to present a proof of the Tsen-Lang theorem on the existence of points on complete intersections and sketch a proof of the Grabber-Harris-Starr theorem giving the existence of a section to a fibration of a rationally connected variety over a curve. Time permitting, recent work of de Jong and Starr on rationally simply connected varieties will be discussed with applications to the number theory of hypersurfaces.

An old conjecture of Hardy and Littlewood posits that on average, the number of representations of a positive integer N as a sum of k, k-th powers is "very small." Recently, it has been observed that this conjecture is closely related to properties of a discrete fractional integral operator in harmonic analysis. This talk will give a basic introduction to the two key problems, describe the  correspondence between them, and show how number theoretic methods, in particular the circle method and mean values of Weyl sums, can be used to say something new in abstract harmonic analysis.

In this talk I will introduce some of the basic ideas linking the theory of complex multiplication for elliptic curves and class field theory. Time permitting, I'll mention Shimura and Taniyama's work on the case of abelian varieties.

The Siegel-Walfisz theorem gives an asymptotic estimate for the number of primes in an arithmetic progression, provided the modulus of the progression is small in comparison with the length of the progression. Counting primes is harder when the modulus is not so small compared to the length, but estimates such as Linnik's constant and the Brun-Titchmarsh theorem give us some information. We aim to look in particular at upper bounds for the number of primes in such a progression, and improving the Brun-Titchmarsh bound.

(Note that the talk will be in L2 and not the usual SR1)

Many interesting features of algebraic varieties are encoded in the spaces of rational curves that they contain. For instance, a smooth cubic surface in complex projective three-dimensional space contains exactly 27 lines; exploiting the configuration of these lines it is possible to find a (rational) parameterization of the points of the cubic by the points in the complex projective plane.

After a general overview, we focus on the Fermat quintic threefold X, namely the hypersurface in four-dimensional projective space with equation x^5+y^5+z^5+u^5+v^5=0. The space of lines on X is well-known. I will explain how to use a mix of algebraic geometry, number theory and computer-assisted calculations to study the space of conics on X.

This talk is based on joint work with R. Heath-Brown.

We have seen that L-functions of elliptic curves of conductor N coincide exactly with L-functions of weight 2 newforms of level N from the Modularity Theorem. We will show how, using modular symbols, we can explicitly compute bases of newforms of a given level, and thus investigate L-functions of an elliptic curve of given conductor. In particular, such calculations allow us to numerically test the Birch-Swinnerton-Dyer conjecture.

In the first half of the talk we explain - in very broad terms - how the objects defined in the previous meetings are linked with each other. We will motivate this 'big picture' by briefly discussing class field theory and the Artin conjecture for L-functions. In the second part we focus on a particular aspect of the theory, namely the L-function preserving construction of elliptic curves from weight 2 newforms via Eichler-Shimura theory. Assuming the Modularity theorem we obtain a proof of the Hasse-Weil conjecture.

This talk is the second in a series of an elementary introduction to the ideas unifying elliptic curves, modular forms and Galois representations. I will discuss what it means for an elliptic curve to be modular and what type of representations one associates to such objects.

Let $X$ be a smooth hypersurface in projective space over a field $K$ of characteristic zero and let $U$ denote the open complement. Then the elements of the algebraic de Rham cohomology group $H_{dR}^n(U/K)$ can be represented by $n$-forms of the form $Q \Omega / P^k$ for homogeneous polynomials $Q$ and integer pole orders $k$, where $\Omega$ is some fixed $n$-form. The problem of finding a unique representative is computationally intensive and typically based on the pre-computation of a Groebner basis. I will present a more direct approach based on elementary linear algebra. As presented, the method will apply to diagonal hypersurfaces, but it will clear that it also applies to families of projective hypersurfaces containing a diagonal fibre. Moreover, with minor modifications the method is applicable to larger classes of smooth projective hypersurfaces.

Suppose that $C$ and $C'$ are cubic forms in at least 19 variables over a

$p$-adic field $k$. A special case of a conjecture of Artin is that the

forms $C$ and $C'$ have a common zero over $k$. While the conjecture of

Artin is false in general, we try to argue that, in this case, it is

(almost) correct! This is still work in progress (joint with

Heath-Brown), so do not expect a full answer.

As a historical note, some cases of Artin's conjecture for certain

hypersurfaces are known. Moreover, Jahan analyzed the case of the

simultaneous vanishing of a cubic and a quadratic form. The approach

we follow is closely based on Jahan's approach, thus there might be

some overlap between his talk and this one. My talk will anyway be

self-contained, so I will repeat everything that I need that might

have already been said in Jahan's talk.

How many integer-points lie in a circle of radius $\sqrt{x}$?

A poor man's approximation might be $\pi x$, and indeed, the aim-of-the-game is to estimate

$$P(x) = \sharp\{(m, n) \in\mathbb{Z}: \;\; m^{2} + n^{2} \leq x\} -\pi x,$$

Once one gets the eye in to show that $P(x) = O(x^{1/2})$, the task is to graft an innings to reduce this bound as much as one can. Since the cricket-loving G. H. Hardy proved that $P(x) = O(x^{\alpha})$ can only possible hold when $\alpha \geq 1/4$ there is some room for improvement in the middle-order.

In this first match of the Junior Number Theory Seminar Series, I will present a summary of results on $P(x)$.

Suppose a power series $f(x):= \sum_{n=0}^{\infty} a_{n} x^{n}$ has radius of convergence equal to $1$ and that $lim_{x\rightarrow 1}f(x) = s$. Does it therefore follow that $\sum_{n=0}^{\infty} a_{n} = s$? Tauber's Theorem answers in the affirmative, \textit{if} one imposes a certain growth condition (a \textit{Tauberian Condition}) on the coefficients $a_{n}$. Without such a condition it is clear that this cannot be true in general - take, for example, $f(x) = \sum_{n=0}^{\infty} (-1)^{n} x^{n}.$

In the previous talks we have seen the formulation of the Birch--Swinnerton-Dyer conjecture. This talk will focus on a fundamental question in diophantine geometry. Namely, given an algebraic curve \textit{C} defined over $\mathbb{Q}$ possessing at least one rational point, what is

the probability that \textit{C} has infinitely many rational points?

For curves of genus 0, the answer has been known ever since the ancient Greeks roamed the earth, and for genus &gt; 1 the answer is also known (albeit for a much shorter time). The remaining case is genus 1, and this question has a history filled with tension and

conflict between data and conjecture.

I shall describe the heuristics behind the conjectures, taking into account the Birch--Swinnerton-Dyer Conjecture and the Parity Conjecture. I shall go on to outline the contrary numeric data, both in families of elliptic curves and for all elliptic curves of increasing conductor.

If one instead considers elliptic curves over function fields $\mathbb{F}_{q} (t)$, then, via a conjecture of Artin and Tate, one can compute the rank (and more) of elliptic curves of extremely large discriminant degree. I shall briefly describe the interplay between Artin--Tate and

Birch--Swinnerton-Dyer, and give new evidence finally supporting the conjecture.

This is the second (of two) talks concerning the Birch--Swinnerton-Dyer Conjecture.

This is the first (of two) talks which will be given concerning the Birch--Swinnerton-Dyer Conjecture.

The goal of this talk is to give sufficient conditions for the existence of points on certain varieties defned over finite fields.

This talk will mention methods of testing whether a given integer is prime. Included topics are Carmichael numbers, Fermat and Euler pseudo-primes and results contingent on the Generalised Riemann Hypothesis.

Aside from a few tangential problems, this seminar will include a proof of Ostrowski's Theorem. This states than any norm over the rationals is equivalent to either the Euclidean norm or the $p$-adic norm, for some prime $p$.