![[Picture of Prof Graham R Everest]](http://www.mth.uea.ac.uk/~h090/twobeaks.jpg)
Prof Graham Everest
School of Mathematics,
University of East Anglia,
Norwich, NR4 7TJ, UK
This page gives access to various other links which include slides of talks and publications, also some pages about elliptic divisibility sequences.
Elliptic Curves and Hilbert's Tenth Problem A talk at the Inter-city Number Theory Seminar held in Delft on 5/10/2007. This talk contains new results about applications of elliptic descent to Hilbert's Tenth Problem over subrings of the rational nmbers, as well as giving an overview of the Primality Conjecture for elliptic curves.
The Divisibility of Some Divisibility Sequences A talk at University College Dublin on 8/11/2006. This is an overview of recent results about the divisibility properties of elliptic divisibility sequences.
Divisibility Sequences The first of two talks at the Universidad de Sevilla on 14/12/2006 (my birthday). This is an expanded overview which contains several computations as well as some new examples.
The Structure of Rational Points on Elliptic Curves This is the second talk at Seville on 15/12/2006. The slides give details of the proof of the primality conjecture in a special case, as well as a proof of the finiteness of pure power denominators of rational points.
L'arithmetique des Suites Elliptiques a Divisibilite A Mathematics talk at the Universite de Rennes on 16/2/2007. This is an overview of the arithmetic of elliptic divisibility sequences which also contains some new results in the function field case.
Elliptic Divisibility Sequences A talk at the ICMS workshop on Number Theory and Computability on 25/06/2007. The slides show how the existence of primitive divisors in EDSs is used in Poonen's proof of the undecidability of Hilbert's Tenth Problem for some large subrings of the rationals.
A generalization of Siegel's Theorem A talk given by Jonathan Reynolds to the Number Theory group at UEA in November 2006.
I maintain a web site about elliptic divisibility sequences. It has five main pages with links to others. The pages are
Elliptic divisibility sequences
Prime values of elliptic divisibility sequences
Elliptic divisibility sequences and the elliptic Lehmer problem
Zsigmondy's Theorem for elliptic curves
A combinatorial interpretation of elliptic divisibility sequences
My time at UEA is divided between doing research in mathematics and giving lectures to under-graduates.
My main area of research interest is number theory. Specifically:
1. The arithmetic of Elliptic Curves,
2. Interactions between Number Theory and Dynamical Systems,
3. Diophantine Equations,
4. Computational Number Theory, particularly the appearance of primes in classical integer sequences which satisfy recurrence relations.
Now follow some comments on each area.
1. and 2. Recently, I have been trying to interpret the known properties of the canonical height for rational points on elliptic curves in terms of dynamical systems. A paper jointly authored with Tom Ward and Manfred Einsiedler is going to appear in the Journal of Number Theory. This area of research is a flourishing area of over-lap between arithmetic and dynamical systems. You can read more about this my book, published jointly with Tom Ward. The methods needed to develop elliptic dynamical systems took us into the dynamics of iterates of rational maps. Our methods shed some light on the connections between heights and the structure of periodic points in that setting also. A report of this will appear in the New York Number Theory Seminar proceedings.
3. For a discussion of the distribution of solutions of Decomposable Form Equations, see my papers with Gyory. I am particularly interested in the 'astronomy' of these equations. That is, thinking of the solutions as stars in the night sky, what 'constellations' are visible from the earth?
4. I have one paper concerned with prime appearance in sequences which generalise the Mersenne sequence. Another paper is concerned with prime appearance in elliptic divisibility sequences. Both appear in the LMS Journal of Computation and Mathematics. A less technical account of these topics can be found in the paper in the journal El Cubo. For more information about primes in general (including Mersenne primes), consult Chris Caldwell's excellent Prime Page.
My former research student Peter Panayi wrote an interesting thesis on computational number theory. In it he computed values of Leopoldt's p-adic regulator.
More of my teaching material is going to appear on the web. For the moment, you can read Christian Rottger's notes on my course on Analytic Number Theory in ps or pdf format. These formed the basis for a book. Also, I have some notes on my course on Arithmetic in ps or WORD format.