When it comes to birds and frogs, I fall into the latter category. I enjoy simple-seeming questions that lead to richness of structure. Areas of interest include: arithmetic geometry, elliptic curves, algebraic divisibility sequences, recurrence sequences, Diophantine geometry, cryptography, arithmetic dynamics, game theory. Ribbit.

The famous Fibonacci numbers (first described by a Sanskrit poet) and Mersenne numbers (studied by Euclid around the same time), are both examples of Lucas sequences of the first kind, i.e. sequences satisfying a recurrence relation of the form

    L_n = aL_n-1 + bL_n-2,   L₀=0,  L₁=1,

for some coefficients a and b. Many of the wonderful number theoretical properties of these sequences are reflections of their relationship to twists of the multiplicative group, G_m. For example, any prime p (except 5) appears for the first time as a divisor of some term F_n of the Fibonaccis, where n is a divisor of p² -1, a fact related to the size of the multiplicative group of the finite field of p² elements.

Replacing G_m with an elliptic curve, we come to a new kind of recurrence relation: elliptic divisibility sequences. An elliptic divisibility sequence is a sequence satisfying this recurrence relation:

Over any field, such a sequence can be generated by a point on an elliptic curve: the sequence is obtained by evaluating the sequence of division polynomials at that point. It is a remarkable theorem of Morgan Ward (1948) that any elliptic divisibility sequence can be obtained in this way. Furthermore, the arithmetic geometry of the underlying elliptic curve shines through the sequence: we can see such things as the type and orders of reduction of the curve and point.

I find this dual perspective enchanting: arithmetic geometry on the one hand, and the number theory of recurrence sequences on the other. For my thesis work under Joseph H. Silverman, I extended the relationship to pairs, or n-tuples, of points on elliptic curves. Elliptic nets are n-dimensional arrays of numbers (an example is the picture at the top right of my webpage). They satisfy a generalisation of the recurrence relation above, given here:

 

Working in this area is full of surprises. For example, starting from a question of Chris Smyth concerning when n divides the n^th term of a divisibility sequence, Joseph H. silverman and I came to the definition of an amicable pair of primes for an elliptic curve: a pair (p,q) such that the curve has p points modulo q and q points modulo p; it turns out these pairs are surprisingly frequent for CM curves!

Some of the questions I'm asking: How does all this generalise for abelian varieties? What is the moduli space of elliptic divisibility sequences (or elliptic nets) and how does it relate to the usual modular curves? What do the sequences tell us about the number of integral points on a curve?

Take a look at my papers on this topic. If you prefer slides, you can learn about elliptic nets from a talk at the ICMS, or, for cyptographers, slides from Elliptic Curve Cryptography 2007.