Number theory is just as relevant today as it was two millennia ago, with the advent of high-powered computing and cryptography. An elliptic curve is a cubic equation of the form E : y² = x³ + Ax + B, where A and B are any rational numbers such that 4A³ + 27B² ≠ 0. The study of these curves can be traced all the way back to the ancient Greeks, and they are currently a very actively researched area of pure mathematics (for example, the arithmetic properties of these curves were pivotal in the proof of Fermat’s Last Theorem).

The famous Birch and Swinnerton-Dyer Conjecture asserts that the number of points needed to generate the group of K-rational points, E(K), on an elliptic curve should equal the order of vanishing at s=1 of its Hasse-Weil L-function L(E/K,s) over the number field. The most fruitful approach in attacking this problem has been to break it down, prime number by prime number, and then to apply the descent machinery developed by Kenkichi Iwasawa in the 1960s.

Daniel's research focusses primarily on the Iwasawa theory of Galois representations, and on how their associated p-adic L-functions relate to Selmer groups. He is also extremely interested in the special values of L-functions, as there is a rich vein of conjectures connecting these L-values with elements in K-groups. He has worked on symmetric square L-functions, on the p-adic version of the BSD Conjecture, and on the ordinary deformation theory of Euler systems. Finally, Daniel's efforts over the last few years have mainly centred on ways to deduce the Iwasawa Main Conjecture when it is already known for another p-congruent Galois representation.

PhD Students Supervised: