Together with
Christopher Skinner, Urban proved many cases of
Iwasawa–Greenberg main conjectures for a large class of
modular forms. As a consequence, for a
modular elliptic curve over the
rational numbers, they prove that the vanishing of the
Hasse–Weil L-function L(
E,
s) of
E at
s = 1 implies that the p-adic
Selmer group of
E is infinite. Combined with theorems of
Gross-
Zagier and
Kolyvagin, this gave a conditional proof (on the
Tate–Shafarevich conjecture) of the conjecture that
E has infinitely many rational points if and only if
L(
E, 1) = 0, a (weak) form of the
Birch–Swinnerton-Dyer conjecture. These results were used (in joint work with
Manjul Bhargava and
Wei Zhang) to prove that a positive proportion of elliptic curves satisfy the
Birch–Swinnerton-Dyer conjecture. ==Awards==