Research Projects

The arithmetic theory of automorphic L-functions has been a subject of mathematical research for some time. The Birch & Swinnerton–Dyer Conjecture (BSD) and its generalization, the Bloch–Kato Conjecture, predict a mysterious relationship between the arithmetic of an automorphic form and the special values of its L-function. For example, if E is an elliptic curve over the field of rational numbers, BSD predicts that the order of vanishing of the Hasse–Weil L-function at s = 1 equals the Mordell–Weil rank of the group of rational points. In scenarios beyond Complex Multiplication, Henri Darmon used p-adic methods to construct local points in the Mordell–Weil group of the curve, known as Stark–Heegner points. These points are conjectured to be global points on the elliptic curve E and satisfy a reciprocity law under Galois automorphisms similar to those satisfied by Heegner points. To prove the Bloch–Kato conjecture, a systematic construction of classes in the Selmer group should be conducted when the L-function is known to vanish. In a joint joint with Chris Williams, the authors proposed the construction of Stark—Heegner cycles for automorphic forms on GL(2) over an arbitrary number field, construct Plectic analogues of Stark—Heegner cycles, construct Stark—Heegner cycles for automorphic forms defined over Orthogonal groups O(p,q) and Unitary groups U(n), and investigate the behavior of Stark—Heegner cycles under theta lifts. The authors hope that these projects will be the first step towards resolving the Bloch—Kato conjectures for automorphic forms and shed light on the Iwasawa theory of automorphic forms by exploiting the Euler system machinery.