Gross–stark Units, Stark–heegner Points, and Class Fields of Real Quadratic Fields

Gross–Stark units, Stark–Heegner points, and class fields of real quadratic fields by Samit Dasgupta Doctor of Philosophy in Mathematics University of California, Berkeley Professor Kenneth Ribet, Chair We present two generalizations of Darmon’s construction of Stark–Heegner points on elliptic curves defined overQ. First, we provide a lifting of Stark–Heegner points from elliptic curves to certain modular Jacobians which parameterize them. This construction involves a generalization of a theorem of Greenberg and Stevens that proves the Mazur–Tate–Teitelbaum conjecture. Next, we replace the modular symbols attached to an elliptic curve with those attached to a modular unit α of level N > 1. For a real quadratic field K in which the rational prime p is inert, this allows the definition of certain numbers u(α, τ) ∈ K× p attached to α and τ ∈ K −Q. The elements u(α, τ) are analogous to classical elliptic units arising from α. In this vein, we conjecture that the elements u(α, τ) belong to specific abelian extensions of K. Although this conjecture is still open, we prove a formula relating the p-adic valuation and p-adic logarithm of u(α, τ) to the leading terms at s = 0 of certain partial zeta functions (classical and p-adic, respectively). The existence of units satisfying these properties is equivalent to the p-adic Gross–Stark conjecture; thus our construction gives an analytic construction of Gross’s units, minus a proof of their algebraicity. Professor Kenneth Ribet Dissertation Committee Chair