Benjamin Howard

Variation of Heegner points in Hida families

Given a weight two modular form f with associated p-adic Galois representation Vf, for certain quadratic imaginary fields K one can construct canonical classes in the Galois cohomology of Vf by taking the Kummer images of Heegner points on the modular abelian variety attached to f. We show that these classes can be interpolated as f varies in a Hida family and construct an Euler system of big Heegner points for Hida’s universal ordinary deformation of Vf. We show that the specialization of this big Euler system to any form in the Hida family is nontrivial, extending results of Cornut and Vatsal from modular forms of weight two and trivial character to all ordinary modular forms, and propose a horizontal nonvanishing conjecture for these cohomology classes. The horizontal nonvanishing conjecture implies, via the theory of Euler systems, a conjecture of Greenberg on the generic ranks of Selmer groups in Hida families.

Rapoport-Zink spaces for spinor groups

We develop a theory of Hodge type Rapoport-Zink formal schemes, which uniformize certain formal completions of the canonical integral models of Shimura varieties of Hodge type at primes of good reduction. We then apply the general theory to the special case of Shimura varieties associated with groups of spinor similitudes, and, in the basic case, determine explicitly the reduced scheme underlying the Rapoport-Zink formal scheme.

Heights of Kudla–Rapoport divisors and derivatives of \(L\)-functions

We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature

On the supersingular locus of the GU(2,2) Shimura variety

(n-1,1)

Intersections of Hirzebruch-Zagier divisors and CM cycles

(n-1,1). We construct an arithmetic theta lift from harmonic Maass forms of weight

Bipartite Euler systems

2-n