David Savitt

The Buzzard-Diamond-Jarvis conjecture for unitary groups

Let p > 2 be prime. We prove the weight part of Serre’s conjecture for rank two unitary groups for mod p representations in the unramified case (that is, the Buzzard–Diamond–Jarvis conjecture for unitary groups), by proving that any Serre weight which occurs is a predicted weight. Our methods are purely local, using the theory of (φ, Ĝ)-modules to determine the possible reductions of certain two-dimensional crystalline representations.

Serre weights for mod p Hilbert modular forms: the totally ramified case

Abstract We study the possible weights of an irreducible 2-dimensional modular mod p representation of Gal(/F), where F is a totally real field which is totally ramified at p, and the representation is tamely ramified at the prime above p. In most cases we determine the precise list of possible weights; in the remaining cases we determine the possible weights up to a short and explicit list of exceptions.

Modularity of some potentially Barsotti-Tate Galois representations

We prove a portion of a conjecture of Conrad, Diamond, and Taylor, yielding some new cases of the Fontaine–Mazur conjectures, specifically, the modularity of certain potentially Barsotti–Tate Galois representations. The proof follows the template of Wiles, Taylor–Wiles, and Breuil–Conrad–Diamond–Taylor, and relies on a detailed study of the descent, across tamely ramified extensions, of finite flat group schemes over the ring of integers of a local field. This makes crucial use of the filtered

Polynomials, meanders, and paths in the lattice of noncrossing partitions

\phi_1

Harmonic Algebraic Curves and Noncrossing Partitions

-modules of Breuil.

ON A CONJECTURE OF CONRAD, DIAMOND, AND TAYLOR

For every polynomial f of degree n with no double roots, there is an associated family C(f) of harmonic algebraic curves, fibred over the circle, with at most n- 1 singular fibres. We study the combinatorial topology of C(f) in the generic case when there are exactly n - 1 singular fibres. In this case, the topology of C(f) is determined by the data of an n-tuple of noncrossing matchings on the set {0,1,...,2n-1} with certain extra properties. We prove that there are 2(2n) n-2 such n-tuples, and that all of them arise from the topology of C(f) for some polynomial f.

Lattices in the cohomology of Shimura curves

Motivated by Gausss first proof of the Fundamental Theorem of Algebra, we study the topology of harmonic algebraic curves. By the maximum principle, a harmonic curve has no bounded components; its topology is determined by the combinatorial data of a noncrossing matching. Similarly, every complex polynomial gives rise to a related combinatorial object that we call a basketball, consisting of a pair of noncrossing matchings satisfying one additional constraint. We prove that every noncrossing matching arises from some harmonic curve, and deduce from this that every basketball arises from some polynomial.