Ehud de Shalit

Residues on buildings and de Rham cohomology of p-adic symmetric domains

The cohomology of Drinfeld’s p-adic symmetric domain was computed by P. Schneider and U. Stuhler in 1991. Here we propose a more explicit and combinatorial approach based on a notion of residue of a closed form along simplices in the BruhatTits building. We identify the cohomology with a certain space of harmonic cochains on the building. We also answer a few questions left open in the original approach.

The p -adic monodromy-weight conjecture for p -adically uniformized varieties

A p -adically uniformized variety is a smooth projective variety whose associated rigid analytic space admits a uniformization by Drinfelds p -adic symmetric domain. For such a variety we prove the monodromy-weight conjecture, which asserts that two independently defined filtrations on the log-crystalline cohomology of the special fiber in fact coincide. The proof proceeds by reducing the conjecture to a combinatorial statement about harmonic cochains on the Bruhat–Tits building, which was verified in our previous work.

Cohomology of discrete groups in harmonic cochains on buildings

Modules of harmonic cochains on the Bruhat-Tits building of the projective general linear group over ap-adic field were defined by one of the authors, and were shown to represent the cohomology of Drinfel’d’sp-adic symmetric domain. Here we define certain non-trivial natural extensions of these modules and study their properties. In particular, for a quotient of Drinfel’d’s space by a discrete cocompact group, we are able to define maps between consecutive graded pieces of its de Rham cohomology, which we show to be (essentially) isomorphisms. We believe that these maps are graded versions of the Hyodo-Kato monodromy operatorN.

Hecke Rings and Universal Deformation Rings

Wiles’ proof of the Shimura-Taniyama-Weil conjecture for semi-stable elliptic curves is based on the “modularity” of certain universal deformation rings.

p-adic periods and modular symbols of elliptic curves of prime conductor

Under certain assumptions, we prove a conjecture of Mazur and Tate describing a relation between the modular symbol attached to an elliptic curve with split multiplicative reduction atp, and itsp-adic period. We generalize this relation to modular forms of weight 2 with coefficients not necessarily in.

A theta operator on Picard modular forms modulo an inert prime

We study the reduction of Picard modular surfaces modulo an inert prime, mod p and p-adic modular forms. We construct a theta operator on such modular forms and study its poles and its effect on Fourier-Jacobi expansions.

On the cohomology of Drinfel’d’sp-adic symmetric domain

There are, by now, three approaches to the de-Rham cohomology of Drinfel’d’sp-adic symmetric domain: the original work of Schneider and Stuhler, and more recent work of Iovita and Spiess, and of de Shalit. In the first part of this paper we compare all three approaches and clarify a few points which remained obscure. In the second half we give a short proof of a conjecture of Schneider and Stuhler, previously proven by Iovita and Spiess, on a Hodge-like decomposition of the cohomology ofp-adically uniformized varieties.

Differentials of the second kind on mumford curves

We define ap-adic analytic Hodge decomposition for the cohomology of Mumford curves, with values in a local system. When the local system is trivial, we give a new proof of Gerritzen’s theorem, that this decomposition forms a variation of Hodge structure, in a family of Mumford curves.

Artin L Functions

Let χ : (ℤ/mℤ)× → ℂ× be a primitive Dirichlet character modulo m. Let K = ℚ(ζ), where ζ = e 2πi/m . The identification G = Gal(K/ℚ) ≃ (ℤ/mℤ)× allows us to attach to χ a character χ Gal : G → ℂ× satisfying 1.1 if (p, m) = 1 and σ p is the Frobenius automorphism at p (the canonical generator of the decomposition group of p in G, which induces on the residue field of any prime of K above p the automorphism x ↦ x p .) The Kronecker-Weber theorem (Kronecker 1853, Weber 1886) asserts that every 1-dimensional character of G ℚ = Gal(ℚ/ℚ) is of the form χ Gal for an appropriate χ.