Related papers: Local points on P-adically uniformized Shimura var…

We prove that a three-dimensional smooth complete intersection of two quadrics over a field k is k-rational if and only if it contains a line defined over k. To do so, we develop a theory of intermediate Jacobians for geometrically rational…

We present new constructions of complex and p-adic Darmon points on elliptic curves over base fields of arbitrary signature. We conjecture that these points are global and present numerical evidence to support our conjecture.

We consider the Chow ring with rational coefficients of the Jacobian of a curve. Assume D is a divisor in a base point free g^r_d of the curve such that the canonical divisor K is a multiple of the divisor D. We find relations between…

One proves the Crew-Tsuzuki "p-adic local monodromy conjecture" (for local fields of characteristic p>0).

We study CM points on the Shimura curves $X_0^D(N)_{/\mathbb{Q}}$ and $X_1^D(N)_{/\mathbb{Q}}$, parametrizing abelian surfaces with quaternionic multiplication and extra level structure. A description of the locus of points with CM by a…

We prove analogues of several well-known results concerning rational morphisms between quadrics for the class of so-called quasilinear $p$-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric…

The Kodaira dimension of Shimura varieties has been studied by many people. Kondo and Gritsenko-Hulek-Sankaran studied the singularities of orthogonal Shimura varieties related to the moduli spaces of polarized K3 surfaces. They proved that…

We prove that the quotient by SL(2)\timesSL(2) of the space of bidegree (a, b) curves on P^1\timesP^1 is rational when ab is even and a\not=b.

We prove the geometrical Satake isomorphism for a reductive group defined over F=k((t)), and split over a tamely ramified extension. As an application, we give a description of the nearby cycles on certain Shimura varieties via the…

We prove a conjecture of Pappas and Rapoport about the existence of ''canonical'' integral models of Shimura varieties of Hodge type with quasi-parahoric level structure at a prime $p$. For these integral models, we moreover show…

Generalized Pauli's theorem, proved by D. S. Shirokov for two sets of anticommuting elements of a real or complexified Clifford algebra of dimension $2^n$, is extended to the case, when both sets of elements depend smoothly on points of…

For an odd prime p, we construct integral models over p for Shimura varieties with parahoric level structure, attached to Shimura data (G,X) of abelian type, such that G splits over a tamely ramified extension of Q_p. The local structure of…

We determine the isogeny classes of abelian surfaces over F_q whose group of F_q-rational points has order divisible by q^2. We also solve the same problem for Jacobians of genus-2 curves.

We investigate $p$-adic automorphic forms on unitary groups through the geometry of infinite-level unitary Shimura varieties and the Hodge-Tate period map. We first develop a perfectoid construction of overconvergent automorphic forms.…

We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional…

Let $X$ be a regular geometrically integral variety over an imperfect field $K$. Unlike the case of characteristic $0$, $X':=X\times_{\mathrm{Spec}\,K}\mathrm{Spec}\,K'$ may have singular points for a (necessarily inseparable) field…

In this paper we investigate complex uniruled varieties $X$ whose rational curves of minimal degree satisfy a special property. Namely, we assume that the tangent directions to such curves at a general point $x\in X$ form a linear subspace…

We consider Shimura varieties associated to a unitary group of signature $(n-s,s)$ where $n$ is even. For these varieties, by using the spin splitting models from Zachos-Zhao, we construct flat, Cohen-Macaulay, and normal $p$-adic integral…

We give an explicit description of fundamental domains associated to the $p$-adic uniformisation of families of Shimura curves of discriminant $Dp$ and level $N\geq 1$, for which the one-sided ideal class number $h(D,N)$ is $1$. The…

Let $A$ be an abelian variety with commutative endomorphism algebra over a finite field $k$. The $k$-isogeny class of $A$ is uniquely determined by a Weil polynomial $f_A$ without multiple roots. We give a classification of the groups of…