Related papers: Shimura curves embedded in Igusa's threefold

In previous work, the first author developed an algorithm for the computation of Hilbert modular forms. In this paper, we extend this to all totally real number fields of even degree and nontrivial class group. Using the algorithm over…

We present an explicit basis for orders of arbitrary level N>1 in definite rational quaternion algebras. These orders have applications to computations of spaces of elliptic and quaternionic modular forms.

This paper studies a class of Abelian varieties that are of $\GL_2$-type and with isogenous classes defined over a number field $k$. We treat the cases when their endomorphism algebras are either (1) a totally real field $K$ or (2) a…

The moduli space of principally polarized abelian varieties with real structure and with level $N=4m$ structure (with $m \ge 1$) is shown to coincide with the set of real points of a quasi-projective algebraic variety defined over $\mathbb…

We construct explicit generating series of arithmetic extensions of Kudla's special divisors on integral models of unitary Shimura varieties over CM fields with arbitrary split levels and prove that they are modular forms valued in the…

This is an addition to a series of papers [FL1, FL2, FL3, FL4], where we develop quaternionic analysis from the point of view of representation theory of the conformal Lie group and its Lie algebra. In this paper we develop split…

We prove a higher dimensional generalization of Gross and Zagier's theorem on the factorization of differences of singular moduli. Their result is proved by giving a counting formula for the number of isomorphisms between elliptic curves…

Let $K$ be a quartic CM field, that is, a totally imaginary quadratic extension of a real quadratic number field. In a 1962 article titled On the classfields obtained by complex multiplication of abelian varieties, Shimura considered a…

We study surfaces constructed from groups of units in quaternion orders $\Lambda$ over the integers in real quadratic fields k. A short presentation of some general theory of such surfaces is given, in particular, we construct certain…

Generalizing the work of Atkin and Kaneko-Zagier in the elliptic case, we describe the non-ordinary locus of a genus-zero non-compact curve $Y$ in a Hilbert modular variety in terms of the zeros of generalized Atkin's orthogonal…

Let $p$ and $q$ be two positive primes. Let $\ell$ be an odd positive prime integer and $F$ a quadratic number field. Let $K$ be an extension of $F$ such that $K$ is a dihedral extension of $\Q$ of degree $\ell$ over $F$ or $K$ is an…

In this paper, we present an algorithm to compute a basis of the space of algebraic modular forms on the maximal order of the definite quaternion algebra of discriminant $2$, and provide a database of such bases. One of our motivations is…

An abelian surface A over a field K has potential quaternionic multiplication if the ring End_\bar K (A) of geometric endomorphisms of A is an order in an indefinite rational division quaternion algebra. In this brief note, we study the…

Let $F$ be a number field, and $D$ be a quaternion $F$-algebra. We show that the class number of any residually unramified $O_F$-order (e.g. an Eichler order) in $D$ is divisible by the class number of $F$.

In this article we compute the mass associated to any unimodular lattice in a Hermitian space over an arbitrary CM field under a condition at 2. We study the geometry and arithmetic of the basic locus of the GU(r,s)-Shimura variety…

Poonen and Stoll have shown that the reduced Shafarevich-Tate group of a principally polarized abelian variety over a global field can have order twice a square (the odd case) as well as a square (the even case). For a curve over a global…

Let L be a quadratic imaginary field, inert at the rational prime p. Fix an integer n at least 3, and let M be the moduli space (in characteristic p) of principally polarized abelian varieties of dimension n equipped with an action by O_L…

Quaternionic tori are defined as quotients of the skew field $\mathbb{H}$ of quaternions by rank-4 lattices. Using slice regular functions, these tori are endowed with natural structures of quaternionic manifolds (in fact quaternionic…

For every fixed genus $g\geq 1$, we consider all quadruples $Q=(w_0,w_1,w_2,d)\in\mathbb{Z}^4_{>0}$ with the property that any smooth degree-$d$ curve embedded in the weighted projective plane $\mathbb{P}^2(w_0,w_1,w_2)$ has genus $g$. We…

Given a newform f, we extend Howard's results on the variation of Heegner points in the Hida family of f to a general quaternionic setting. More precisely, we build big Heegner points and big Heegner classes in terms of compatible families…