Singular Moduli of Shimura Curves

The purpose of this note is to introduce a method for proving the non-existence of rational points on a coarse moduli space X of abelian varieties over a given number field K in cases where the moduli problem is not fine and points in X(K) may not be represented by an abelian variety (with additional structure) admitting a model over the field K. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired by the work of Ellenberg and Skinner on the modularity of ℚ-curves, is that one may still attach a Galois representation of Gal(/K) with values in the quotient group GL(Tℓ(A))/ Aut(A) to a point P = [A] ∈ X(K) represented by an abelian variety A/, provided Aut(A) lies in the centre of GL(Tℓ(A)). We exemplify our method in the cases where X is a Shimura curve over an imaginary quadratic field or an Atkin–Lehner quotient over ℚ.

We consider some Diophantine problems of mixed modular-multiplicative type associated with the Zilber-Pink conjecture. In particular, we prove a finiteness statement for the number of multiplicative relations between singular moduli (j-invariants of elliptic curves with complex multiplication.)

Let S-1 and S-2 be two Shimura curves over Q attached to rational indefinite quaternion algebras B-1Q and B-2Q with maximal orders B-1 and B-2 respectively. We consider an irreducible closed algebraic curve C in the product (S-1 x S-2)(C) such that C(C) boolean AND (S-1 x S-2)(C) contains infinitely many complex multiplication points. We prove, assuming the Generalized Riemann Hypothesis (GRH) for imaginary quadratic fields. that C is of Hedge type.

We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational point of order $N$, refining results of Silverberg. Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic isogenies of a $K$-CM elliptic curve $E$ defined over $K(j(E))$.

The arithmetic of quaternions is recalled from a constructive point of view. A Hecke module is introduced, defined as a free abelian group on right ideal classes of a quaternion order, together with a natural action of Hecke operators. An equivalent construction in terms of Shimura curves is then introduced, and the quaternion construction is applied to the analysis of specific modular and Shimura curves.

We determine over which fields twisted Mumford quotients have rational points. Using the $p$-adic uniformization, we apply these results to Shimura curves, and show some new cases for which the jacobians are even in the sense of [PS].

Rational points on elliptic curves with complex multiplication by the ring of integers in [formula omitted](√−7)

For any field K, a “K-curve” is an elliptic curve E defined over some finite separable extension of K such that every Galois conjugate of E is isogenous with E over a separable closure of K. A K-curve must either have complex multiplication (CM) or be isogenous to a collection of K-curves parametrized by a K-rational point on some modular curve X* (N). We give the first proof of this result, obtained in 1993 but not heretofore published. We also indicate some of the reasons for interest in K-curves, and in particular in Q-curves, and give some computational examples and open questions.

The values of the elliptic modular $j$-invariant at imaginary quadratic arguments are algebraic integers, known as singular moduli of level one. If $d_1$ and $d_2$ are imaginary quadratic discriminants, then we may consider a generalized resultant of the class polynomials of the orders of discriminant $d_1$ and $d_2$; this is the norm of the differences of singular moduli of the corresponding orders, denoted here by $J(d_1,d_2)$. These resultants are highly factorizable; Gross-Zagier established a closed formula for $J(d_1,d_2)^2$ when $d_1$ and $d_2$ are fundamental discriminants, with $(d_1,d_2)=1$. In this paper we present a conjectural extension of the Gross-Zagier formula to the case when $d_1$ and $d_2$ are not necessarily fundamental, and $(d_1,d_2)=l^e$, where 1 is a prime not dividing the product of the conductors of $d_1$ and $d_2$.

We study the group of automorphisms of Shimura curves X0(D, N) attached to an Eichler order of square-free level N in an indefinite rational quaternion algebra of discriminant D>1. We prove that, when the genus g of the curve is greater than or equal to 2, Aut (X0(D, N)) is a 2-elementary abelian group which contains the group of Atkin–Lehner involutions W0(D, N) as a subgroup of index 1 or 2. It is conjectured that Aut (X0(D, N))=W0(D, N) except for finitely many values of (D, N) and we provide criteria that allow us to show that this is indeed often the case. Our methods are based on the theory of complex multiplication of Shimura curves and the Cerednik–Drinfeld theory on their rigid analytic uniformization at primes p| D.

We give an explicit procedure for constructing Shimura curves analogous to the modular Curves X-0(N) that are counterexamples to the Hasse principle over imaginary quadratic fields. These counterexamples are accounted for by the Manin obstruction.

We report on a new approach, as well as some related experiments, to construct families of K3 surfaces having real or complex multiplication. The approach is based on an explicit description of the transcendental part of the cohomology in a topological way, using topological tori. Fundamental ideas include considering the period space of marked K3 surfaces, determining the periods by numerical integration, as well as tracing the modular curve by a numerical continuation method.

This paper discusses a broadcasting multi-digital signature algorithm based on the elliptic curve cryptosystem. The method overcomes the problem of poor speed digital signature verification due to more complex operations, reduces relatively complex multiplication and point multiplication in the original broadcast multiple digital signature scheme, and avoids matrix inversion. It improves the speed of verification at the same time, and does not affect the security of the signature.

Let $X_0^6(1)/W_6$ be the Atkin-Lehner quotient of the Shimura curve $X_0^6(1)$ associated to a maximal order in an indefinite quaternion algebra of discriminant $6$ over $\mathbb Q$. By realizing modular forms on $X_0^6(1)/W_6$ in two ways, one in terms of hypergeometric functions and the other in terms of Borcherds forms, and using Schofer's formula for values of Borcherds forms at CM-points, we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over $\overline\mathbb Q$ with complex multiplication.

The finiteness of the group of rational points of the elliptic curve X0 (32) in the infinite cyclotomic two-extension field over the rational numbers is proved. For each finite step, the finiteness of the two-component of the Shafarevich-Tate group is proved and its structure is determined.