Non-elliptic Shimura Curves of Genus One
نویسندگان
چکیده
We present explicit models for non-elliptic genus one Shimura curves X0(D, N) with Γ0(N)-level structure arising from an indefinite quaternion algebra of reduced discriminant D, and Atkin-Lehner quotients of them. In addition, we discuss and extend Jordan’s work [10, Ch. III] on points with complex multiplication on Shimura curves.
منابع مشابه
Shimura curves of genus at most two
We enumerate all Shimura curves XD 0 (N) of genus at most two: there are exactly 858 such curves, up to equivalence. The elliptic modular curve X0(N) is the quotient of the completed upper halfplane H∗ by the congruence subgroup Γ0(N) of matrices in SL2(Z) that are upper triangular modulo N ∈ Z>0. The curve X0(N) forms a coarse moduli space for (generalized) elliptic curves equipped with a cycl...
متن کاملExplicit modular towers
We give a general recipe for explicitly constructing asymptotically optimal towers of modular curves such as {X0(l)}n>1. We illustrate the method by giving equations for eight towers with various geometric features. We conclude by observing that such towers are all of a specific recursive form and speculate that perhaps every tower of this form which attains the Drinfeld-Vlăduţ bound is modular...
متن کاملAppendix to the preceding paper: A rank 3 generalization of the Conjecture of Shimura and Taniyama
The Conjecture of Shimura and Taniyama is a special case of a general philosophy according to which a motive of a certain type should correspond to a special type of automorphic forms on a reductive group. Now familiar extensions of the conjecture include (i) essentially the same statement for elliptic curves over totally real fields and (ii) the statement that for each irreducible abelian surf...
متن کاملOn equations defining fake elliptic curves par Pilar BAYER et Jordi GUÀRDIA
Shimura curves associated to rational nonsplit quaternion algebras are coarse moduli spaces for principally polarized abelian surfaces endowed with quaternionic multiplication. These objects are also known as fake elliptic curves. We present a method for computing equations for genus 2 curves whose Jacobian is a fake elliptic curve with complex multiplication. The method is based on the explici...
متن کاملA descent method for explicit computations on curves
It is shown that the knowledge of a surjective morphism $Xto Y$ of complex curves can be effectively used to make explicit calculations. The method is demonstrated by the calculation of $j(ntau)$ (for some small $n$) in terms of $j(tau)$ for the elliptic curve with period lattice $(1,tau)$, the period matrix for the Jacobian of a family of genus-$2$ curves complementing the classi...
متن کامل