Non-elliptic Shimura curves of genus one

Abstract

We present explicit models for non-elliptic genus one Shimura curves X 0 (D, N ) with Γ 0 (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.

The kernel of Ribet's isogeny for genus three Shimura curves

“…Hence, F v = ∅, |F u | = 4 and |F w | = 8. For ι = v, the number field generated by the coordinates of the projection π ι (P ) of a point P ∈ F ι can be found in [6]. For instance, if U ∈ F u we have Table 1.…”

“…The Jacobian of the curve X (u) has conductor D because it is a quotient of the new part of the Jacobian of X 0 (D) defined over Q and its Q-isomorphism class can be determined by using the Cěrednik-Drinfeld description of the fibers of X D at primes p | D (cf. [17] and the footnote in page 938 of [6]). Next, we show the set X (u) (Q), given as group when X (u) is elliptic over Q, and Cremona's label of the elliptic curve Jac(X (u) ):…”

“…When X (u) (Q) = ∅, i.e. for D = 35, 51, 57 and 82, see Theorem 4.2 of [6]. For D = 51, 57 and 82, we proceed as in [6].…”

“…for D = 35, 51, 57 and 82, see Theorem 4.2 of [6]. For D = 51, 57 and 82, we proceed as in [6]. We take a subset {P 1 , · · · , P n } ⊂ Jac(X (u) )(Q) of representative elements of the group Jac(X (u) )(Q)/2 Jac(X (u) )(Q) and for each P i we attach an equation y 2 = F i (x) as in subsection 2.1 of [6].…”

“…For D = 51, 57 and 82, we proceed as in [6]. We take a subset {P 1 , · · · , P n } ⊂ Jac(X (u) )(Q) of representative elements of the group Jac(X (u) )(Q)/2 Jac(X (u) )(Q) and for each P i we attach an equation y 2 = F i (x) as in subsection 2.1 of [6]. The equation in the statement corresponds to the unique equivalence class such that the number field generated by the roots of F i (x) agrees with the number field generated by the coordinates of the points in π u (F w ).…”