Maps between Jacobians of Modular Curves

On maps between modular Jacobians and Jacobians of Shimura curves

Fix a squarefree integer N , divisible by an even number of primes, and let Γ be a congruence subgroup of level M , where M is prime to N . For each D dividing N and divisible by an even number of primes, the Shimura curve X(Γ0(N/D)∩Γ ) associated to the indefinite quaternion algebra of discriminant D and Γ0(N/D) ∩ Γ -level structure is well defined, and we can consider its Jacobian J(Γ0(N/D) ∩...

On Relations between Jacobians of Certain Modular Curves

We confirm a conjecture of Merel describing a certain relation between the jacobians of various quotients of X(p) in terms of specific correspondences.

Endomorphisms of Jacobians of Modular Curves

Endomorphisms of Jacobians of Modular Curves

Let XΓ = Γ\H∗ be the modular curve associated to a congruence subgroup Γ of level N with Γ1(N) ≤ Γ ≤ Γ0(N), and let X = XΓ,Q be its canonical model over Q. The main aim of this paper is to show that the endomorphism algebra End0Q(JX) of its Jacobian JX/Q is generated by the Hecke operators Tp, with p N , together with the “degeneracy operators” DM,d, D t M,d, for dM |N . This uses the fundament...

On Component Groups of Jacobians of Drinfeld Modular Curves

Assume N is prime, so X0(N)Q has two cusps; these are labelled 0 and ∞. The two cusps are Q-rational points on X0(N)Q and the divisor (0) − (∞) on X0(N)Q generates a finite cyclic subgroup C in J0(N)(Q) called the cuspidal divisor group. Denote by C/Z the finite flat subgroup scheme of J generated by C ⊂ J0(N)(Q). Let C be the FN -valued points of CZ (“the specialization” of C in J × FN ). It i...