On Relations between Jacobians of Certain Modular Curves

Arithmetic Properties of the Shimura-shintani-waldspurger Correspondence

We prove that the theta correspondence for the dual pair (S̃L2, PB×), for B an indefinite quaternion algebra over Q, preserves rationality and p-integrality in both directions. As a consequence, we deduce the rationality of certain period ratios of modular forms and even p-integrality of these ratios under the assumption that p does not divide a certain L-value. The rationality is applied to giv...

Endomorphisms of Jacobians of Modular Curves

Visibility of the Shafarevich–Tate Group at Higher Level

We study visibility of Shafarevich–Tate groups of modular abelian varieties in Jacobians of modular curves of higher level. We prove a theorem about the existence of visible elements at a specific higher level under certain hypothesis which can be verified explicitly. We also provide a table of examples of visible subgroups at higher level and state a conjecture inspired by our data. 2000 Mathe...

On the Torsion of Jacobians of Principal Modular Curves of Level 3

We demonstrate that the 3-power torsion points of the Jacobians of the principal modular curves X(3n) are fixed by the kernel of the canonical outer Galois representation of the pro-3 fundamental group of the projective line minus three points. The proof proceeds by demonstrating the curves in question satisfy a two-part criterion given by Anderson and Ihara. Two proofs of the second part of th...

Exhibiting Sha[2] on Hyperelliptic Jacobians

We discuss approaches to computing in the Shafarevich-Tate group of Jacobians of higher genus curves, with an emphasis on the theory and practice of visualisation. Especially for hyperelliptic curves, this often enables the computation of ranks of Jacobians, even when the 2-Selmer bound does not bound the rank sharply. This was previously only possible for a few special cases. For curves of gen...