Proving the triviality of rational points on Atkin–Lehner quotients of Shimura curves

Rational Points on Atkin-Lehner Quotients of Shimura Curves

We study three families of Atkin-Lehner quotients of quaternionic Shimura curves: X, X 0 (N), and X D+ 1 (N), which serve as moduli spaces of abelian surfaces with potential quaternionic multiplication (PQM) and level N structure. The arithmetic geometry of these curves is similar to, but even richer than, that of the classical modular curves. Two important differences are the existence of a no...

On Atkin-lehner Quotients of Shimura Curves

We study the Čerednik-Drinfeld p-adic uniformization of certain AtkinLehner quotients of Shimura curves over Q. We use it to determine over which local fields they have rational points and divisors of a given degree. Using a criterion of Poonen and Stoll we show that the Shafarevich-Tate group of their jacobians is not of square order for infinitely many cases. In [PSt] Poonen and Stoll have sh...

Heights of Heegner Points on Shimura Curves

Introduction 2 1. Shimura curves 6 1.1. Modular interpretation 6 1.2. Integral models 13 1.3. Reductions of models 17 1.4. Hecke correspondences 18 1.5. Order R and its level structure 25 2. Heegner points 29 2.1. CM-points 29 2.2. Formal groups 34 2.3. Endomorphisms 37 2.4. Liftings of distinguished points 40 3. Modular forms and L-functions 43 3.1. Modular forms 44 3.2. Newforms on X 49 3.3. ...

Lectures on Shimura Curves 8: Real Points

To be more precise, let F be a totally real field of degree g, of narrow class number 1, and B/F a quaternion algebra split at exactly one infinite place ∞1 of F . Let N be an integral ideal prime to the discriminant of B. Let us write Γ, Γ0(N ), Γ1(N ), Γ(N ) for the corresponding congruence subgroups of Γ, the image in PGL2(R) of the positive norm units of a maximal order O (unique up to conj...

Rational points on curves