Hasse principle violations for Atkin-Lehner twists of Shimura curves
نویسندگان
چکیده
منابع مشابه
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...
متن کاملRational 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...
متن کاملRational Points on Atkin-lehner Twists of Modular Curves
These are the (more detailed) notes accompanying a talk that I am to give at the University of Pennsylvania on July 21, 2006. The topic is rational points on Atkin-Lehner twists of the modular curves X0(N). Apart from being an interesting Diophantine problem in its own right, there is an ulterior motive: Q-rational points correspond to “elliptic Q-curves” and thus to projective Galois represent...
متن کاملOn the Hasse Principle for Shimura Curves
Let C be an algebraic curve defined over a number field K, of positive genus and without K-rational points. We conjecture that there exists some extension field L over which C violates the Hasse principle, i.e., has points everywhere locally but not globally. We show that our conjecture holds for all but finitely many Shimura curves of the form XD 0 (N)/Q or X D 1 (N)/Q, where D > 1 and N are c...
متن کاملGalois Groups via Atkin-lehner Twists
Using Serre’s proposed complement to Shih’s Theorem, we obtain PSL2(Fp) as a Galois group over Q for at least 614 new primes p. Assuming that rational elliptic curves with odd analytic rank have positive rank, we obtain Galois realizations for 3 8 of the primes that were not covered by previous results; it would also suffice to assume a certain (plausible, and perhaps tractable) conjecture conc...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 2018
ISSN: 0002-9939,1088-6826
DOI: 10.1090/proc/14001