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 shown that if the Shafarevich-Tate group of a principally polarized abelian variety A over a global field K is finite, then its order can be twice a square as well as a square. They call A/K even if the quotient of its Shafarevich-Tate group by the maximal divisible subgroup has order a square and odd otherwise. They prove [PSt, Corollary 10] that the jacobian of a curve X/K of genus g is odd if and only if the number of places v of K where X fails to have a Kv-rational divisor of degree g−1 is odd. They also show that infinitely many hyperelliptic jacobians over Q are odd for every even genus. Let B be an indefinite rational quaternion division algebra of discriminant DiscB. Let VB/Q be the Shimura curve corresponding to a maximal order in B. In [JL1] and [JL2] we determined when Shimura curves VB have rational points, rational divisors of a given degree, and rational divisor classes of a given degree over any local field. The key ingredient of our analysis was the explicit p-adic uniformization of these curves given by Drinfeld [Dr]. The criterion of Poonen and Stoll then immediately yields that the jacobian of VB/Q is even ([PSt, Theorem 23]). Atkin-Lehner involutions wd for d|DiscB act on the Shimura curves VB. Set V (d) B = V (d) B /Q = VB/wd. We will show that these AtkinLehner quotients V (d) B /Q, in contrast to VB/Q, yield examples of odd nonhyperelliptic jacobians of arbitrarily high genus. For simplicity we consider quaternion algebras B in which exactly two odd primes ramify and study the Atkin-Lehner quotient V (p ) B /Q where p is a prime. For a related study of more general Shimura curves over totally real fields see [JLV]. In this paper we will prove the following: Date: February 4, 1998. 1991 Mathematics Subject Classification. 11G18,11G20,14G20,14G35,14H40.