Local diophantine properties of Shimura curves
This paper makes the observation that in contrast to the classical elliptic modular curves, Shimura curves can fail to have points over nonarchimedean local fields. Let B be an indefinite rational quaternion division algebra of discriminant Disc B and denote by \(V_ B=V_ B/{\mathbb{Q}}\) the corresponding Shimura curve. The question of whether \(V_ B(L)=\emptyset\) for a nonarchimedean local field L is reduced by Hensel's lemma to a question about the special fiber in a regular model of \(V_ B\) over Spe\(c({\mathcal O}_ L)\). The authors analyze this question about the special fiber using the Eichler-Selberg trace formula in the case of good reduction and Drinfeld's results in the case of bad reduction. Necessary and sufficient conditions are deduced for \(V_ B\) to have points rational over a nonarchimedean local field: Theorem. Let K be a finite extension of \({\mathbb{Q}}_ p\) with \(e=e(K/{\mathbb{Q}}_ p),\quad f=f(K/{\mathbb{Q}}_ p).\) Then (I) If f is even, \(V_ B(K)\neq \emptyset\). - (II) Suppose f is odd. - (1) If p \(\nmid Disc B\), \(V_ B(K)=\emptyset\) if and only if for every \(\alpha\) satisfying \(x^ 2+sx+p^ f=0\) with \(a\in {\mathbb{Z}}\), \(| s|<2p^{f/2}\), either \({\mathbb{Q}}(\alpha)\) fails to split B or \(p| \alpha\) and p splits in \({\mathbb{Q}}(\alpha)\). - (2) Suppose p \(| Disc B\). (i) If e is even, \(V_ B(K)\neq \emptyset\) if and only if either \({\mathbb{Q}}(\sqrt{-p})\) splits B or \(p=2\) and \({\mathbb{Q}}(\sqrt{-1})\) splits B. (ii) If e is odd, \(V_ B(K)\neq \emptyset\) if and only if either Disc B\(=2p\) with \(p\equiv 1 mod 4\) or \(p=2\) and \(Disc B=2\cdot q_ 1...q_{2r-1}\) with the primes \(q_ i\), 1\(\leq i\leq 2r-1\), satisfying \(q_ i\equiv 3 mod 4.\)
no points over nonarchimedean local fields
scientific article; zbMATH DE number 3851263
scientific article