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Local diophantine properties of Shimura curves. (English) Zbl 0536.14018

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 \(q_ i\equiv 3 mod 4.\)

MSC:

14H25 Arithmetic ground fields for curves
14G20 Local ground fields in algebraic geometry
11D88 \(p\)-adic and power series fields
14G05 Rational points
14H45 Special algebraic curves and curves of low genus
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References:

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