Local diophantine properties of Shimura curves
type: Item
instance of: scholarly article
title: Local diophantine properties of Shimura curves
zbMATH Open document ID: 0536.14018
DOI: 10.1007/BF01456184
author: Bruce W. Jordan
author: Ron Livné
published in: Mathematische Annalen
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/182956
review text: 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 \(q_ i\equiv 3 mod 4.\)
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 11D88
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 3851263
zbMATH Keywords: no points over nonarchimedean local fields
zbMATH Keywords: Shimura curves
Wikidata QID: Q104439613
MaRDI profile type: Publication
cites work: Q3235396
cites work: Hecke polynomials as congruence \(\zeta\) functions in elliptic modular case
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cites work: Q3871006
cites work: Q5635548
cites work: Construction of class fields and zeta functions of algebraic curves
cites work: On the real points of an arithmetic quotient of a bounded symmetric domain
cites work: Q3941526
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description: scientific article; zbMATH DE number 3851263