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
Full Text:

References:

 [1] ?erednik, I.V.: Uniformization of algebraic curves by discrete arithmetic subgroups of PGL2(k w ) with compact quotients. (in Russian). Mat. Sb. (N.S.)100, 59-88 (1976), Math. USSR Sb.29, 55-78 (1976) [2] Drinfeld, V.G.: Coverings ofp-adic symmetric regions (in Russian). Functional. Anal. i Prilo?en.10, 29-40 (1976), Functional Anal. Appl.10, 107-115 (1976) [3] Eichler, M.: Modular correspondences and their representations. Report of an International Colloquium on Zeta-Functions. Bombay, 1956 · Zbl 0073.26501 [4] Grothendieck, A.: Etude locale des schémas et des morphismes de schémas (EGAIV). Publ. Math. IHES32 (1967) [5] Ihara, Y.: Hecke polynomials as congruence ? functions in elliptic modular case. Ann. Math.85, 267-295 (1967) · Zbl 0181.36501 [6] Jordan, B.: On the diophantine arithmetic of Shimura curves. Thesis, Harvard University, 1981. [7] Kurihara, A.: On some examples of equations defining Shimura curves and the Mumford uniformization. J. Fac. Sci. Univ. Tokyo, Sec. IA25, 277-301 (1979) · Zbl 0428.14012 [8] Michon, J.-F.: Courbes de Shimura de genre I Séminaire Delange-Pisot-Poitou, 1980/1981 [9] Milne, J.S.: Étale cohomology, Princeton, NJ: Princeton University Press 1980 · Zbl 0433.14012 [10] Morita, Y.: Ihara’s conjectures and moduli space of abelian varieties Master’s Thesis. Univ. Tokyo, 1970 [11] Mumford, D.: An analytic construction of degenerating curves over complete local rings. Compositio Math.24, 129-174 (1972) · Zbl 0228.14011 [12] Serre, J.-P.: Arbres, amalgames, SL2. Astérisque46. Soc. Math. France 1977 [13] Shimura, G.: Construction of class fields and zeta functions of algebraic curves. Ann. Math.85, 58-159 (1967) · Zbl 0204.07201 [14] Shimura, G.: On the real points of an arithmetic quotient of a bounded symmetric domain. Math. Ann.215, 135-164 (1975) · Zbl 0394.14007 [15] Zink, T.: Über die schlechte Reduktion einiger Shimuramannigfaltigkeiten. Compositio Math.45, 15-107 (1981) · Zbl 0483.14006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.