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<p>In this paper we study the geodesic continued fraction in the case of the Shimura curve coming from the (2,&#xa0;3,&#xa0;7)-triangle group. We construct a certain continued fraction expansion of real numbers using the so-called coding of the geodesics on the Shimura curve, and prove the Lagrange type periodicity theorem for the expansion which captures the fundamental relative units of quadratic extensions of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1159_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {Q}}(\cos (2\pi /7))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <mo>cos</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">/</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with rank one relative unit groups. We also discuss the convergence of these continued fractions.</p>

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The geodesic continued fraction for Shimura curves and its periodicity: the case of the (2, 3, 7)-triangle group

  • 期刊论文

  • 发表日期：

    2025年7月21日

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  • Hohto Bekki

摘要

In this paper we study the geodesic continued fraction in the case of the Shimura curve coming from the (2, 3, 7)-triangle group. We construct a certain continued fraction expansion of real numbers using the so-called coding of the geodesics on the Shimura curve, and prove the Lagrange type periodicity theorem for the expansion which captures the fundamental relative units of quadratic extensions of \({\mathbb {Q}}(\cos (2\pi /7))\) Q ( cos ( 2 π / 7 ) ) with rank one relative unit groups. We also discuss the convergence of these continued fractions.