In this article we consider compact Riemann surfaces that are uniquely determined by the property of possessing a group of automorphisms of a prescribed order, strengthening uniqueness results proved by Nakagawa. More precisely, we deal with the cases in which such an order is 3g and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1618_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$3g+3,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mi>g</mi> <mo>+</mo> <mn>3</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where g is the genus. We prove that if g is odd (respectively g even and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1618_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">$g \not \equiv 2 \text{ mod } 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>≢</mo> <mn>2</mn> <mspace width="0.333333em" /> <mtext>mod</mtext> <mspace width="0.333333em" /> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>) then there exists a unique Riemann surface of genus g with a group of automorphisms of order 3g (respectively <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1618_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$3g+3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mi>g</mi> <mo>+</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>). A similar conclusion can be derived in terms of orientably-regular hypermaps. In addition, we determine the full automorphism group of such Riemann surfaces and provide decompositions of their Jacobians.

Classifying compact Riemann surfaces by number of symmetries

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发表日期：

2025年2月18日

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Sebastián Reyes-Carocca,
Pietro Speziali

摘要

In this article we consider compact Riemann surfaces that are uniquely determined by the property of possessing a group of automorphisms of a prescribed order, strengthening uniqueness results proved by Nakagawa. More precisely, we deal with the cases in which such an order is 3g and $3g+3,$ $3 g + 3,$ where g is the genus. We prove that if g is odd (respectively g even and $g \not \equiv 2 \text{ mod } 3$ $g ≢ 2 mod 3$ ) then there exists a unique Riemann surface of genus g with a group of automorphisms of order 3g (respectively $3g+3$ $3 g + 3$ ). A similar conclusion can be derived in terms of orientably-regular hypermaps. In addition, we determine the full automorphism group of such Riemann surfaces and provide decompositions of their Jacobians.