We reinterpret ideas in Klein’s paper on transformations of degree 11 from the modern point of view of dessins d’enfants, and extend his results by considering dessins of type (3, 2, p) and degree p or <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, where p is prime. In many cases, we determine the passports and monodromy groups of these dessins, and in a few small cases we give drawings which are topologically (or, in certain examples, even geometrically) correct. We use the Bateman–Horn conjecture and extensive computer searches to support the conjecture that there are infinitely many primes of the form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p=(q^n-1)/(q-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mi>n</mi> </msup> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for some prime power q, in which case infinitely many groups <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{PSL}_n(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>PSL</mtext> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> arise as permutation groups and monodromy groups of degree p (an open problem in group theory).

KLEIN’S TEN “DESSINS D’ENFANTS” OF DEGREE 11: THEME WITH VARIATIONS

期刊论文
发表日期：

2026年1月1日

查看全文

G. A. Jones,
A. K. Zvonkin

摘要

We reinterpret ideas in Klein’s paper on transformations of degree 11 from the modern point of view of dessins d’enfants, and extend his results by considering dessins of type (3, 2, p) and degree p or $p+1$ $p + 1$ , where p is prime. In many cases, we determine the passports and monodromy groups of these dessins, and in a few small cases we give drawings which are topologically (or, in certain examples, even geometrically) correct. We use the Bateman–Horn conjecture and extensive computer searches to support the conjecture that there are infinitely many primes of the form $p=(q^n-1)/(q-1)$ $p = (q^{n} - 1) / (q - 1)$ for some prime power q, in which case infinitely many groups $\textrm{PSL}_n(q)$ ${PSL}_{n} (q)$ arise as permutation groups and monodromy groups of degree p (an open problem in group theory).