1981/01/01 by Larissa Queen · 4 citations
Computer Science · #Algorithm #Artificial intelligence #Coding theory and cryptography #Computer science
paper · pdf · doi:10.1090/s0025-5718-1981-0628715-7
openalex publication_date 1981/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22
In [2] Conway and Norton have assigned a "Thompson series" of the form <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q Superscript negative 1 Baseline plus upper H 1 q plus upper H 2 q squared plus ellipsis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mi>q</mml:mi> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>q</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo>+</mml:mo> <mml:mo> … </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">q - 1 + H1q + H2q2 + …</mml:annotation> </mml:semantics> </mml:math> </disp-formula> to each element <italic>m</italic> of the Fischer-Griess "Monster" group <italic>M</italic> and conjectured that these functions are Hauptmoduls for certain genus-zero modular groups. We have found, for a large number of values of <italic>N</italic> , all the genus-zero groups between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma 0 left-parenthesis upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Γ 0(N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P upper S upper L left-parenthesis 2 comma upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">PSL(2,R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that have Hauptmoduls of the above form, and this provides the necessary verification that the series assigned in [2] to particular elements of <italic>M</italic> really are such Hauptmoduls. (Atkin and Fong [1] have recently verified that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript n Baseline left-parenthesis m right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Hn(m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> really is a character of <italic>M</italic> for all <italic>n</italic> .) We compute Thompson series for various finite groups and discuss the differences between these groups and <italic>M</italic> . We find that the resulting Thompson series are all Hauptmoduls for suitable genus-zero subgroups of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P upper S upper L left-parenthesis 2 comma upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">PSL(2,R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .