An explicit modular equation in two variables for 𝑄(√3)

Abstract

A system of modular equations of norm 2 had been found for the Hilbert modular function field of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Q left-parenthesis StartRoot 2 EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf {Q}}(\sqrt 2 )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in an earlier issue of this journal. Here an analogous system is found for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Q left-parenthesis StartRoot 3 EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msqrt> <mml:mn>3</mml:mn> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf {Q}}(\sqrt 3 )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> but with the help of MACSYMA. There are special difficulties in the fact that two spaces of Hilbert modular functions exist for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Q left-parenthesis StartRoot 3 EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msqrt> <mml:mn>3</mml:mn> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf {Q}}(\sqrt 3 )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that can be interchanged by the modular equations. The equations are also a remarkable example of hidden symmetries in the algebraic manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper V 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">V</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {V}}_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is defined in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper C Superscript 4"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {C}}^4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by the modular equation.

Locations

• Mathematics of Computation - View - PDF