The cusp forms of weight 3 on Γ₂(2,4,8)

Bert van Geemen, Duco van Straten

Type: Article

Publication Date: 1993-01-01

Citations: 10

DOI: https://doi.org/10.1090/s0025-5718-1993-1181333-5

Abstract

The congruence subgroup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma 2 left-parenthesis 2 comma 4 comma 8 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>8</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Gamma _2}(2,4,8)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Gamma _2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4 times 4"> <mml:semantics> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>×</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">4 \times 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> integral symplectic matrices is contained in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma 2 left-parenthesis 4 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>4</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Gamma _2}(4)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and contains <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma 2 left-parenthesis 8 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>8</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Gamma _2}(8)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma 2 left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mn>2</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">{\Gamma _2}(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the principal congruence subgroup of level <italic>n</italic>. The Satake compactification of the quotient of the three-dimensional Siegel upper half space by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma 2 left-parenthesis 2 comma 4 comma 8 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>8</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Gamma _2}(2,4,8)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is shown to be a complete intersection of ten quadrics in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P Superscript 13"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>13</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbb {P}^{13}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We determine the space of global holomorphic three forms on this space, which coincides with the space of cusp forms of weight 3 on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma 2 left-parenthesis 2 comma 4 comma 8 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>8</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Gamma _2}(2,4,8)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; it has dimension 2283. Finally, we study the action of the Hecke operators on this space and consider the Andrianov <italic>L</italic>-functions of some eigenforms.

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