Orbits in unimodular Hermitian lattices

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a unimodular indefinite hermitian lattice over the integers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German o"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">o</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {o}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of an algebraic number field, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N left-parenthesis upper L comma c right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo>,</mml:mo> <mml:mi>c</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">N(L,c)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the number of primitive representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c element-of German o"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">o</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">c \in \mathfrak {o}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that are inequalivant modulo the action of the integral special unitary group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper U left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">SU(L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The value of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N left-parenthesis upper L comma c right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo>,</mml:mo> <mml:mi>c</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">N(L,c)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is determined from the local representations via a product formula.

Locations

• Transactions of the American Mathematical Society - View - PDF