Regularized inner products and modular invariants for real quadratic fields

In this paper, we study the regularized inner products of Borcherds-Zagier bases of the space of weakly holomorphic modular forms of weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 slash 2"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">1/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We prove a general result, which, in particular, will imply that the regularized inner products of these forms can be expressed in terms of traces of integrals of Klein’s <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j"> <mml:semantics> <mml:mi>j</mml:mi> <mml:annotation encoding="application/x-tex">j</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -function over real quadratic geometric invariants (certain modular surfaces) and Hurwitz-Kronecker class numbers. This complements a result of Duke, Imamoğlu and Tóth, describing the regularized inner products of (dual) weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3 slash 2"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">3/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> weakly holomorphic modular forms in terms of real quadratic analog of traces of singular moduli. Further, we establish an identity relating the traces of harmonic Maass forms <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J overTilde"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>J</mml:mi> <mml:mo stretchy="false"> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\tilde {J}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and sesquiharmonic Maass forms <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction partial-differential Over partial-differential z EndFraction ModifyingAbove double-struck upper J With caret Subscript m"> <mml:semantics> <mml:mrow> <mml:mfrac> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mrow> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>z</mml:mi> </mml:mrow> </mml:mfrac> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">J</mml:mi> </mml:mrow> <mml:mo stretchy="false"> ^ </mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\frac {\partial }{\partial z}\hat {\Bbb J}_{m}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , both of weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which will play an important role in proving the above results. The form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove double-struck upper J With caret Subscript 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">J</mml:mi> </mml:mrow> <mml:mo stretchy="false"> ^ </mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\hat {\Bbb J}_{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is essentially the pre-image of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j"> <mml:semantics> <mml:mi>j</mml:mi> <mml:annotation encoding="application/x-tex">j</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -function under the hyperbolic Laplacian while <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J overTilde"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>J</mml:mi> <mml:mo stretchy="false"> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\tilde {J}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is related to the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j"> <mml:semantics> <mml:mi>j</mml:mi> <mml:annotation encoding="application/x-tex">j</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -function as a dual object via the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="xi"> <mml:semantics> <mml:mi> ξ </mml:mi> <mml:annotation encoding="application/x-tex">\xi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -operator. As an application, we also interpret the regularized inner products of Borcherds-Zagier bases in ter