An exact family of bivariate polynomials and variants of Chinburg’s Conjectures

Abstract This article provides partial solutions to Chinburg’s conjectures by studying a sequence of multivariate polynomials. These conjectures assert that for every odd quadratic Dirichlet character of conductor f , $$\chi _{-f}=\left( \frac{-f}{.}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>χ</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mfenced> <mml:mfrac> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:mo>.</mml:mo> </mml:mfrac> </mml:mfenced> </mml:mrow> </mml:math> , there exists a bivariate polynomial (or a rational function in the weak version) whose Mahler measure is a rational multiple of $$L'(\chi _{-f},-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>χ</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We prove that the Mahler measure of a polynomial family, denoted by $$P_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:math> , can be expressed as a linear combination of the derivatives of Dirichlet L -functions. Specifically, this family provides solutions to the conjectures for conductors $$f=3,4,8,15,20$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>8</mml:mn> <mml:mo>,</mml:mo> <mml:mn>15</mml:mn> <mml:mo>,</mml:mo> <mml:mn>20</mml:mn> </mml:mrow> </mml:math> , and 24. We further generalize Chinburg’s conjectures from real primitive odd Dirichlet characters to all primitive odd characters. For this generalized version, the polynomials $$P_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:math> provide solutions for conductors 5, 7, and 9.