2024/02/11 by T. D. Browning, Tim Browning, Browning, Tim +4
Mathematics · #Analytic Number Theory Research #Meromorphic and Entire Functions
paper · pdf · doi:10.1017/fms.2026.10259
Abstract We use a function field version of the circle method to prove that a positive proportion of elements in <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">[</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:math> \mathbb Fq[t] double struck upper F Subscript q Baseline left bracket t right bracket are representable as a sum of three cubes of minimal degree from <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">[</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:math> \mathbb Fq[t] double struck upper F Subscript q Baseline left bracket t right bracket , assuming a suitable form of the Ratios Conjecture and that <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:mrow> <mml:mi>char</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>></mml:mo> <mml:mn>3</mml:mn> </mml:math> \operatorname \mathrm char(\mathbb Fq)>3 char left parenthesis double struck upper F Subscript q Baseline right parenthesis greater than 3 . The analogue of this conjecture for quadratic Dirichlet L -functions is known for large fixed q , via recent developments in homological stability.