2023/01/11 by Daniel Delbourgo, Heiko Knospe
Mathematics · #Algebraic Geometry and Number Theory #advanced mathematical theories #Advanced Algebra and Geometry
paper · doi:10.1090/mcom/3823
Following both Ernvall-Metsänkylä and Ellenberg-Jain-Venkatesh, we study the density of the number of zeroes (i.e. the cyclotomic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda"> <mml:semantics> <mml:mi> λ </mml:mi> <mml:annotation encoding="application/x-tex">λ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -invariant) for the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic zeta-function twisted by a Dirichlet character <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi"> <mml:semantics> <mml:mi> χ </mml:mi> <mml:annotation encoding="application/x-tex">χ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of any order. We are interested in two cases: (i) the character <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi"> <mml:semantics> <mml:mi> χ </mml:mi> <mml:annotation encoding="application/x-tex">χ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is fixed and the prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> varies, and (ii) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="o r d left-parenthesis chi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>o</mml:mi> <mml:mi>r</mml:mi> <mml:mi>d</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi> χ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">ord(χ )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are both fixed but <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi"> <mml:semantics> <mml:mi> χ </mml:mi> <mml:annotation encoding="application/x-tex">χ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is allowed to vary. We predict distributions for these <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda"> <mml:semantics> <mml:mi> λ </mml:mi> <mml:annotation encoding="application/x-tex">λ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -invariants using <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic random matrix theory and provide numerical evidence for these predictions. We also study the proportion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi"> <mml:semantics> <mml:mi> χ </mml:mi> <mml:annotation encoding="application/x-tex">χ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -regular primes, which depends on how <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> splits inside <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q left-parenthesis chi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> χ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb Q(χ )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Finally, we tabulate the values of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda"> <mml:semantics> <mml:mi> λ </mml:mi> <mml:annotation encoding="application/x-tex">λ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -invariant for every character <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi"> <mml:semantics> <mml:mi> χ </mml:mi> <mml:annotation encoding="application/x-tex">χ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of conductor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="less-than-or-slanted-equals 1000"> <mml:semantics> <mml:mrow> <mml:mo> ⩽ </mml:mo> <mml:mn>1000</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\leqslant 1000</