{"title":"Large values of quadratic Dirichlet 𝐿-functions over monic irreducible polynomial in 𝔽_{𝕢}[𝕥]","authors":"Pranendu Darbar, Gopal Maiti","doi":"10.1090/proc/16828","DOIUrl":null,"url":null,"abstract":"<p>We prove an <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega\">\n <mml:semantics>\n <mml:mi mathvariant=\"normal\">Ω</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\Omega</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-result for the quadratic Dirichlet <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\">\n <mml:semantics>\n <mml:mi>L</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">L</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-function <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue upper L left-parenthesis 1 slash 2 comma chi Subscript upper P Baseline right-parenthesis EndAbsoluteValue\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n <mml:mi>L</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>χ</mml:mi>\n <mml:mi>P</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">|L(1/2, \\chi _P)|</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> over irreducible polynomials <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P\">\n <mml:semantics>\n <mml:mi>P</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">P</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> associated with the hyperelliptic curve of genus <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g\">\n <mml:semantics>\n <mml:mi>g</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">g</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> over a fixed finite field <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper F Subscript q\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">F</mml:mi>\n </mml:mrow>\n <mml:mi>q</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {F}_q</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in the large genus limit. In particular, we showed that for any <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon element-of left-parenthesis 0 comma 1 slash 2 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>ϵ</mml:mi>\n <mml:mo>∈</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\epsilon \\in (0, 1/2)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <disp-formula content-type=\"math/mathml\">\n\\[\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"max Underscript StartLayout 1st Row upper P element-of script upper P Subscript 2 g plus 1 Baseline EndLayout Endscripts StartAbsoluteValue upper L left-parenthesis 1 slash 2 comma chi Subscript upper P Baseline right-parenthesis EndAbsoluteValue much-greater-than exp left-parenthesis left-parenthesis StartRoot left-parenthesis 1 slash 2 minus epsilon right-parenthesis ln q EndRoot plus o left-parenthesis 1 right-parenthesis right-parenthesis StartRoot StartFraction g ln Subscript 2 Baseline g Over ln g EndFraction EndRoot right-parenthesis comma\">\n <mml:semantics>\n <mml:mrow>\n <mml:munder>\n <mml:mo movablelimits=\"true\" form=\"prefix\">max</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mstyle scriptlevel=\"1\">\n <mml:mtable rowspacing=\"0.1em\" columnspacing=\"0em\" displaystyle=\"false\">\n <mml:mtr>\n <mml:mtd>\n <mml:mi>P</mml:mi>\n <mml:mo>∈</mml:mo>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">P</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mn>2</mml:mn>\n <mml:mi>g</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:msub>\n </mml:mtd>\n </mml:mtr>\n </mml:mtable>\n </mml:mstyle>\n </mml:mrow>\n </mml:munder>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n <mml:mi>L</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>χ</mml:mi>\n <mml:mi>P</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n <mml:mo>≫</mml:mo>\n <mml:mi>exp</mml:mi>\n <mml:mo></mml:mo>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:msqrt>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n <mml:mo>−</mml:mo>\n <mml:mi>ϵ</mml:mi>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mi>ln</mml:mi>\n <mml:mo></mml:mo>\n <mml:mi>q</mml:mi>\n </mml:msqrt>\n <mml:mo>+</mml:mo>\n <mml:mi>o</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:msqrt>\n <mml:mfrac>\n <mml:mrow>\n <mml:mi>g</mml:mi>\n <mml:msub>\n <mml:mi>ln</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo></mml:mo>\n <mml:mi>g</mml:mi>\n </mml:mrow>\n <mml:mrow>\n <mml:mi>ln</mml:mi>\n <mml:mo></mml:mo>\n <mml:mi>g</mml:mi>\n </mml:mrow>\n </mml:mfrac>\n </mml:msqrt>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mo>,</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\max _{\\substack {P\\in \\mathcal {P}_{2g+1}}}|L(1/2, \\chi _P)|\\gg \\exp \\left (\\left (\\sqrt {\\left (1/2-\\epsilon \\right )\\ln q}+o(1)\\right )\\sqrt {\\frac {g \\ln _2 g}{\\ln g}}\\right ),</mml:annotation>\n </mml:semantics>\n</mml:math>\n\\]\n</disp-formula> where <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper P Subscript 2 g plus 1\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">P</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mn>2</mml:mn>\n <mml:mi>g</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {P}_{2g+1}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is the set of all monic irreducible polynomials of degree <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2 g plus 1\">\n <mml:semantics>\n <mml:mrow>\n <mml:mn>2</mml:mn>\n <mml:mi>g</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">2g+1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. This matches with the order of magnitude of the Bondarenko–Seip bound.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16828","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We prove an Ω\Omega-result for the quadratic Dirichlet LL-function |L(1/2,χP)||L(1/2, \chi _P)| over irreducible polynomials PP associated with the hyperelliptic curve of genus gg over a fixed finite field Fq\mathbb {F}_q in the large genus limit. In particular, we showed that for any ϵ∈(0,1/2)\epsilon \in (0, 1/2),
\[
maxP∈P2g+1|L(1/2,χP)|≫exp(((1/2−ϵ)lnq+o(1))gln2glng),\max _{\substack {P\in \mathcal {P}_{2g+1}}}|L(1/2, \chi _P)|\gg \exp \left (\left (\sqrt {\left (1/2-\epsilon \right )\ln q}+o(1)\right )\sqrt {\frac {g \ln _2 g}{\ln g}}\right ),
\]
where P2g+1\mathcal {P}_{2g+1} is the set of all monic irreducible polynomials of degree 2g+12g+1. This matches with the order of magnitude of the Bondarenko–Seip bound.
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