{"title":"旋积场上某些赫克字符的 L 函数非消失","authors":"Keunyoung Jeong , Yeong-Wook Kwon , Junyeong Park","doi":"10.1016/j.jnt.2024.06.002","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above 2. As an application, we show that for each isogeny factor of the Jacobian of the <em>p</em>-th Fermat curve where 2 is a quadratic residue modulo <em>p</em>, there are infinitely many twists whose analytic rank is zero. Also, for a certain hyperelliptic curve over the 11-th cyclotomic field whose Jacobian has complex multiplication, there are infinitely many twists whose analytic rank is zero.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Nonvanishing of L-function of some Hecke characters on cyclotomic fields\",\"authors\":\"Keunyoung Jeong , Yeong-Wook Kwon , Junyeong Park\",\"doi\":\"10.1016/j.jnt.2024.06.002\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above 2. As an application, we show that for each isogeny factor of the Jacobian of the <em>p</em>-th Fermat curve where 2 is a quadratic residue modulo <em>p</em>, there are infinitely many twists whose analytic rank is zero. Also, for a certain hyperelliptic curve over the 11-th cyclotomic field whose Jacobian has complex multiplication, there are infinitely many twists whose analytic rank is zero.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-07-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24001483\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24001483","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
在这篇论文中,我们展示了循环域上一些赫克特征的非消失性。本文的主要内容是计算一些素数(包括 2 以上的素数)的特征函数和 Weil 表示的作用。作为应用,我们证明了对于第-次费马曲线的雅各布因子的每个等元因子,其中 2 是二次残差模,有无穷多个捻的解析秩为零。另外,对于第 11 个旋回域上的某条超椭圆曲线,其雅各布因子具有复乘法,则有无穷多个阶数为零的捻。
Nonvanishing of L-function of some Hecke characters on cyclotomic fields
In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above 2. As an application, we show that for each isogeny factor of the Jacobian of the p-th Fermat curve where 2 is a quadratic residue modulo p, there are infinitely many twists whose analytic rank is zero. Also, for a certain hyperelliptic curve over the 11-th cyclotomic field whose Jacobian has complex multiplication, there are infinitely many twists whose analytic rank is zero.