Modular symbol




In mathematics, modular symbols, introduced independently by Bryan John Birch and by Manin (1972), span a vector space closely related to a space of modular forms, on which the action of the Hecke algebra can be described explicitly. This makes them useful for computing with spaces of modular forms.



Definition


The abelian group of (universal weight 2) modular symbols is spanned by symbols {α,β} for α, β in the rational projective line Q∪ ∞ subject to the relations


  • {α,β} + {β,γ} = {α,γ}

Informally, {α,β} represents a homotopy class of paths from α to β in the upper half-plane.


The group GL2(Q) acts on the rational projective line, and this induces an action on the modular symbols.


There is a pairing between cusp forms f of weight 2 and modular symbols given by integrating the cusp form, or rather fdτ, along the path corresponding to the symbol.



References




  • Manin, Ju. I. (1972), "Parabolic points and zeta functions of modular curves", Math. USSR-Izv., 6: 19–64, doi:10.1070/IM1972v006n01ABEH001867, ISSN 0373-2436, MR 0314846.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  • Manin, Yuri Ivanovich (2009), "Lectures on modular symbols", Arithmetic geometry, Clay Math. Proc., 8, Providence, R.I.: American Mathematical Society, pp. 137–152, ISBN 978-0-8218-4476-2, MR 2498060


  • Cremona, J.E. (1997), Algorithms for modular elliptic curves (2nd ed.), Cambridge: Cambridge University Press, ISBN 0-521-59820-6, Zbl 0872.14041




Comments

Post a Comment

Popular posts from this blog

Information security

Image
This article is part of a series on Information security Related security categories Internet security Cyberwarfare Computer security Mobile security Network security Threats Computer crime Vulnerability Eavesdropping Malware Spyware Ransomware Trojans Viruses Worms Rootkits Bootkits Keyloggers Screen scrapers Exploits Backdoors Logic bombs Payloads Denial of service Defenses Computer access control Application security Antivirus software Secure coding Secure by default Secure by design Secure operating systems Authentication Multi-factor authentication Authorization Data-centric security Encryption Firewall Intrusion detection system Mobile secure gateway Runtime application self-protection (RASP) v t e Information security , sometimes shortened to InfoSec , is the practice of preventing unauthorized access, use, disclosure, disruption, modification, inspection, recording or destruction of information. Th...
Read more

Lambak Kiri

Image
For Lambak Kiri public housing area, see Lambak Kiri Landless Indigenous Citizens' Housing Scheme. Village in Brunei-Muara, Brunei Lambak Kiri Village Location in Brunei Coordinates: 4°59′26″N 114°57′04″E  /  4.990472°N 114.951114°E  / 4.990472; 114.951114 Coordinates: 4°59′26″N 114°57′04″E  /  4.990472°N 114.951114°E  / 4.990472; 114.951114 Country Brunei District Brunei-Muara Mukim Berakas A Government  • Village head Hamdani Omar Postcode BB1214 Lambak Kiri is a populated place in Brunei-Muara District, Brunei. It is officially a village subdivision under Berakas A, a mukim in the district. Lambak Kiri may also refer to the greater area which includes the Lambak Kiri public housing area of the Landless Indigenous Citizens' Housing Scheme (Malay: Skim Tanah Kurnia Rakyat Jati ), although the latter is a separate subdivision. Contents 1 Name 2 Administration 3 See also 4 References Name ...
Read more

章鱼与海女图

Image
本条目 可参照英語維基百科相應條目来扩充 。 （2014/5/25） 若您熟悉来源语言和主题，请协助参考外语维基扩充条目。请勿直接提交机械翻译，也不要翻译不可靠、低品质内容。依版权协议，译文需在编辑摘要注明来源，或于讨论页顶部标记 {{Translated page}} 标签。 本条目 需要擴充。 （2013年8月15日） 请協助改善这篇條目，更進一步的信息可能會在討論頁或扩充请求中找到。请在擴充條目後將此模板移除。 《 章鱼与海女 》（日语： 蛸と海女 ／ たことあま   Tako to ama ）是一幅由日本畫家葛饰北斋创作的浮世绘，出自1814年（文化4年）出版的艳本《喜能会之故真通》（日语： 喜能会之故真通 ／ きのえのこまつ   Kinoe no Komatsu ）。畫中有兩隻章鱼以觸手攻擊一名採珠的海女，並進行擬似戀獸癖的交媾行為。作品尺寸为6½" × 8¾" （16.51 cm × 22.23 cm）。 [1] 在此之前的类似作品还有北尾重政的艳本《謡曲色番組》及勝川春潮的艳本《艳本千夜多女志》等，有被北斋作为参考的可能性。 《章鱼与海女图》有不同英文翻译，一般称The Dream of the Fisherman's Wife，又作Girl Diver and Octopi、Diver and Two Octopi，Richard Douglas Lane称其为 Girl Diver and Octopi [2] ，Mathi Forrer 称其为 Pearl Diver and Two Octopi [3] ，Danielle Talerico 称其为 Diver and Two Octopi 。 [4] 目录 1 作品介绍 1.1 画中对话 1.2 解释 2 评价 3 影响 3.1 模仿作品 4 另见 5 参考来源 6 外部連結 作品介绍 画中对话 原文 译文 大蛸： いつぞハいつぞハとねらいすましてゐたかいがあつて、けふといふけふ、とうとうとらまへたア。てもむつくりとしたいいぼぼだ。いもよりハなをこうぶつだ。サアサア、すつてすつてすいつくして...
Read more