Modular form: Difference between revisions

Revision as of 22:36, 9 February 2025

Revision as of 22:39, 9 February 2025

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In [[mathematics]], a '''modular form''' is a (complex) [[analyticholomorphic function]] on the [[Upper half-plane#Complex plane|complex upper half-plane]], <math>\,\mathcal{H}\,</math>, that roughly satisfies a [[functional equation]] with respect to the [[Group action (mathematics)|group action]] of the [[modular group]] and a growth condition. The theory of modular forms has origins in [[complex analysis]], with important connections with [[number theory]]. Modular forms also appear in other areas, such as [[algebraic topology]], [[sphere packing]], and [[string theory]].

Modular form theory is a special case of the more general theory of [[automorphic form]]s, which are functions defined on [[Lie group]]s that transform nicely with respect to the action of certain [[discrete subgroup]]s, generalizing the example of the modular group <math>\mathrm{SL}_2(\mathbb Z) \subset \mathrm{SL}_2(\mathbb R)</math>. Every modular form is attached to a [[Galois representation]].<ref name=":0">{{Cite news |last=Van Wyk |first=Gerhard |date=July 2023 |title=Elliptic Curves Yield Their Secrets in a New Number System |work=Quanta |url=https://www.quantamagazine.org/elliptic-curves-yield-their-secrets-in-a-new-number-system-20230706/?mc_cid=e612def96e&mc_eid=506130a407}}</ref>