- Tags: - #arithmetic-geometry/Langlands #NT/analytic - Refs: - Course notes: https://www.math.columbia.edu/~phlee/CourseNotes/ModularForms.pdf#page=1 #resources/course-notes - Apostol, Modular Functions and Dirichlet Series in Number Theory. - See also previous book in series, Introduction to Analytic Number Theory - Harvard summer tutorial: https://people.math.harvard.edu/~smarks/mod-forms-tutorial/index.html - Links: - Hecke operator - Eisenstein series - Weight of a modular form - modular curve - Siegel modular forms - Hecke operator - automorphic form

Another common setup: can realize certain affine modular curves \(\dcoset{\Gamma}{{\mathbb{H}}}{1}\) as \(\dcoset{\operatorname{GL}_2({\mathbf{Q}})}{\operatorname{GL}_2({\mathbf{A}}_{\mathbf{Q}})}{K}\) where \(K = K^\infty \times K_\infty\) and \(K^\infty \leq \operatorname{GL}_2({\mathbf{A}}_{\mathbf{Q}}^\infty)\) is a compact open and \(K_\infty \coloneqq{\mathbf{R}}_{>0}\times {\operatorname{SO}}_2({\mathbf{R}})\).