modular form

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- 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

modular form

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Ways to think of a modular form:
- Functions (automorphic forms) on $\operatorname{GL}_2$
- Functions on ${\mathbb{H}}$
- Sections of a line bundle over a moduli of curves
Related to q series
- Many classical modular forms are generating function for integer partitions in interesting ways

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L functions

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Examples

Ramanujan delta: $\Delta(q)=q \prod_{n \geq 1}\left(1-q^{n}\right)^{24}$, a holomorphic cusp form of weight 12 and level 1.

Notes

Modularity theorem: If $E \in \mathrm{Ell} _{{\mathbf{Q}}}$, then $E$ admits a rational parameterization. Proved by Wiles et al.

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Relation to Weil Conjectures: attachments/Pasted%20image%2020220204094635.png

Links to this page

unresolved links output

Hecke operator in Quick_Notes/2021-04-15, -Unsorted/modular form

automorphic forms in Quick_Notes/2021-09-14, -Unsorted/modular form, -Unsorted/special values of L functions

level of a modular form in Unsorted/modular form

weight of a modular form in Quick_Notes/2021-04-18, -Quick_Notes/2021-08-05 quick_note, -Unsorted/chromatic homotopy theory, -Unsorted/modular form
stable homotopy

modular form
j invariant

modular form
chromatic homotopy theory

The stack $M_{\ell}$ admits a compactification ${ \mathcal{M}_{\mathrm{ell}} }$ whose $R$ points are generalized elliptic curves. The space of integral modular forms of weight $k$ is defined to be the space of sections \begin{align*} H ^ { 0 } \left( \mkern 1.5mu\overline{\mkern-1.5mu{ \mathcal{M}_{\mathrm{ell}} }\mkern-1.5mu}\mkern 1.5mu; \omega{ {}^{ \scriptscriptstyle\otimes_{k}^{n} } } \right) \end{align*} Motivated by the definition of integral modular forms and the descent spectral sequence in the case of $U = M_\ell$ , the spectrum $\mathrm{TMF} $ (see TMF) is defined to be its global sections: \begin{align*} \mathrm{TMF} \coloneqq{\mathcal{O}}^ { {\mathsf{Top}}} \left( \mkern 1.5mu\overline{\mkern-1.5mu{ \mathcal{M}_{\mathrm{ell}} }\mkern-1.5mu}\mkern 1.5mu \right) \end{align*}
automorphic form

modular form
Langlands

modular form
L function

L function of a modular form
Fermat's Last Theorem

modularity
Birch and Swinnerton-Dyer conjecture

- Tags: - #AG/elliptic-curves #open/conjectures - Refs: - Swinnerton-Dyer, Notes on elliptic curves. II - J. T. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. - Links: - elliptic curve - modularity - L function - motivic L function - Application: congruent number problem - Mazur’s theorem - Faltings theorem
AWS 2022: Automorphic Forms Beyond $\operatorname{GL}_2$

modular form

#arithmetic-geometry/Langlands #NT/analytic #resources/course-notes