Modular form: Difference between revisions

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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>.

The term "modular form", as a systematic description, is usually attributed to Hecke.

Each 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>

== Definition ==

In general,<ref>{{Cite web|last=Lan|first=Kai-Wen|title=Cohomology of Automorphic Bundles|url=http://www-users.math.umn.edu/~kwlan/articles/iccm-2016.pdf|url-status=live|archive-url=https://web.archive.org/web/20200801235440/http://www-users.math.umn.edu/~kwlan/articles/iccm-2016.pdf|archive-date=1 August 2020}}</ref> given a subgroup <math>\Gamma \subset \text{SL}_2(\mathbb{Z})</math> of [[finite index]], called an [[arithmetic group]], a '''modular form of level <math>\Gamma</math> and weight <math>k</math>''' is a holomorphic function <math>f:\mathcal{H} \to \mathbb{C}</math> from the [[upper half-plane]] such that two conditions are satisfied:

* Automorphy condition: For any <math>\gamma \in \Gamma</math> there is the equality<ref group="note">Some authors use different conventions, allowing an additional constant depending only on <math>\gamma</math>, see e.g. {{Cite web |title=DLMF: §23.15 Definitions ‣ Modular Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions |url=https://dlmf.nist.gov/23.15#E5 |access-date=2023-07-07 |website=dlmf.nist.gov}}</ref> <math>f(\gamma(z)) = (cz + d)^k f(z)</math>

* Growth condition: For any <math>\gamma \in \text{SL}_2(\mathbb{Z})</math> the function <math>(cz + d)^{-k} f(\gamma(z))</math> is bounded for <math>\text{im}(z) \to \infty</math>

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There are a number of other usages of the term "modular function", apart from this classical one; for example, in the theory of [[Haar measure]]s, it is a function {{math|Δ(''g'')}} determined by the conjugation action.

'''[[Maass forms]]''' are [[Analytic function|real-analytic]] [[eigenfunction]]s of the [[Laplacian]] but need not be [[Holomorphic function|holomorphic]]. The holomorphic parts of certain weak Maass wave forms turn out to be essentially Ramanujan's [[mock theta function]]s. Groups which are not subgroups of {{math|SL(2, '''Z''')}} can be considered.

'''[[Hilbert modular form]]s''' are functions in ''n'' variables, each a complex number in the upper half-plane, satisfying a modular relation for 2&times;2 matrices with entries in a [[totally real number field]].

'''[[Siegel modular form]]s''' are associated to larger [[symplectic group]]s in the same way in which classical modular forms are associated to {{math|SL(2, '''R''')}}; in other words, they are related to [[abelian variety|abelian varieties]] in the same sense that classical modular forms (which are sometimes called ''elliptic modular forms'' to emphasize the point) are related to elliptic curves.

'''[[Jacobi form]]s''' are a mixture of modular forms and elliptic functions. Examples of such functions are very classical - the Jacobi theta functions and the Fourier coefficients of Siegel modular forms of genus two - but it is a relatively recent observation that the Jacobi forms have an arithmetic theory very analogous to the usual theory of modular forms.

'''[[Automorphic form]]s''' extend the notion of modular forms to general [[Lie group]]s.

'''[[Modular integral]]s'''integrals of weight {{mvar|k}} are meromorphic functions on the upper half plane of moderate growth at infinity which ''fail to be modular of weight {{mvar|k}}'' by a rational function.

'''[[Automorphic factor]]s''' are functions of the form <math>\varepsilon(a,b,c,d) (cz+d)^k</math> which are used to generalise the modularity relation defining modular forms, so that

:<math>f\left(\frac{az+b}{cz+d}\right) = \varepsilon(a,b,c,d) (cz+d)^k f(z).</math>

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* In the 1960s, as the needs of number theory and the formulation of the [[modularity theorem]] in particular made it clear that modular forms are deeply implicated.

Taniyama and Shimura identified a 1-to-1 matching between certain modular forms and elliptic curves. Langlands built on this idea in the construction of his expansive Langlands program, which has become one of the most far-reaching and consequential research programs in math.

The term "modular form", as a systematic description, is usually attributed to Hecke.

In 1994 [[Andrew Wiles]] used modular forms to prove [[Fermat’s Last Theorem]]. In 2001 all elliptic curves were proven to be modular over the rational numbers. In 2013 elliptic curves were proven to be modular over real [[quadratic fields]]. In 2023 elliptic curves were proven to be modular over about half of imaginary [[quadratic fields]], including fields formed by combining the [[rational numbers]] with the [[square root]] of integers down to −5. <ref name=":0" />

== Notes ==