Everything about Modular form
Analytic function on the upper half-plane with a certain behavior under the modular group
In number theory and complex analysis, a modular form is a type of function of a complex number variable that possesses a high degree of symmetry, of a certain kind. Similarly to a periodic function of a real variable, a modular form repeats or transforms in a certain way when its argument is subjected to a particular transformation. Unlike an ordinary periodic function, its symmetries include transformations such as replacing a complex number z by −1/z, and the transformation law is not an exact symmetry of the function, but more like the transformation law of a quasiperiodic function: the function picks up an additional factor, depending on the transformation. Modular forms serve as an important bridge between complex analysis, number theory, and geometry. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Modular formA 5-minute orientation
- Skim a recent paper: Blockchain-Enabled IoT-BIM Platform for Supply Chain Management in Modular ConstructionInfluential work since 2022
- Pick up a classic book: The algebraic theory of modular systemsThe most-reprinted book on the subject
- Read the landmark paper: Rethinking the Modular Nation Form: Toward a Sociohistorical Conception of NationalismCited 93 times
- Go sideways: Maass wave formA closely related topic
- Go sideways: Hecke operatorA closely related topic
📖 Definitions
Noun A complex analytic function satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition.
🏆 Landmark papers (most cited)
Rethinking the Modular Nation Form: Toward a Sociohistorical Conception of Nationalism
Modular exaptation: A missing link in the synthesis of artificial form
Structure−Function Studies of Modular Aromatics That Form Molecular Organogels
Modular growth and form of corals: a matter of metamers?
LINEAR RELATIONS BETWEEN MODULAR FORM COEFFICIENTS AND NON-ORDINARY PRIMES
On a modular form of Zaremba’s conjecture
Subconvexity for modular form L-functions in the t aspect
A “strange” vector-valued quantum modular form
🆕 Influential since 2022
Blockchain-Enabled IoT-BIM Platform for Supply Chain Management in Modular Construction
Magnetically tightened form-stable phase change materials with modular assembly and geometric conformality features
The three-loop equal-mass banana integral in ε-factorised form with meromorphic modular forms
Design and optimization of free-form surfaces for modular concrete 3D printing
FireAntV3: A Modular Self-Reconfigurable Robot Toward Free-Form Self-Assembly Using Attach-Anywhere Continuous Docks
📚 Books worth reading
The algebraic theory of modular systemsF. S. Macaulay · 1916
Compactification of Siegel moduli schemesChing-Li Chai · 1984
Geometric Modular Forms and Elliptic CurvesHaruzo Hida · 2000
Modular forms on schiermonnikoogB. Edixhoven, Gerard van der Geer · 2008
Invitation to the mathematics of Fermat-WilesYves Hellegouarch · 2001
Some applications of modular formsPeter Sarnak · 1990
Galois theory and modular formsK. Hashimoto, Katsuya Miyake · 2003
Modular Forms and FunctionsRobert A. Rankin · 2008
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
🔗 Related topics
- Maass wave form In mathematics, Maass forms or Maass wave forms are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the…
- Hecke operator In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by Erich Hecke (1937a,1937b), is a certain kind of "averaging"…
- Cusp form In number theory, a cusp form is a particular kind of modular form with zero constant coefficient in its Fourier series expansion. A cusp form is distinguished…
- Modular curve In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of…
- Picard modular surface In mathematics, a Picard modular surface is a complex algebraic surface obtained as an arithmetic quotient of the two-dimensional complex unit ball by…
- Dedekind eta function In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane…
- Siegel modular form In mathematics, Siegel modular forms are a type of automorphic form that generalize conventional elliptic modular forms, which are closely related to elliptic…
- Automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G to the complex numbers (or complex vector…