Concept

Modular curve

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Modular group

In mathematics, the modular group is the projective special linear group of 2 × 2 matrices with integer coefficients and determinant 1. The matrices A and −A are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic. The modular group Γ is the group of linear fractional transformations of the upper half of the complex plane, which have the form where a, b, c, d are integers, and ad − bc = 1.

J-invariant

In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for SL(2, Z) defined on the upper half-plane of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such that Rational functions of j are modular, and in fact give all modular functions. Classically, the j-invariant was studied as a parameterization of elliptic curves over C, but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine).

Fuchsian group

In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces.

Moduli stack of elliptic curves

In mathematics, the moduli stack of elliptic curves, denoted as or , is an algebraic stack over classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves . In particular its points with values in some field correspond to elliptic curves over the field, and more generally morphisms from a scheme to it correspond to elliptic curves over . The construction of this space spans over a century because of the various generalizations of elliptic curves as the field has developed.

Hecke operator

In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations. used Hecke operators on modular forms in a paper on the special cusp form of Ramanujan, ahead of the general theory given by .

Klein quartic

In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 168 × 2 = 336 automorphisms if orientation may be reversed. As such, the Klein quartic is the Hurwitz surface of lowest possible genus; see Hurwitz's automorphisms theorem. Its (orientation-preserving) automorphism group is isomorphic to PSL(2, 7), the second-smallest non-abelian simple group after the alternating group A5.

Congruence subgroup

In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the off-diagonal entries are even. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer.

Shimura variety

In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties are not algebraic varieties but are families of algebraic varieties. Shimura curves are the one-dimensional Shimura varieties. Hilbert modular surfaces and Siegel modular varieties are among the best known classes of Shimura varieties.

Belyi's theorem

In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve C, defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only. This is a result of G. V. Belyi from 1979. At the time it was considered surprising, and it spurred Grothendieck to develop his theory of dessins d'enfant, which describes non-singular algebraic curves over the algebraic numbers using combinatorial data.

Monstrous moonshine

In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the j function. The initial numerical observation was made by John McKay in 1978, and the phrase was coined by John Conway and Simon P. Norton in 1979. The monstrous moonshine is now known to be underlain by a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, which has the monster group as its group of symmetries.