mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a Fricke involution is the

involution Involution may refer to: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiati ...

of the

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 the complex upper half-plane H by the action of a congruence subgroup Γ of the modular g ...

''X''₀(''N'') given by τ → –1/''N''τ. It is named after Robert Fricke. The Fricke involution also acts on other objects associated with the modular curve, such as spaces of

modular form In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...

s and the Jacobian ''J''₀(''N'') of the modular curve. The quotient of ''X''₀(''N'') by the Fricke involution is a curve called ''X''₀⁺(''N''), and for ''N'' prime this has genus zero only for a finite list of primes, called supersingular primes, which are the primes that divide the order of the

Monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group; it has order : : = 2463205976112133171923293 ...

References

* Modular forms {{numtheory-stub

See also

References