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In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation To meet Wikipedias quality standards, this article or section may require cleanup. ...
In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ...
- Φn(x, y)=0,
where for the j-invariant j(τ), Real part of the j-invariant as a function of the nome q on the unit disk In mathematics, Kleins j-invariant, regarded as a function of a complex variable Ï„, is a modular function defined on the upper half-plane of complex numbers. ...
- x=j(n τ), y=j(τ)
is a point on the curve. The curve is sometimes called X0(n), though often that is used for the abstract algebraic curve for which there exist various models. A related object is the classical modular polynomial, a polynomial in one variable defined as Φn(x, x). In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ...
Geometry of the modular curve
 Knot at infinity of X 0(11) The classical modular curve, which we will call X0(n), is of degree greater than or equal to 2n when n>1, with equality if and only if n is a prime. The polynomial Φn has integer coefficients, and hence is defined over every field. However, the coefficients are sufficiently large that computational work with the curve can be difficult. As a polynomial in x with coefficients in Z[y], it has degree ψ(n), where ψ is the Dedekind psi function. Since Φn(x, y) = Φn(y, x), X0(n) is symmetrical around the line y=x, and has singular points at the repeated roots of the classical modular polynomial, where it crosses itself in the complex plane. These are not the only singularities, and in particular when n>2, there are two singularites at infinity, where x=0, y=∞ and x=∞, y=0, which have only one branch and hence have a knot invariant which is a true knot, and not just a link. Image File history File links Modknot11. ...
Image File history File links Modknot11. ...
In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by where the product is taken over all primes p dividing n. ...
Parametrization of the modular curve When n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, or 25, X0(n) has genus zero, and hence can be parametrized by rational functions. The simplest nontrivial example is X0(2), where if In mathematics, the geometric genus in algebraic geometry is a basic birational invariant pg of algebraic varieties, defined for non-singular complex projective varieties (and more generally for complex manifolds) as the Hodge number hn,0 (equal to h0,n by Serre duality). ...
 is (up to the constant term) the McKay-Thompson series for the class 2B of the Monster, and η is the Dedekind eta function, then In mathematics, monstrous moonshine is a term devised by John Conway and Simon P. Norton in 1979, used to describe the (then totally unexpected) connection between the monster group M and modular functions (particularly, the j function). ...
In mathematics, the Monster group M is a group of order 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808017424794512875886459904961710757005754368000000000 ≈ 8 · 1053. ...
The Dedekind eta function is a function defined on the upper half plane of complex numbers whose imaginary part is positive. ...
 parametrizes X0(2) in terms of rational functions of j2. It is not necessary to actually compute j2 to use this parametrization; it can be taken as an arbitrary parameter.
Mappings A curve C over the rationals Q such that there exists a surjective morphism from X0(n) to C for some n, given by a rational map with integer coefficients - φ:X0(n) → C,
is a modular curve. The famous modularity theorem tells us that all elliptic curves over Q are modular. In mathematics, a modular curve is a Riemann surface, or corresponding algebraic curve, constructed as HΓ where H is the upper half-plane in the complex numbers, and Γ is a Fuchsian group acting on H, with Γ a subgroup of the modular group of integral 2×2 matrices. ...
The Taniyama–Shimura theorem (also called the modularity theorem) establishes an important connection between elliptic curves, which are objects from algebraic geometry, and modular forms, which are certain periodic holomorphic functions investigated in number theory. ...
In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ...
Mappings also arise in connection with X0(n) since points on it correspond to n-isogenous pairs of elliptic curves. Two elliptic curves are isogenous if there is a morphism of varieties (defined by a rational map) between the curves which is also a group homomorphism, respecting the group law on the elliptic curves, and hence which sends the point at infinity (serving as the identity of the group law) to the point at infinity. The isogenies with cyclic kernel of degree n, the cyclic isogenies, correspond to points on X0(n).
When X0(n) has genus one, it will itself be isomorphic to an elliptic curve, which will have the same j-invariant. For instance, X0(11) has j-invariant -122023936/161051 = - 21211-5313, and is isomorphic to the curve y2+y = x3-x2-10x-20. If we substitute this value of j for y in X0(5), we obtain two rational roots and a factor of degree four. The two rational roots correspond to isomorphism classes of curves with rational coefficients which are 5-isogenous to the above curve, but not isomorphic, having a different function field. Real part of the j-invariant as a function of the nome q on the unit disk In mathematics, Kleins j-invariant, regarded as a function of a complex variable Ï„, is a modular function defined on the upper half-plane of complex numbers. ...
Specifically, we have the six rational points x=-122023936/161051, y=-4096/11, x=-122023936/161051, y=-52893159101157376/11, and x=-4096/11, y=-52893159101157376/11, plus the three points exchanging x and y, all on X0(5), corresponding to the six isogenies between these three curves. If in the curve y2+y = x3-x2-10x-20 isomorphic to X0(11) we substitute
 and  and factor, we get an extraneous factor of a rational function of x, and the curve y^2+y=x^3-x^2, with j-invariant -4096/11. Hence both curves are modular of level 11, having mappings from X0(11).
By a theorem of Henri Carayol, if an elliptic curve E is modular then its conductor, an isogeny invariant described originally in terms of cohomology, is the smallest integer n such that there exists a rational mapping φ:X0(n) → E. Since we now know all elliptic curves over Q are modular, we also know that the conductor is simply the level n of its minimal modular parametrization. In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ...
Galois theory of the modular curve The Galois theory of the modular curve was investigated by Erich Hecke. Considered as a polynomial in x with coefficients in Z[y], the modular equation Φ0(n) is a polynomial of degree ψ(n) in x, whose roots generate a Galois extension of Q(y). In the case of X0(p) with p prime, where the characteristic of the field is not p, the Galois group of Erich Hecke (September 20, 1887 – February 13, 1947) was a German mathematician. ...
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions (described below); one also says that the extension is Galois. ...
In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...
In mathematics, a Galois group is a group associated with a certain type of field extension. ...
- Q(x, y)/Q(y)
is PGL2(p), the projective general linear group of linear fractional transformations of the projective line of the field of p elements, which has p+1 points, the degree of X0(p). The projective linear group of a vector space V over a field F is the quotient group PGL(V) = GL(V)/Z(V) where GL(V) is the general linear group on V and Z(V) is the group of all nonzero scalar transformations of V. The projective special linear...
In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ...
In mathematics, a projective line is a one-dimensional projective space. ...
This extension contains an algebraic extension
 of Q. If we extend the field of constants to be F, we now have an extension with Galois group PSL2(p), the projective special linear group of the field with p elements, which is a finite simple group. By specializing y to a specific field element, we can, outside of a thin set, obtain an infinity of examples of fields with Galois group PSL2(p) over F, and PGL2(p) over Q. The projective linear group of a vector space V over a field F is the quotient group PGL(V) = GL(V)/Z(V) where GL(V) is the general linear group on V and Z(V) is the group of all nonzero scalar transformations of V. The projective special linear...
When n is not a prime, the Galois groups can be analyzed in terms of the factors of n as a wreath product. In mathematics, the wreath product of group theory is a specialized product of two groups, based on a semidirect product. ...
See also In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ...
Real part of the j-invariant as a function of the nome q on the unit disk In mathematics, Kleins j-invariant, regarded as a function of a complex variable Ï„, is a modular function defined on the upper half-plane of complex numbers. ...
In mathematics, a modular curve is a Riemann surface, or corresponding algebraic curve, constructed as HΓ where H is the upper half-plane in the complex numbers, and Γ is a Fuchsian group acting on H, with Γ a subgroup of the modular group of integral 2×2 matrices. ...
In mathematics, modular functions are certain kinds of mathematical functions mapping complex numbers to complex numbers. ...
External links - [1]Genus of X0(n)
- [2]Coefficients of X0(n)
References - Erich Hecke, Die eindeutige Bestimmung der Modulfunktionen q-ter Stufe durch algebraische Eigenschaften, Math. Ann. 111 (1935), 293-301, reprinted in Mathematische Werke, third edition, Vandenhoeck & Ruprecht, Göttingen, 1983, 568-576[3]
- Anthony Knapp, Elliptic Curves, Princeton, 1992
- Serge Lang, Elliptic Functions, Addison-Wesley, 1973
- Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1972
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