Modular curve

2025-05-25T17:50:14Z

2A02:A459:8FB6:0:844E:3072:E9D7:7446: /* Relation with the Monster group */ Fixed sentence.

{{Short description|Algebraic variety}}
In [[number theory]] and [[algebraic geometry]], a '''modular curve''' ''Y''(Γ) is a [[Riemann surface]], or the corresponding [[algebraic curve]], constructed as a [[Quotient by a group action|quotient]] of the complex [[upper half-plane]] '''H''' by the [[Group action (mathematics)|action]] of a [[congruence subgroup]] Γ of the [[modular group]] of integral 2×2 matrices SL(2, '''Z'''). The term modular curve can also be used to refer to the '''compactified modular curves''' ''X''(Γ) which are [[compactification (mathematics)|compactification]]s obtained by adding finitely many points (called the '''cusps of Γ''') to this quotient (via an action on the '''extended complex upper-half plane'''). The points of a modular curve [[moduli problem|parametrize]] isomorphism classes of [[elliptic curve]]s, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to [[complex number]]s, and, moreover, prove that modular curves are [[field of definition|defined]] either over the field of [[rational number]]s '''Q''' or a [[cyclotomic field]] '''Q'''(ζ''n''). The latter fact and its generalizations are of fundamental importance in number theory.

== Analytic definition ==
The modular group SL(2, '''Z''') acts on the upper half-plane by [[fractional linear transformation]]s. The analytic definition of a modular curve involves a choice of a congruence subgroup Γ of SL(2, '''Z'''), i.e. a subgroup containing the [[principal congruence subgroup|principal congruence subgroup of level ''N'']] for some positive integer ''N'', which is defined to be

:<math>\Gamma(N)=\left\{
\begin{pmatrix}
a & b\\
c & d\\
\end{pmatrix} : \ a \equiv d \equiv 1 \mod N \text{ and } b, c \equiv0 \mod N \right\}.</math>

The minimal such ''N'' is called the '''level of Γ'''. A [[Complex manifold|complex structure]] can be put on the quotient Γ\'''H''' to obtain a [[noncompact]] Riemann surface called a '''modular curve''', and commonly denoted ''Y''(Γ).

=== Compactified modular curves ===
A common compactification of ''Y''(Γ) is obtained by adding finitely many points called the cusps of Γ. Specifically, this is done by considering the action of Γ on the '''extended complex upper-half plane''' '''H'''* = {{nowrap|'''H''' ∪ '''Q''' ∪ {∞}}}. We introduce a topology on '''H'''* by taking as a basis:
* any open subset of '''H''',
* for all ''r'' > 0, the set <math>\{\infty\}\cup\{\tau\in \mathbf{H} \mid\text{Im}(\tau)>r\}</math>
* for all [[coprime integers]] ''a'', ''c'' and all ''r'' > 0, the image of <math>\{\infty\}\cup\{\tau\in \mathbf{H} \mid\text{Im}(\tau)>r\}</math> under the action of
::<math>\begin{pmatrix}a & -m\\c & n\end{pmatrix}</math>
:where ''m'', ''n'' are integers such that ''an'' + ''cm'' = 1.

This turns '''H'''* into a topological space which is a subset of the [[Riemann sphere]] '''P'''1('''C'''). The group Γ acts on the subset {{nowrap|'''Q''' ∪ {∞}}}, breaking it up into finitely many [[Orbit (group theory)|orbits]] called the '''cusps of Γ'''. If Γ acts transitively on {{nowrap|'''Q''' ∪ {∞}}}, the space Γ\'''H'''* becomes the [[Alexandroff compactification]] of Γ\'''H'''. Once again, a complex structure can be put on the quotient Γ\'''H'''* turning it into a Riemann surface denoted ''X''(Γ) which is now [[Compact space|compact]]. This space is a compactification of ''Y''(Γ).<ref>{{citation|last=Serre|first= Jean-Pierre|title=Cours d'arithmétique|edition=2nd|series= Le Mathématicien|volume= 2|publisher= Presses Universitaires de France|year= 1977}}</ref>

== Examples ==
The most common examples are the curves ''X''(''N''), ''X''0(''N''), and ''X''1(''N'') associated with the subgroups Γ(''N''), Γ0(''N''), and Γ1(''N'').

The modular curve ''X''(5) has genus 0: it is the Riemann sphere with 12 cusps located at the vertices of a regular [[icosahedron]]. The covering ''X''(5) → ''X''(1) is realized by the action of the [[icosahedral symmetry|icosahedral group]] on the Riemann sphere. This group is a simple group of order 60 isomorphic to ''A''5 and PSL(2, 5).

The modular curve ''X''(7) is the [[Klein quartic]] of genus 3 with 24 cusps. It can be interpreted as a surface with three handles tiled by 24 heptagons, with a cusp at the center of each face. These tilings can be understood via [[dessins d'enfants]] and [[Belyi function]]s – the cusps are the points lying over ∞ (red dots), while the vertices and centers of the edges (black and white dots) are the points lying over 0 and 1. The Galois group of the covering ''X''(7) → ''X''(1) is a simple group of order 168 isomorphic to [[PSL(2,7)|PSL(2, 7)]].

There is an explicit classical model for ''X''0(''N''), the [[classical modular curve]]; this is sometimes called ''the'' modular curve. The definition of Γ(''N'') can be restated as follows: it is the subgroup of the modular group which is the kernel of the reduction [[Modular arithmetic|modulo]] ''N''. Then Γ0(''N'') is the larger subgroup of matrices which are upper triangular modulo ''N'':

:<math>\left \{ \begin{pmatrix} a & b \\ c & d\end{pmatrix} : \ c\equiv 0 \mod N \right \},</math>

and Γ1(''N'') is the intermediate group defined by:

:<math>\left \{ \begin{pmatrix} a & b \\ c & d\end{pmatrix} : \ a\equiv d\equiv 1\mod N, c\equiv 0 \mod N \right \}.</math>

These curves have a direct interpretation as [[moduli space]]s for [[elliptic curve]]s with ''[[level structure (algebraic geometry)|level structure]]'' and for this reason they play an important role in [[arithmetic geometry]]. The level ''N'' modular curve ''X''(''N'') is the moduli space for elliptic curves with a basis for the ''N''-[[torsion (algebra)|torsion]]. For ''X''0(''N'') and ''X''1(''N''), the level structure is, respectively, a cyclic subgroup of order ''N'' and a point of order ''N''. These curves have been studied in great detail, and in particular, it is known that ''X''0(''N'') can be defined over '''Q'''.

The equations defining modular curves are the best-known examples of [[modular equation]]s. The "best models" can be very different from those taken directly from [[elliptic function]] theory. [[Hecke operator]]s may be studied geometrically, as [[Correspondence (algebraic geometry)|correspondence]]s connecting pairs of modular curves.

Quotients of '''H''' that ''are'' compact do occur for [[Fuchsian group]]s Γ other than subgroups of the modular group; a class of them constructed from [[quaternion algebra]]s is also of interest in number theory.

== Genus ==
The covering ''X''(''N'') → ''X''(1) is Galois, with Galois group SL(2, ''N'')/{1, −1}, which is equal to PSL(2, ''N'') if ''N'' is prime. Applying the [[Riemann–Hurwitz formula]] and [[Gauss–Bonnet theorem]], one can calculate the genus of ''X''(''N''). For a [[prime number|prime]] level ''p'' ≥ 5,

:<math>-\pi\chi(X(p)) = |G|\cdot D,</math>

where χ = 2 − 2''g'' is the [[Euler characteristic]], |''G''| = (''p''+1)''p''(''p''−1)/2 is the order of the group PSL(2, ''p''), and ''D'' = π − π/2 − π/3 − π/''p'' is the [[defect (geometry)|angular defect]] of the spherical (2,3,''p'') triangle. This results in a formula

:<math>g = \tfrac{1}{24}(p+2)(p-3)(p-5).</math>

Thus ''X''(5) has genus 0, ''X''(7) has genus 3, and ''X''(11) has genus 26. For ''p'' = 2 or 3, one must additionally take into account the ramification, that is, the presence of order ''p'' elements in PSL(2, '''Z'''), and the fact that PSL(2, 2) has order 6, rather than 3. There is a more complicated formula for the genus of the modular curve ''X''(''N'') of any level ''N'' that involves divisors of ''N''.

===Genus zero===
In general a '''modular function field''' is a [[Function field of an algebraic variety|function field]] of a modular curve (or, occasionally, of some other [[moduli space]] that turns out to be an [[irreducible variety]]). [[Genus (mathematics)|Genus]] zero means such a function field has a single [[transcendental function]] as generator: for example the [[J-invariant|j-function]] generates the function field of ''X''(1) = PSL(2, '''Z''')\'''H'''*. The traditional name for such a generator, which is unique up to a [[Möbius transformation]] and can be appropriately normalized, is a '''Hauptmodul''' ('''main''' or '''principal modular function''', plural '''Hauptmoduln''').

The spaces ''X''1(''n'') have genus zero for ''n'' = 1, ..., 10 and ''n'' = 12. Since each of these curves is defined over '''Q''' and has a '''Q'''-rational point, it follows that there are infinitely many rational points on each such curve, and hence infinitely many elliptic curves defined over '''Q''' with ''n''-torsion for these values of ''n''. The converse statement, that only these values of ''n'' can occur, is [[Mazur's torsion theorem]].

=== ''X''0(''N'') of genus one ===

The modular curves <math>\textstyle X_0(N)</math> are of genus one if and only if <math>\textstyle N</math> equals one of the 12 values listed in the following table.<ref>{{cite book |editor-last1=Birch |editor-first1=Bryan |editor-last2=Kuyk |editor-first2=Willem |date=1975 |title=Modular functions of one variable IV |location=Berlin, Heidelberg |series=Lecture Notes in Mathematics |volume=476 |publisher=Springer-Verlag|page=79 |isbn=3-540-07392-2}}</ref> As [[elliptic curve]]s over <math>\mathbb{Q}</math>, they have minimal, integral Weierstrass models <math>y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6</math>. This is, <math>\textstyle a_j\in\mathbb{Z}</math> and the absolute value of the discriminant <math>\Delta</math> is minimal among all integral Weierstrass models for the same curve. The following table contains the unique ''reduced'', minimal, integral Weierstrass models, which means <math>\textstyle a_1, a_3\in\{0,1\}</math> and <math>\textstyle a_2\in\{-1,0,1\}</math>.<ref>{{cite journal |last1=Ligozat |first1=Gerard |date=1975 |title=Courbes modulaires de genre 1 |url=http://www.numdam.org/article/MSMF_1975__43__5_0.pdf |journal=Bulletin de la Société Mathématique de France |volume=43 |issue= |pages=44–45 |access-date=2022-11-06}}</ref> The last column of this table refers to the home page of the respective elliptic modular curve <math>\textstyle X_0(N)</math> on ''[[The L-functions and modular forms database (LMFDB)]]''.
{| class="wikitable"
|+ <math>X_0(N)</math> of genus 1
|-
! colspan="4"| <math>y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6</math>
|-
|<math>N</math> || <math>[a_1,a_2,a_3,a_4,a_6]</math> || <math>\Delta</math> || LMFDB
|-
| 11 || [0, -1, 1, -10, -20] || <math>\textstyle -11^5</math> || [https://www.lmfdb.org/EllipticCurve/Q/11a1/ link]
|-
| 14 || [1, 0, 1, 4, -6] || <math>\textstyle -2^6\cdot 7^3</math> || [https://www.lmfdb.org/EllipticCurve/Q/14a1/ link]
|-
| 15 || [1, 1, 1, -10, -10] || <math>\textstyle 3^4\cdot 5^4</math> || [https://www.lmfdb.org/EllipticCurve/Q/15a1/ link]
|-
| 17 || [1, -1, 1, -1, -14] || <math>\textstyle -17^4</math> || [https://www.lmfdb.org/EllipticCurve/Q/17a1/ link]
|-
| 19 || [0, 1, 1, -9, -15] || <math>\textstyle -19^3</math> || [https://www.lmfdb.org/EllipticCurve/Q/19a1/ link]
|-
| 20 || [0, 1, 0, 4, 4] || <math>\textstyle -2^8\cdot 5^2</math> || [https://www.lmfdb.org/EllipticCurve/Q/20a1/ link]
|-
| 21 || [1, 0, 0, -4, -1] || <math>\textstyle 3^4\cdot 7^2</math> || [https://www.lmfdb.org/EllipticCurve/Q/21a1/ link]
|-
| 24 || [0, -1, 0, -4, 4] || <math>\textstyle 2^8\cdot 3^2</math> || [https://www.lmfdb.org/EllipticCurve/Q/24a1/ link]
|-
| 27 || [0, 0, 1, 0, -7] || <math>\textstyle -3^9</math> || [https://www.lmfdb.org/EllipticCurve/Q/27a1/ link]
|-
| 32 || [0, 0, 0, 4, 0] || <math>\textstyle -2^{12}</math> || [https://www.lmfdb.org/EllipticCurve/Q/32a1/ link]
|-
| 36 || [0, 0, 0, 0, 1] || <math>\textstyle -2^4\cdot 3^3</math> || [https://www.lmfdb.org/EllipticCurve/Q/36a1/ link]
|-
| 49 || [1, -1, 0, -2, -1] || <math>\textstyle -7^3</math> || [https://www.lmfdb.org/EllipticCurve/Q/49a1/ link]
|}
== Relation with the Monster group ==
Modular curves of genus 0, which are quite rare, turned out to be of major importance in relation with the [[monstrous moonshine]] conjectures. The first several coefficients of the ''q''-expansions of their Hauptmoduln were computed already in the 19th century, but it came as a shock that the same large integers show up as dimensions of representations of the largest sporadic simple group Monster.

Another connection is that the modular curve corresponding to the [[normalizer]] Γ0(''p'')+ of [[modular group Gamma0|Γ0]](''p'') in SL(2, '''R''') has genus zero if and only if ''p'' is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71, and these are precisely [[supersingular prime (moonshine theory)|supersingular primes in moonshine theory]], i.e. the prime factors of the order of the [[monster group]]. The result about Γ0(''p'')+ is due to [[Jean-Pierre Serre]], [[Andrew Ogg]] and [[John G. Thompson]] in the 1970s, and the subsequent observation relating it to the monster group is due to Ogg, who wrote up a paper offering a bottle of [[Jack Daniel's]] whiskey to anyone who could explain this fact, which was a starting point for the theory of monstrous moonshine.<ref>{{harvtxt|Ogg|1974}}</ref>

The relation runs very deep and, as demonstrated by [[Richard Borcherds]], it also involves [[generalized Kac–Moody algebra]]s. Work in this area underlined the importance of [[modular function|modular ''functions'']] that are meromorphic and can have poles at the cusps, as opposed to [[modular form|modular ''forms'']], that are holomorphic everywhere, including the cusps, and had been the main objects of study for the better part of the 20th century.

== See also ==
*[[Manin–Drinfeld theorem]]
*[[Moduli stack of elliptic curves]]
*[[Modularity theorem]]
*[[Shimura variety]], a generalization of modular curves to higher dimensions

==References==
{{reflist}}

* Steven D. Galbraith - [https://www.math.auckland.ac.nz/~sgal018/thesis.pdf Equations For Modular Curves]
{{refbegin}}
*{{Citation
| last=Shimura
| first=Goro
| author-link=Goro Shimura
| title=Introduction to the arithmetic theory of automorphic functions
| publisher=[[Princeton University Press]]
| series=Publications of the Mathematical Society of Japan
| volume=11
| year=1994
| orig-year=1971
| mr=1291394
| isbn=978-0-691-08092-5
| postscript=, Kanô Memorial Lectures, '''1'''
}}
* {{citation | url = https://encyclopediaofmath.org/wiki/Modular_curve
| work = Encyclopaedia of Mathematics
| isbn = 1-4020-0609-8
| title = Modular curve
| author-link2 = A.N. Parshin
| first1 = A.A. | last1 = Panchishkin
| first2 = A.N. | last2 = Parshin
}}
*{{citation | last = Ogg | first = Andrew P. | author-link = Andrew Ogg |year= 1974 |chapter = Automorphismes de courbes modulaires | title=Seminaire Delange-Pisot-Poitou. Theorie des nombres, tome 16, no. 1 (1974–1975), exp. no. 7 |chapter-url=http://archive.numdam.org/ARCHIVE/SDPP/SDPP_1974-1975__16_1/SDPP_1974-1975__16_1_A4_0/SDPP_1974-1975__16_1_A4_0.pdf |mr =0417184 | language=fr}}
{{refend}}

[[Category:Algebraic curves]]
[[Category:Modular forms]]
[[Category:Riemann surfaces]]

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