Hurwitz bound on automorphism groups of curves

E262119

mathematical theorem result in algebraic geometry result in the theory of Riemann surfaces

The Hurwitz bound on automorphism groups of curves is a classical result in algebraic geometry stating that a compact Riemann surface of genus at least 2 has at most 84(g − 1) automorphisms.

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All labels observed (5)

Label	Occurrences
Hurwitz bound	4
Hurwitz's automorphisms theorem	2
Hurwitz automorphism bound	1
Hurwitz bound on automorphism groups of curves canonical	1
Hurwitz’s automorphism theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T2394218 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Hurwitz bound on automorphism groups of curves
Context triple: [Riemann–Hurwitz formula, relatedTo, Hurwitz bound on automorphism groups of curves]

A. Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
B. Hasse bound for elliptic curves
The Hasse bound for elliptic curves is a fundamental result in number theory that gives tight limits on how far the number of points on an elliptic curve over a finite field can deviate from the size of the field plus one.
C. Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
D. Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
E. Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Hurwitz bound on automorphism groups of curves
Target entity description: The Hurwitz bound on automorphism groups of curves is a classical result in algebraic geometry stating that a compact Riemann surface of genus at least 2 has at most 84(g − 1) automorphisms.

A. Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
B. Hasse bound for elliptic curves
The Hasse bound for elliptic curves is a fundamental result in number theory that gives tight limits on how far the number of points on an elliptic curve over a finite field can deviate from the size of the field plus one.
C. Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
D. Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
E. Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in algebraic geometry ⓘ result in the theory of Riemann surfaces ⓘ
appliesTo	compact Riemann surfaces ⓘ curves of genus at least 2 ⓘ smooth projective algebraic curves over the complex numbers ⓘ
assumption	genus g is at least 2 ⓘ
attainedBy	Hurwitz curves ⓘ linked to: Klein quartic Klein quartic curve of genus 3 ⓘ
concerns	finite groups acting as algebraic automorphisms of curves ⓘ finite groups acting as conformal automorphisms of Riemann surfaces ⓘ
context	automorphism groups of algebraic curves ⓘ automorphism groups of compact Riemann surfaces ⓘ
dependsOn	genus of the curve ⓘ
doesNotApplyTo	Riemann surfaces of genus 0 ⓘ Riemann surfaces of genus 1 ⓘ
equivalentFormulation	If G is a finite group of automorphisms of a compact Riemann surface of genus g ≥ 2, then \|G\| ≤ 84(g − 1) ⓘ
example	The Klein quartic has 168 = 84(3 − 1) automorphisms ⓘ linked to: Klein quartic
field	algebraic geometry ⓘ complex analysis ⓘ geometric group theory ⓘ
generalizationOf	bounds on automorphism groups of algebraic curves over fields of characteristic zero ⓘ
givesUpperBoundFor	order of the automorphism group of a compact Riemann surface ⓘ order of the automorphism group of a smooth projective algebraic curve of genus g ≥ 2 over C ⓘ
hasGeneralization	Arakelov-type inequalities for families of curves ⓘ bounds on automorphism groups of curves in positive characteristic ⓘ
holdsOver	complex numbers ⓘ
implies	Automorphism groups of curves of genus g ≥ 2 are finite ⓘ
mathematicsSubjectClassification	14H37 ⓘ 30F10 ⓘ
namedAfter	Adolf Hurwitz ⓘ
proofUses	Riemann–Hurwitz formula ⓘ orbifold Euler characteristic ⓘ ramified coverings of the Riemann sphere ⓘ
relatedConcept	Hurwitz curve ⓘ linked to: Klein quartic Hurwitz group ⓘ triangle group (2,3,7) ⓘ linked to: (2,3,7) triangle group
relatedResult	Accola–Maclachlan bound ⓘ Wiman bound ⓘ
sharpness	The bound 84(g − 1) is attained for infinitely many genera ⓘ
statement	A compact Riemann surface of genus g ≥ 2 has at most 84(g − 1) automorphisms ⓘ
typicalNotation	\|Aut(X)\| ≤ 84(g(X) − 1) ⓘ
upperBoundExpression	84(g − 1) ⓘ
usedIn	classification of Riemann surfaces with large automorphism groups ⓘ construction of curves with many symmetries ⓘ study of moduli spaces of curves ⓘ
yearProved	1893 ⓘ

How these facts were elicited