Automorphisms of Riemann surfaces of genus g > or = 2
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TextPublication details: 2012Description: 45pSubject(s): Mathematics | Automorphisms | Riemann SurfacesOnline resources: Click here to access online Dissertation note: 2012M.ScHBNI Abstract: It is shown that automorphism group of any Riemann surface X of genus g > or = 2 is finite. Also given a bound to the cardinality of the automorphism group, depending on the genus, speci fically Aut(X) < or = 84(g-1). This bound will be obtained by applying Hurwitz formula to the natural holomorphic map from a Riemann surface to it's quotient under action of the finite group Aut(X). The finiteness is proved by considering a homomorphism from Aut(X) to the permutation group of a finite set and showing that the kernel is finite. The finite set under consideration is the set of Weierstass points. p is a Weierstass point, if the set of integers n, such that there is no f {element of} M(X) whose only pole is p with order n, is not {1, ... g}. All these are explained in Chapter 4. Riemann-Roch Theorem is heavily used which is proved in Chapter 3. Proof of Riemann-Roch Theorem requires existence of non-constant meromorphic functions on a Riemann surface, which is proved in Chapter 2. Basics are dealt with in Chapter 1.
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2012
M.Sc
HBNI
It is shown that automorphism group of any Riemann surface X of genus g > or = 2 is finite. Also given a bound to the cardinality of the automorphism group, depending on the genus, speci fically Aut(X) < or = 84(g-1). This bound will be obtained by applying Hurwitz formula to the natural holomorphic map from a Riemann surface to it's quotient under action of the finite group Aut(X). The finiteness is proved by considering a homomorphism from Aut(X) to the permutation group of a finite set and showing that the kernel is finite. The finite set under consideration is the set of Weierstass points. p is a Weierstass point, if the set of integers n, such that there is no f {element of} M(X) whose only pole is p with order n, is not {1, ... g}. All these are explained in Chapter 4. Riemann-Roch Theorem is heavily used which is proved in Chapter 3. Proof of Riemann-Roch Theorem requires existence of non-constant meromorphic functions on a Riemann surface, which is proved in Chapter 2. Basics are dealt with in Chapter 1.
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