308
X – The Riemann Surface of an Algebraic Function
Ultimately, the points of a Riemann surface of a field L of algebraic functions are, interchangeably, the places, the valuations rings or the valuations
of L. We go from a valuation ring o to a place by observing that the x ∈ o
that are not invertible in o form an ideal p of o and that the quotient o/p
is isomorphic to C ; for x ∈ o, x(P ) is then the class of x mod p, and set
x(P ) = ∞ if x /
∈ o. On the other hand, it can be shown that the only ideals
of o are the pairwise distinct powers p
n of p; for x ∈ o, v P (x) is then the
smallest n such that x ∈ p
n , and for x /
∈ o set v P (x) = −v P (x
−1 ) . In fact,
p is the set of the x ∈ L that vanish at the point P of the Riemann surface,
and p
n the set of x having a zero of order ≥ n at P .
The relations with the theory of algebraic curves also need to be mentioned.
The algebraic point of view was invented by Richard Dedekind and Heinrich Weber
15 about thirty years after Riemann. His rather obscure and vague
constructions, a fortiori the “ verbosity ” of his far less brilliant successors,
must have annoyed the crystal clear minds of these two algebraists. As
Dedekind had already provided a clear and in many respects final form of
the theory of algebraic number fields,
16 he naturally tried to apply similar
methods to algebraic functions fields of one variable by replacing Q by C(X).
About thirty years later, Hermann Weyl, Die Idee der Riemannschen Fl¨ ache,
introduced the first correct ideas about “ abstract ” 1-dimensional complex
manifolds into this question and used Dirichlet’s principle to prove a priori the existence of “ many ” meromorphic functions on Riemann surfaces.
This result, which is obvious for surfaces associated to algebraic functions,
requires even now a long proof in the general case. In the 1930s and 40s, in
particular thanks to Andr´ e Weil and Oscar Zarisky, what a n-dimensional algebraic variety over an arbitrary field is was beginning to be understood and
a purely algebraic mechanism was being set up. It was an improvement on the
doubtful geometric arguments of the Italian school (which nonetheless had
discovered results and introduced very important ideas since 1870). Starting with the notion of a place and ending with that of a complex analytic
manifold, Claude Chevalley published the first post-war modern presentation on algebraic functions in one variable. Reading it is still advisable. Serge
Lang’s book goes much further in about a hundred pages, which indicates
how concise the proofs are. . .
15 author of a Lehrbuch der Algebra where almost everything that was known in
algebra around 1900 can be found.
16 It has been substantially improved on and generalized, but without fundamentally changing its point of view other than by introducing the notion of valuation.
Reading his main articles that can be found in his complete works, is still a most
advisable exercise.
X – The Riemann Surface of an Algebraic Function
Ultimately, the points of a Riemann surface of a field L of algebraic functions are, interchangeably, the places, the valuations rings or the valuations
of L. We go from a valuation ring o to a place by observing that the x ∈ o
that are not invertible in o form an ideal p of o and that the quotient o/p
is isomorphic to C ; for x ∈ o, x(P ) is then the class of x mod p, and set
x(P ) = ∞ if x /
∈ o. On the other hand, it can be shown that the only ideals
of o are the pairwise distinct powers p
n of p; for x ∈ o, v P (x) is then the
smallest n such that x ∈ p
n , and for x /
∈ o set v P (x) = −v P (x
−1 ) . In fact,
p is the set of the x ∈ L that vanish at the point P of the Riemann surface,
and p
n the set of x having a zero of order ≥ n at P .
The relations with the theory of algebraic curves also need to be mentioned.
The algebraic point of view was invented by Richard Dedekind and Heinrich Weber
15 about thirty years after Riemann. His rather obscure and vague
constructions, a fortiori the “ verbosity ” of his far less brilliant successors,
must have annoyed the crystal clear minds of these two algebraists. As
Dedekind had already provided a clear and in many respects final form of
the theory of algebraic number fields,
16 he naturally tried to apply similar
methods to algebraic functions fields of one variable by replacing Q by C(X).
About thirty years later, Hermann Weyl, Die Idee der Riemannschen Fl¨ ache,
introduced the first correct ideas about “ abstract ” 1-dimensional complex
manifolds into this question and used Dirichlet’s principle to prove a priori the existence of “ many ” meromorphic functions on Riemann surfaces.
This result, which is obvious for surfaces associated to algebraic functions,
requires even now a long proof in the general case. In the 1930s and 40s, in
particular thanks to Andr´ e Weil and Oscar Zarisky, what a n-dimensional algebraic variety over an arbitrary field is was beginning to be understood and
a purely algebraic mechanism was being set up. It was an improvement on the
doubtful geometric arguments of the Italian school (which nonetheless had
discovered results and introduced very important ideas since 1870). Starting with the notion of a place and ending with that of a complex analytic
manifold, Claude Chevalley published the first post-war modern presentation on algebraic functions in one variable. Reading it is still advisable. Serge
Lang’s book goes much further in about a hundred pages, which indicates
how concise the proofs are. . .
15 author of a Lehrbuch der Algebra where almost everything that was known in
algebra around 1900 can be found.
16 It has been substantially improved on and generalized, but without fundamentally changing its point of view other than by introducing the notion of valuation.
Reading his main articles that can be found in his complete works, is still a most
advisable exercise.