Mathematics

# 39b. Roch on the Riemann-Roch Theorem^{1}

If *w* is an algebraic function, defined by the [irreducible polynomial] equation

then any function

*ω* having the same ramifications as

*w* ["wie

*w* verzweigte"] can be expressed as a rational function

of

*w* and

*z*.

^{2} If

*ω* has simple infinities ["wird

*ω* unendlich erster Ordnung"] at the

*m* branch points of its [Riemann] surface

*T*, then by §5 of Riemann 39

*a* [the expression for]

*ω* contains

arbitrary constants. Already Riemann’s condition for the existence of functions becoming infinite in fewer than

points shows that the number of essential ["wirklich vorhandenen"] constants can be greater.

However, this can also be the case when

For example, if

then

*ω* becomes infinite in the

points where

(see §10 of Riemann’s paper),

^{3} and contains as many arbitrary constants as

*φ*_{1}, namely

*p*, whereas

in this case. On the other hand, if

are four functions which are infinitesimals of the second order at

points, whose square roots Riemann calls "Abelian functions," then there exist [a certain number of] expressions

which are rational functions of

*z* and

*w*. These indeed depend linearly on

parameters, a theorem stemming from Riemann which is a special case of the following result.

The most general Abelian integral of the second kind, which becomes simply infinite at *m* points *e*_{j}, is

(see [39*a*], §5). Here

are special [Abelian] integrals of the second kind which become infinite like

where

In order for

*v* to become algebraic, its moduli of periodicity must vanish.

^{4} This gives 2

*p* linear equations to be satisfied by the

constants

*β*_{k} and

*α*_{l}, thus leaving

arbitrary constants. Only if these 2

*p* equations are linearly independent can

*ω* contain more than

constants.

To investigate this possibility, we let the special expressions

be the nowhere infinite ["endlich bleibende"] integrals introduced by Riemann as arguments for his *θ*-function. The integral *u*_{μ} has the modulus of periodicity *πi* across the cut *α*_{μ}, and moduli of periodicity zero across the other

cuts

*a*_{k}, and across the cuts

*b*_{v} the moduli of periodicity

*a*_{μ, v-} Near the point

*e*_{k}, let us suppose that

In general—that is, when the *e*_{k} are neither at ∞ nor branch points—we have

The [Abelian] integrals of the second kind can be so chosen that their moduli of periodicity vanish across the cuts *α*_{μ}. For, given *t*_{k} with the moduli

across these cuts, one need only set

to get *t*_{k} with the desired properties. This reduces the conditions for the moduli of periodicity of the expression

to vanish across the cuts *α*_{μ} to the equations

The remaining constants *β* correspond simply to the *p* conditions for the moduli of periodicity to vanish across the cuts *b*_{v}, as will be explained below.

The whole ["ganze"] integral ∫ *u*_{μ} dv extended over the bounded, simply connected domain formed from *T* by taking its cuts *α*_{μ} and *b*_{v} as boundaries is equal to

where

and

are the values of

*u*_{μ} . on the positive and negative sides of the cuts. This gives [Roch sketches the reasoning]

which, since *μ* ranges over the integers

gives

*p* equations which express the dependence of the moduli of periodicity on the points

*e*_{k}.

From now on, we shall assume for simplicity that all *e*_{k} are finite and that none is a branch point. This reduces (1) to

This formula transforms the *p* equations

noted above, into

These equations are linearly dependent if and only if one can so choose coefficients

in [the expression]

that

for

This is the condition on

which permits φ to vanish at all branch points other than the

*r* coincident ones.

^{5} Nevertheless, two or more of the

*p* equations can follow from the others, and the general result can be stated as follows:

*If a function w has in simple poles, and if q linearly independent functions*

*can vanish at these poles, then ω depends on*
*arbitrary constants.*
For example, let

be the equation of a quintic curve. Then in general, that is, unless the curve has a double point,

the integrals remaining bounded are

hence every quadratic polynomial function is [such] a function *φ*_{k}. The function

has poles at the five points where the straight line

intersects the given curve. At these five points, three linearly independent functions φ vanish, namely:

and

Hence

*ω* contains

arbitrary constants:

*a, b,* and

*c*.

If the given quintic curve has a double-point, then

Hence, if

and

represent two straight lines passing through the double-point, only

and

are functions φ which are zero at the five points where

*ω* becomes infinite, and

*ω* again contains

constants.

If some of the points *e*_{k} are infinite, then one must ask about the number of [linearly independent] functions which are of lower degree than the highest possible one and which vanish at the points *e*_{k} which lie in the finite plane. [The result is:]

*Let*

*be the constants in question, and let those* [Abelian]

*integrals which remain bounded be represented by the series* (α). Let

*there exist q linearly independent functions*

*which vanish at the in points e*_{k}. *Then an algebraic function which becomes simply infinite at the points e*_{k}*contains*

*arbitrary constants.*
[Roch concludes by sketching a way to treat higher-order infinities, and observing that his formula (1) generalizes an earlier formula of Weierstrass for hyperelliptic integrals.]

^{1} G. Roch "Uber die Anzahl willkürlichen Constanten in algebraischen Funktionen," J. reine ang. Math. 64 (1864), 372–376.

^{2} Roch wrote *s*, *s*′ where we have written *w*, *ω*.

^{3} Roch here refers to the paper by Riemann from which Selection 39*a* is taken.

^{4} "Damit *v* in eine algebraische Funktion übergehe, müssen sämmtliche Periodicitätsmoduln yon *v* verschwinden." Today, "moduli of periodicity" are called "periods."

^{5} "oder wenn die *m* Punkte *e*_{k} solche sind, in denen amuser den *r* zusammengefallenen sich auf-hebenden Verzweigungspunkte ... φ verschwinden kann."