Background on Curves and Jacobians
4.4 Arithmetic of curves
4.4.1 Local rings and smoothness
Definition 4.92 LetPbe a point on an affine curveC. The set of rational functions that are regular atP form a subringOP ofK(C).
In fact,OPis a local ring with maximal ideal
mP ={f ∈ OP |f(P) = 0}. It is called thelocal ring ofP.
Theresidue field ofPis defined asOP/mP.
One hasK(P) =OP/mP, hence,deg(P) = [K(P) :K].
ForS ⊂CdefineOS :=
P∈S OP. It is thering of regular functions onS. IfSis closed then OSis the localization ofK[C]with respect to the ideal definingS.
A rational functionronCis a morphism if and only ifr∈ OC =K[C].
For a projective curve, the ring of rational functions onC that are regular atP is equal toOP, the local ring ofPin a nonempty affine part ofC.
Definition 4.93 LetP∈Cfor a projective curveC. The pointPisnonsingularifOPis integrally closed inK(C). Otherwise the point is calledsingular. A curve is callednonsingularorsmoothif every point ofC(K)is nonsingular.
A smooth curve satisfies thatK[Ca]is integrally closed in K(C)for any choice ofCa. IfC is projective but not smooth we take an affine coveringCiand defineCias affine curve corresponding to the integral closure ofK[Ci]. By the uniqueness of the integral closure we can glue together the curvesCito a projective curveCcalled thedesingularization of the curveC. Note that in general even forCa plane curve,Cshall not be plane.
There is a morphismϕ : C → C that is a bijection on the nonsingular points of C. Hence projective smooth curves that are birationally equivalent are isomorphic.
Therefore, irreducible projective nonsingular curves are in one-to-one correspondence to function fields of dimension1overK.
To have a criterion for smoothness that can be verified more easily we restrict ourselves to affine parts of curves.
Lemma 4.94 (Jacobi criterion) LetCa ⊆Anbe an affine curve, letf1, . . . , fd∈ K[x]be gener- ators ofI(Ca), and letP ∈Ca(K). If the rank of the matrix
(∂fi/∂xj)(P)
i,jisn−1then the curve is nonsingular atP.
§ 4.4 Arithmetic of curves 65
Using this lemma one can show that there are only finitely many singular points on a curve.
For a nonsingular pointP the dimension ofmP/m2P is one. Therefore, the local ringOP is a discrete valuation ring.
Definition 4.95 LetCbe a curve andP ∈Cbe nonsingular. Thevaluation atPonOPis given by vP :OP → {0,1,2, . . .} ∪ {∞}, vP(f) = max{i∈Z|f ∈miP}.
The valuation is extended toK(C)by puttingvP(g/h) =vP(g)−vP(h). The value group ofvP
is equal toZ.
The valuationvP is a non-archimedean discrete normalized valuation (cf. Chapter 3).
A functiontwithv(t) = 1is calleduniformizer forCatP.
LetP1andP2be nonsingular points. ThenvP1 =vP2 if and only ifP1∈GK·P2.
Example 4.96 LetC = P1/K and chooseP ∈ A1. Letf ∈ K(x). The valuevP(f)off at P = (a)∈Kequals the multiplicity ofaas a root off. Ifais a pole off, the pole-multiplicity is taken with negative sign as it is the zero-multiplicity of1/f.
This leads to a correspondence of Galois orbits of nonsingular points ofCto normalized valua- tions ofK(C)that are trivial onK. For a nonsingular curve this is even a bijection. Namely, to each valuationv ofK(C)corresponds a local ring defined byOv :={f ∈ K(C) | v(f) 0} with maximal idealmv. IfCis smooth, there exists a maximal idealMv ⊂K[Ca], whereCa is chosen such thatK[Ca]⊂ Ov, satisfyingOv =Op. Over the algebraic closure there exist points P1, . . . , Pd such that Ov equalsOPi and thePi form an orbit underGK. The degree of Mv is [K[Ca]/Mv :K] = [Ov/mv :K]. It is equal to the order ofGK·Pi of one of the corresponding points onC.
Two valuations ofv1, v2 ofK(C)are called equivalentif there exists a numberc ∈ R>0 with v1=cv2.
Definition 4.97 The equivalence class of a valuationvofK(C)which is trivial onKis called a placepofK(C). The set of places ofF/Kis denoted byΣF/K.
In every place there is one valuation with value groupZ. It is called thenormalized valuation of pand denoted byvp.
We have seen:
Lemma 4.98 LetF/Kbe a function field and letC/Kbe a smooth projective absolute irreducible curve such thatF K(C)with an isomorphismϕfixing each element ofK.
There is a natural one-to-one correspondence induced byϕbetween the places ofF/K and the Galois orbits of points onC.
Example 4.99 Consider the function fieldK(x1)with associated smooth curveP1/K and affine coordinate ringK[x1]. The normalized valuations inΣK(C)/Kfor which the valuation ring contains K[x1]correspond one-to-one to the irreducible monic polynomials inK[x1]. There is one additional valuation with negative value atx1, calledv∞, which is equal to the negative degree valuation, corresponding to the valuation atp∞(t) =tinK[t] =K[1/x1]. Geometricallyv∞corresponds to P1 A1.
66 Ch. 4 Background on Curves and Jacobians
4.4.2 Genus and Riemann–Roch theorem We want to define a group associated to the points of a curveC.
Definition 4.100 LetC/Kbe a curve. Thedivisor groupDivCof Cis the free abelian group over the places ofK(C)/K. An elementD∈DivCis called adivisor. It is given by
D =
pi∈ΣK(C)/K
nipi,
whereni∈Zandni= 0for almost alli.
The divisorDis called aprime divisorifD=pwithpa place ofK(C)/K.
Thedegreedeg(D)of a divisorDis given by deg : DivC → Z
D → deg(D) =
pi∈ΣK(C)/K
nideg(pi).
A divisor is calledeffectiveif allni0. ByEDone means thatE−Dis effective.
ForD∈DivCput
D0 =
pi∈ΣK(C)/K ni0
nipi and D∞ =
pi∈ΣK(C)/K ni0
−nipi,
thusD=D0−D∞.
Recall that overKeach placepicorresponds to a Galois orbit of points on the projective nonsingular curve attached toK(C). Thus,Dcan also be given in the form
D =
Pi∈C
niPi
withni∈Z, almost allni= 0andni =njifPi∈Pj·GK.
Assume now thatCis absolutely irreducible. Then we can make a base change fromKtoK. As a result we get again an irreducible curveC·K(given by the same equations asCbut interpreted overK) with function fieldK(C)·K.
Applying the results from above we get DivC·K=
Pi∈C
niPi
withni∈Zand almost allni= 0. For all fieldsLbetweenKandKthe Galois groupGLoperates by linear extension of the operation on points.
Proposition 4.101 Assume thatC/Kis a projective nonsingular absolutely irreducible curve. Let Lbe a field betweenK andK and denote byDivC·L the group of divisors of the curve overL obtained by base change fromKtoL. Then
DivC·L={D∈DivC·K|σ(D) =D, for allσ∈GL}. Especially:DivC= DivGK
C·K.
§ 4.4 Arithmetic of curves 67
Important examples of divisors ofCare associated to functions. We use the relation between nor- malized valuations ofK(C)which are trivial onKand prime divisors.
Definition 4.102 LetC/Kbe a curve andf ∈K(C)∗. Thedivisordiv(f)off is given by div :K(C) → DivC
f → div(f) =
pi∈ΣK(C)/K
vpi(f)pi.
A divisor associated to a function is called aprincipal divisor. The set of principal divisors forms a groupPrincC.
We have a presentation ofdiv(f)as difference of effective divisors as above:
div(f) = div(f)0−div(f)∞.
The points occurring indiv(f)0(resp. indiv(f)∞) with nonzero coefficient are calledzeroes (resp.
poles) off.
Example 4.103 Recall the setting of Example 4.99 for the curveC = P1. Since polynomials of degreedover fields havedzeroes (counted with multiplicities) overKwe get immediately from the definition:
deg(f) = 0, for allf ∈K(x1)∗.
Now let C be arbitrary. Take f ∈ K(C)∗. For constant f ∈ K∗ the divisor isdiv(f) = 0.
OtherwiseK(f)is of transcendence degree1over Kand can be interpreted as function field of the projective line (with affine coordinatef) overK. By commutative algebra (cf. [ZASA1976]) we learn about the close connection between valuations inK(f)andK(C), the latter being a finite algebraic extension ofK(f). Namely,div(f)∞is the conorm of the negative degree valuation on K(f)and hence has degree[K(C) :K(f)](cf. [STI 1993, p. 106]).
Sincediv(f)0= div(f−1)∞we get:
Proposition 4.104 Let C be an absolutely irreducible curve with function fieldK(C)andf ∈ K(C)∗.
(i) deg div(f)0
= 0if and only iff ∈K∗. (ii) Iff ∈K(C)Kthen[K(C) :K(f)] = deg
div(f)∞
= deg
div(f)0 . (iii) For allf ∈K(C)∗we get:deg
div(f)
= 0.
So the principal divisors form a subgroup of the groupDiv0Cof degree zero divisors.
To each divisorD we associate a vector space consisting of those functions with pole order at placespibounded by the coefficientsniofD.
Definition 4.105 LetD∈DivC. Define
L(D) :={f ∈K(C)|div(f)−D}.
It is not difficult to see thatL(D)is a finite dimensionalK-vector space. Put(D) = dimK
L(D) . TheTheorem of Riemann–Rochgives a very important connection betweendeg(D)and(D).
We give a simplified version of this theorem, which is sufficient for our purposes. The interested reader can find the complete version in [STI1993, Theorem I.5.15].
68 Ch. 4 Background on Curves and Jacobians
Theorem 4.106 (Riemann–Roch) LetC/Kbe an absolutely irreducible curve with function field K(C). There exists an integerg0such that for every divisorD∈DivC
(D)deg(D)−g+ 1.
For allD∈DivCwithdeg(D)>2g−2one even has equality(D) = deg(D)−g+ 1.
Definition 4.107 The numbergfrom Theorem 4.106 is called thegenus ofK(C)or thegeometric genus ofC. IfCis projective nonsingular thengis called thegenus ofC.
The Riemann–Roch theorem guarantees the existence of functions with prescribed poles and zeroes provided that the number of required zeroes is at most2g−2less than the number of poles. Namely, ifni > 0atpi thenf ∈ L(D)is allowed to have a pole of order at mostni atpi. Vice versa a negativenirequires a zero of multiplicity at leastniatpi.
As an important application we get:
Lemma 4.108 LetC/Kbe a nonsingular curve and letD =
nipibe aK-rational divisor ofC of degreeg. Then there is a functionf ∈K(C)which has poles of order at mostni(hence zeroes of order at least−niifni<0) in the pointsPi∈Ccorresponding topiand no poles elsewhere. In other words: the divisorD+ (f)is effective.
Example 4.109 For the function fieldK(x1), Lagrange interpolation allows to find quotients of polynomials for any given zeroes and poles. This leads to(D) = deg(D) + 1. The curveP1/K has genus0.
The Hurwitz genus formularelates the genus of algebraic extensions F/F/K. It is given in a special case in the following theorem (cf. [STI 1993, Theorem III.4.12] for the general case).
Theorem 4.110 (Hurwitz Genus Formula) LetF/F be a tame finite separable extension of alge- braic function fields having the same constant fieldK. Letg(resp. g) denote the genus ofF/K (resp.F/K). Then
2g−2 = [F:F](2g−2)
p∈ΣF /K
p|p
e(p|p)−1
deg(p).
One of the most important applications of the Riemann–Roch theorem is to find affine equations for a curve with given function field. We shall demonstrate this in two special cases which will be the center of interest later on.