a universal mapping property of generalized jacobian varieties

Annals of Mathematics

A Universal Mapping Property of Generalized Jacobian VarietiesAuthor(s): Maxwell RosenlichtSource: Annals of Mathematics, Second Series, Vol. 66, No. 1 (Jul., 1957), pp. 80-88Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1970118 .

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ANNALS OF MATHEMATICS

Vol. 66, No. 1, July, 1957 Printed in U.S.A.

A UNIVERSAL MAPPING PROPERTY OF GENERALIZED JACOBIAN VARIETIES

BY MAXWELL ROSENLICHT

(Received January 31, 1957)

1. Introduction. Statement of the theorem

The canonical map (oo of a nonsingular algebraic curve C into its jacobian variety Jo is universal for maps of C into abelian varieties in the following sense [8, Th. 21, p. 77]: If A/: C -* A is any rational map of C into an abelian variety A, then there exists a rational homomorphism r: Jo -* A and a point a E A such that for any P E C we have ipP = -rooP + a. In this paper we shall prove an analogous result for rational maps of a curve into any commutative algebraic group.

In [2] it is shown that if C is any algebraic curve that is defined over a field k and o is a semilocal subring of k(C), then o defines an equivalence relation (stricter than ordinary linear equivalence) among divisors of C that are rational over k, and this equivalence relation possesses most of the properties of the classical linear equivalence. It is shown in [5, ?4] that if o is contained in the local ring of each of the singular points of a complete birational model of C over k which is nonsingular with reference to k, and if k' is any overfield of k contained in the universal domain, then there exists a semilocal subring k'o of k'(C) such that the essential properties of the equivalence relations on C got from o and k'o are the same; furthermore we can find a complete birational model C, of C over k such that o is the intersection of the local rings in k(C) of a finite number of points (including all the singular points) of CO, and k'o is gotten in the same way from CO if we replace k by k'. In this case we say that we have an equivalence relation on C (or on any birational model of C) that is defined over k, and we can define a connected commutative algebraic group J, the generalized jacobian variety of C for our equivalence relation (unique to within a biregular isomorphism), and a canonical map : C -? J (unique to within a constant translation on J; note that the term "canonical map" is used in the sense of [3] rather than in the more correct sense of [5, ?4], a usage that will continue through- out this paper), and J and so possess many properties analogous to those of the ordinary jacobian variety Jo of C and the ordinary canonical map (oo, as de- veloped classically in [8]. The points of a complete nonsingular birational model of C that correspond to the points of CO defining the equivalence relation (these correspond to the valuation rings of k(C)/k which contain o) are called the places of the equivalence relation; by [3, Th. 12] and [3, Th. 8, Cor. 1] the points where the canonical map so (or, more precisely, its transfer to a complete nonsingular model of C) is undefined are precisely the places of the equivalence relation that correspond to singular points of C, . If the set of places of the equivalence relation is empty, this equivalence is precisely the ordinary linear equivalence. The theorem we shall prove in this paper is the following.

80



GENERALIZED JACOBIAN VARIETIES 81

THEOREM. Let C :C -+ G be a rational map of a complete nonsingular curve C into a commutative algebraic group G and let P1, . , P8 be the points of C at which 4' is not defined. Then there exists an equivalence relation on C whose places are precisely PI , , P8, such that if J is the generalized jacobian of this equivalence relation and (p: C -* J the corresponding canonical map, then there exists a rational homomorphism r: J -+ G and a point a e G such that, for any P e C distinct from P, eP8., we have pP = TrP + a.

This result has also been proved by Serre (unpublished) and applied by him and Lang to class field theory (cf. S. Lang, Sur les series L d'une vari6te alg6brique, Bull. Soc. Math. France, vol. 84 (1956), pp. 405-6). The theorem can also be used to prove theorems about algebraic groups themselves. For example, it follows di- rectly from [4, Th. 15, Lemma 1] that on any noncomplete variety we can find a closed subset which is a noncomplete curve, so (using the theorem and the struc- tural results of [3] on generalized jacobians) any noncomplete commutative algebraic group contains an algebraic subgroup that is biregularly isomorphic to either the additive or multiplicative group in one variable. Now if one checks the proof we have given in [4] of the theorem of Chevalley to the effect that any connected algebraic group contains a connected normal linear algebraic subgroup that gives rise to a factor group that is an abelian variety, it is clear that we get a proof of this theorem that does not involve the complicated Theorem 15 of [4]. Actually, Chevalley's theorem is used to prove our present result in the case of characteristic p / 0, but the proof we shall give in the case of characteristic zero is quite elementary, if long; thus we get a new proof of Chevalley's theorem in the characteristic zero case, and in fact this new proof long antedates the proof we have given in [4]. The proof of our theorem in the case of characteristic zero breaks down in characteristic p # 0 because of the use we make of differentials, but it seems possible that a suitable use of Dieudonne's semideriva- tions will enable one to extend our characteristic zero method to the general case.

The theorem as given above is incomplete in that it makes no mention of fields of definition, but the following more precise version is an immediate consequence.

Let C be a complete curve that is defined over k and nonsingular with reference to k, let G be a commutative algebraic group that is defined over k, and let 4': C -i G be a rational map satisfying the property that the map from C X C into G given by P X Q -i> ipP - 4Q is defined over k. Then there exists an equivalence relation on C that is defined over k, with places corresponding precisely to the singular points of C and the points at which 4' is not defined, having the following properties: If we take (as we may, by [5, ?4]) the generalized jacobian J of our equivalence relation to be defined over k and the canonical map : C -* J to be such that the map from C X C into J given by P X Q -+ (pP - (pQ is defined over k, then there exists a rational homomorphism r: J -* G, also defined over k, and a point a E G such that for any P e C that does not correspond to a place of our equivalence relation we have ipP = mrP + a.

To show this, first note that the set of singular points of C is k-closed. Simi-



82 MAXWELL ROSENLICHT

larly, the given rationality condition on 41 shows that the set of points of C at which 41 is not defined is k (Q)-closed, for any Q e C at which ' is defined; taking Q to be, in turn, two independent generic points of C over k shows that this latter set is also k-closed. Let C' be a complete nonsingular birational model of C, 4,': C' -* G the rational map corresponding to At, let J', so', r', a' be got by appli- cation of our original statement of the theorem to the map 4,': C' G, let k' D k be a field of definition for everything here, and let o' c k'(C') k'(C) be the semilocal ring giving rise to J'. Let n be an integer > 0 and let o be the set of all elements of k(C) of the form

(element of k) + (element of k(C) of order _ n at each prime divisor of k(C)/k corresponding to a point of C at which /' is not defined or to a singular point of C).

Then o gives rise to an equivalence relation on C that is defined over k, and if we take n sufficiently large we have (by [2, Th. 2]) o c o', so k'o c o'. Let J and so be the generalized jacobian and canonical map associated with o and having the indicated rationality properties. By [3, Th. 8] there exists a rational homomorphism from J to J' under which so and so' are consistent, up to a constant translation on J'. Hence we have a rational homomorphism r: J -* G and a point a E G such that pP = r-oP + a whenever P E C is not a place of the equivalence relation got from o, and we need only show that r is defined over k. Taking Q to be a fixed generic point of C over k and defining Q: C -* J, iQ: C -- G by (pQP = (pP - (pQ, 4t'QP t= 'P - 4tQ, we have p Q and 1J Q defined over k(Q) and t' QP = rp QP. If ir = dim J and M1, , M, are independent generic points of C over k(Q), then (pQMl + + 'pQM, is generic for J over k(Q) and

r((PQMJ + ...+ ?PQMT) = t'QM1 + + t'QMT .

As a consequence r is defined over k(Q). Since this is true for each Q that is generic for C over k, r is defined over k.

2. Reduction of the proof For the remainder of this paper, C will denote a fixed complete nonsingular

algebraic curve, G a fixed connected commutative algebraic group, ;t: C -* G a fixed rational map, and P1, ... , Ps will denote the distinct points of C at which + is not defined. The connectedness condition on G clearly involves no loss of generality.

If 2I is any divisor on C that is independent of P1, , P8 , we define the point ES E G by writing W as a finite sum -=Ej aiM, , where the ai's are integers

and the Mi's are points of C distinct from P1, , P8, and setting 4t'1 = Eiai *Mi . Let D, as in [7], denote the projective line. If f is any nonconstant rational function on C and x E D, the divisor (f)x on C is defined by (f)x = prc[r (C X x)], where r is the graph of f on C X D. The divisor (f) of f is given by (f) = (f)o - (f). , and if x # oo we have (f - x)O = (f)x and (f - x). = (f) . If (f)x is independent of P1, . , Ps , i.e. if x # f(Pl), * * *, f(P,) the ex- pression 4'(f)x makes sense; in fact we shall show that this map x -* 4(f)x is,




where defined, a rational map from D into G. For this purpose, let k be a field of definition for C, G, A/, and f, let x be a generic point of D over k, and let t e D be distinct from f(P1), ., f(P8). Then (f)x = prc[r(C X x)] and (f)t = prc[F-(C X h)], so (by [7, Th. 13, p. 206], taking Us = D and V = C) if we extend the k-specialization x -? t to (f)2, we get the unique specialization (f) -) (f) , and hence there is a unique extension of the k-specialization x -* to a k-specialization (x, 4t(f)x) ---( , 41(f)t). But the divisor (f)x is rational over k(x), so by the main theorem on symmetric functions [8, Th. 1, p. 15] we have 4/(f)x rational over k(x). Hence there exists a rational map Of D -> G, defined over any common field of definition for C, G, 4A, and f, such that, if t is any point of D distinct from f(P), , * * X f(P,), then Of is defined at t and oft = (f)t .

Now let k be an algebraically closed field of definition for C, G, and 4A. Then Pi, -, P, are each rational over k. Let n (to be determined later) be an integer >0 and let o be the set of all functions of k(C) of the form

(element of k) + (element of k(C) of order > n at each of the points P1, , Ps).

o is clearly a semilocal subring of k(C), giving rise to an equivalence relation whose places are precisely Pi, . , P, . Let J be the generalized jacobian of C for this equivalence relation, : C -* J the corresponding canonical map, and take J and (p to be both defined over k. Suppose for the moment that for any divisor a on C that is o-equivalent to zero we have AS =I 0. Then 4' induces a homomorphism from the group of divisor classes on C (under o-equivalence) of degree zero and independent of P1, ... , P8 into G. But the linear extension of (p to divisors on C of degree zero and independent of P1, . , P, induces an isomorphism between this group of divisor classes and J [3, Th. 7]. Thus there exists an (algebraic) homomorphism r: J -+ G such that rnpw = 4AS for any divisor 2f on C of degree zero and independent of P1, , P, . Let M1, , M,, be independent generic points of C over k, where ir dim J is the o-genus of C, and let 2to be a divisor on C of degree ir, independent of P1, - , P8, and rational over k. Then (p(Mi + + - - Mr - 2fo) is generic for J over k and

k((p(M1 + + M., - 2fo)) -= k(M1 + + Mr)

(= the smallest overfield of k over which the divisor M1 + + M7r is rational = the subfield of fixed elements of k(Ml, . , M,) under the k-auto- morphisms that interchange M1, . , Mr). /iM1 + .. ?+ 4'M,, being symmetric in M1, ... , M1r, is rational over k(Ml + + Mr), and the same is true of 4'M1 + .. ?+ 4PMT - 4'Wo r o(Mi + . + MT - 2o). We deduce that the map r: J -* G is a rational map defined over k, at least for generic points of J over k. By [3, Th. 5] (or the identical [4, Th. 3]), r is a rational homomorphism defined over ka. If P, P' E C are distinct from P1, , P, then

Tfp(P - PI) = 4(P -P),

so P- rsoP = 4P'- r5oP' = a constant point of G. Thus the above italicized statement implies the truth of our theorem. Now the integral closure o of o in k(C) consists of all elements of k(C) that are finite at each of the points P1, - , P.s .




The number a of [2] corresponding to o is given by 6 = diMk D/0 = ns - 1 (this is true if s > 0; if s = 0 then a = 0), and the divisor Y of the conductor of o in b [2, p. 179] is nPi + - - - + nP8 (or 0, if s = n = 1). Ifk' D k is also a subfield of the universal domain, it follows from [2, Th. 12] that 8 and E remain unchanged if we extend k to k', and hence we deduce that k'o consists precisely of all functions of the form

(element of k') + (element of k'(C) of order ? n at each of the points

Pi ) ... ,P).

If 2 is a divisor on C that is o-equivalent to zero then there exists a field k' D k and a unit F e k'o such that 9 = (F). If F e k', then W = 0 and AS2 = 0, so suppose F o k'. Let c = F(P1) - * = F(P8) (if s 0, take c to be any non- zero element of k'), and let f = 1/(F - c). Then c 0 0 and f has poles of order >n at each of the points P1, , P8. F = (cf + 1)/f, so

(F) = (cf + 1)- (f) (cf + 1)o -(f)o = (f)-ic -(f)o.

Since f(P1) = = f(P8) oo, the map Of: D -* G is defined at any finite point of D, and i/21 = #(F) = ik(f)-11c - 'i(f)o = Of(-(1/c)) - Of(O). We conclude that to prove our theorem it suffices to show that there exists an integer n such that if f is any (nonconstant) rational function on C having poles of order > n at each of the points P1, - - *, P8, then the map Of: D -> G is a constant map.

The last assertion, and hence the theorem, is easy to prove in the special case that G is an extension of a torus by an abelian variety, and in this case it even suffices to take n = 1: Here G contains as an algebraic subgroup a direct product (Gm)' of multiplicative groups in one variable and G/(Gm)' is an abelian variety. Let f be a rational function on C such that f(P1) = = f(P8) = oo. The compositum of Of and the natural homomorphism from G to G/(Gm)V gives a rational map from D into an abelian variety, and this map must be constant [8, Th. 8, p. 33]; hence 0fD is contained in a coset of (Gm)' on G. Thus we have to prove that any rational map from D into (Gm)', defined everywhere on D except at oc, is constant. This immediately reduces to the case v = 1, which case is settled by observing that any nonconstant rational function on D must have both zeros and poles.

We shall now use Chevalley's theorem (and a few results of Kolchin that appear in [1]) to obtain a further reduction in the case of characteristic p 7 0. For any commutative algebraic group over a field of characteristic p we can define a rational homomorphism p of the group into itself by pg = p * g. G being as above, the sequence of connected algebraic groups G D pG D p2UG D ...

must come to an end, so G contains a connected algebraic subgroup G, such that pG1 = G, and G1 = pG, for some integer v. If G2 is the component of the identity of the kernel of pV on G, then dim G1 + dim G2 = dim G, p G2 = 0, and pV has finite kernel on G1. Thus G1 n G2 is finite and the obvious homomorphism from G1 X G2 to G has finite kernel, whence G = G1G2. Now any




connected commutative linear algebraic group is the direct product of a torus and a connected algebraic subgroup each of whose elements is unipotent, and if the latter factor is of dimension >0 it contains an algebraic subgroup that is biregularly isomorphic to the additive group Ga, Xin which case the homomorphism p applied to our group has kernel of dimension > 1 . Hence the maximal connected linear algebraic subgroup of G1 is a torus. Also, pV applied to any abelian variety, in particular to an abelian variety that is a rational homomorphic image of G2 , has finite kernel, so G2 is linear. But pV applied to a torus is surjective, so G2 consists entirely of unipotent elements. Now let T1, T2 be the natural homomorphisms from G to G/IG and G/G2 respectively, and suppose that our last italicized assertion has been proved for the two cases 7-14: C -? G/G1 and 7T2X: C -* G/G2. Then there exists an integer n such that if f is a (nonconstant) rational function on C with poles of order > n at each of the places P1,

P8 we have 7'1I'(f)$ and 72#(f)s constant, for x E D - ). Choosing fixed elements a, , a2 E G such that 4l'(fX) = r-a1, r24t'(f)x = r2a2, we get f1x 41(f)x E a1Gi n a2G2, which is a finite set, so Of is constant. Now G/G2 =G1G2/G2 is a rational homomorphic image of G1, and its maximal connected linear algebraic subgroup is a rational homomorphic image of that of GC , hence also a torus, so the previous paragraph is applicable to the map 724/': C - G/G2 . On the other hand, G/G1 = G1G2/G1 is a rational homomorphic image of G2 , hence is linear and consists entirely of unipotent elements. But such a group is biregularly isomorphic to an algebraic group of unipotent matrices in triangular form. Hence to prove the last italicized statement for the case of characteristic p 5 0 it suffices to take G to be an algebraic group of unipotent matrices in triangular form.

3. Completion of the proof

Let ,1:C -* G and P1, -* , P8 be as above, assuming in addition that in the case of characteristic p # 0 G is an algebraic group of unipotent matrices in triangular form. We shall complete the proof of our theorem by showing that there exists an integer n such that if f is any (nonconstant) rational function on G having poles of order _ n at each of the points P1, P, F, then the map Of D -* G is constant.

Let V be a fixed affine open subset of G with coordinate functions u1, ,ur (that is, V is an open subset of G and u1, - *, ur are rational functions on G, everywhere defined and finite on V, such that the map g -* (u1(g), . , uT(g)) is a surjective biregular birational map from V to a closed subset of the affine space of dimension r), and suppose V chosen such that A/'C 4 G - V, that is, such that the map 41: C -> V is defined. Then u1#, - , u,4 are rational functions on C. Let y > 0 be an integer such that each of the functions u1i, , urlP has order > -y at each of the points P1, . , P, . Let n > 0 be an integer (to be determined later), let f be a (nonconstant) rational function on C that has a pole of order at least n at each of the points P1, , P8 , and let k be an algebraically closed field of definition for C, G, A/i, u1, , UT X and f. For j = 1, ... , r, the function (uj4,i)I/f' E k(C) is finite at each of the points P1,




* , P8. Let x be a quantity that is transcendental over k. We can write (f) =

Q, + + Qt, where t > 0 and each Qj is a point of C that is algebraic over k(x). Furthermore f(Qi) = x, i= 1, * * , t, so k(x) c k(Qi) and each Qt is a generic point of C over k. k(Q, - , Qt) is an algebraic function field of one variable over the constant field k. Let r denote a fixed k-specialization of

k(Qi, , . Qt)

into (k, oc) such that Tx - oo and let v be the valuation of k(Ql, * , Qt) over k that is canonically associated with r, normalized so that v(x) = -1 (v therefore takes values on a discrete subgroup of the additive group of rational numbers). Since each Qj is generic for C over k, 4/'Qj is defined, and uiVQj, ... , urQi are elements of k(Ql, , Qt). We now try to find a lower bound for v(uttQj), i = 1, ,t; j = 1, . , r. For any i = 1, , r, rQi is a well- defined point of C. First suppose that rQi is one of the points P1, , P8. Then, since (ujiP)n/fM is defined and finite at TQi we have

v ( (u"0QX) / ((QX) 0,

and since f(Qi) = x we deduce v(upPQj) _ -sA/n. On the other hand, if rQi is none of the points P1, ..., P8, then trQj is defined (as a point of G), and if qrQi is contained in V we have v(uppQi) > 0. Hence for any i = 1, , t; j = 1, ... , r, we have v(uppQj) > -,u/n, provided that TQi is not a point of C, distinct from P1, -.- P,F8, whose image under V is contained in G - V.

It is now easy to conclude our arguments in the case that G is an algebraic group of N X N unipotent matrices in triangular form. Here G itself is an affine variety, and we may take V = G and ul, . , ur to be the N2 entries of our matrices. In this case, v(upPQj) > -ys/n for all i, j, i.e. each entry of each matrix 4/Qj has v-order > -y/n. Now if (ai'), ) , (as )) are any unipotent N X N matrices in triangular form (i, j now range from 1 to N), then the ijth entry of the product (a(')) ... (at)) is

EhlE-,h~l~ aihahl2 ***ht-2ht~lh-i = Ei<hl < ... ht-1<j %thj * ht-i X

and (since each diagonal element of (as'), ) , (a8t)) is 1) we get that each entry of the matrix (a)')) (ast)) is a sum of monomials in the a~')'s (i, j- 1, , N; 1 - 1, , t) of total degree ? N.- 1. In particular, each entry of the matrix Ofx =I(f)x = 0Q1 + -.. + VlQt (where "+ " now denotes matrix multipli- cation) is a sum of monomials in the entries of 4,VQ1, , VQt of total degree < N - 1, so each entry of OfX has v-order ? -u(N - 1)/n. Now assume that n > ,(N- 1) (a bound that depends only on C, G, and 4'). Then each entry of OfX has v-order > -1. But OfX is rational over k(x) and v(x) =- 1, so each entry of Of X has v-order > 0, i.e. is finite at r. It follows that the map of: D G-* is defined at the point (oc) E D, hence at all points of D. Thus 6fD is a complete subvariety of an affine variety, hence a single point. This completes the proof if G is an algebraic group of unipotent matrices in triangular form, in particular if the characteristic is p - 0.




To handle the characteristic zero case we need the following simple lemma. The information needed henceforth on derivations and differentials may be found in [6].

LEMMA. Let the nonsingular variety V be a closed subset of an affine space, and let xl, a , Xr be the coordinate functions on V. If co is a differential form on V that is everywhere finite and if k is a field of definition for V and co, then co can be expressed as a polynomial with coefficients in k in x1, , xr and their differentials.

For simplicity we restrict the proof to the case in which c is a form of degree one. For any point P e V we can choose a subset {il, --, iaI of {1, -, r, such that xi,, -.. , xi. are uniformizing coordinates for V at P, and we then have X = E=1 fj dx i , where each fj E k(x) is finite at P. Hence there exists a polynomial Fp(X) E k[X, X - * , Xr] (the Xi's are indeterminates) such that Fp(P) $ 0 and Fp(x)w e k1[x] dx1 + - - + k1[x] dXr. The set of all polynomials F e k[X] such that F(x)w e k[x] dx1 + - * * + k[x] dxr is an ideal in k[X] that contains the ideal of V. Since this ideal has no zero, it is the unit ideal. Hence X ce kfx dxj + * - * + k[x] dx, . Q.E.D.

We now settle the characteristic zero case. In what follows, "differential" will mean "differential form of degree one." Let V, u1, . , Ur be as before and let ko be a field of characteristic zero that is a field of definition for G and ul , -- , ur. Any invariant differential on G is everywhere finite, and if such a differential is defined over ko the lemma shows that it can be expressed in the form X=O Fj duj, where each Fj Eko[u]. Let v be an integer _ 0 such that each invariant differential on G that is defined over ko is of the above form, with each Fj a polynomial of degree ? v. Let As, n (still to be determined), f, k, etc., be as in the second paragraph of this section, and suppose (as we may) that k has been chosen so as to contain ko. Any invariant differential co on G that is defined over k is a linear combination with coefficients in k of ones that are defined over ko, hence is of the form co = >= Fj duj, where each Fj e k[u] has degree _ v. For such an co and any i = I, .. , t, we have a differential co(4Qi) (defined in [6, ?3]) of k(Ql, . , Q,)/k, and we have

Cv(,tQi) Z= =1 Fj(u1,4Qi, ,urtQi) dutitQj .

Supposing that rQi is one of the points P1, S, , we have v(uj4,Qi) > -si/n for j 1, , r, so (noting that v(x) = - 1 and that any element of

k(Ql) ,- , Qt) can be expanded in a fractional power series in 1/x with coefficients in k)

v(dupPQi/dx) ? -,4/n + 1;

hence v(w(4iQ1)/dx) > -,4v/n - ,u/n + 1 = -,u(v + 1)/n + 1. If, on the other hand, TQi is none of the points P1, , , P,, then 4,&TQi is a well-defined point of G and we can write co as a finite sum co = A yet dza , where the i r's and za's are functions in k(G) that are defined at /rQiT. Each y>4'Qi and za'PQi




is then an element of k(Qj, .. , Qt) that is finite at r, hence expandible in series of positive fractional powers of 1/x; the equation co(jPQj) = a Ya('PQ) dza(tyPQi) thus shows that v(w(t/Qj)/dx) > 1. We have therefore shown that for each i = 1, , t we have v(w(4iQj)/ dx) > -,u(v + 1)/n + 1. But by the last result of [6], co(Ofx) = o(V(f)x) = w(6VQi + . + VQt) is equal to co(qQj) + + w(41Qt). Hence v(w(Ofx)/dx) ? -y(v + 1)/n + 1. The point Of1 e G is rational over k(x), so co(0fx) is a differential of k(x)/k. Since v(x) = -1, co(Ofx) is a differential of k(x)/k having at the place x -> oo a pole of order ? ,u(v + 1)/n + 1. Now consider another place of k(x)/k, say the place x -* c, c e k. Then OfC is defined and, if we write X = La. y dza, where the ya's and za's are functions of k(G) that are defined at 0ac, we get c(Ofx) = La y,(Ofx) dza(1fx), which is finite at the place x -* c. Now take n > y(v + 1) (a bound which depends only on C, G, i1 and the choice of V and u1, . , Ur). Then c(0fx) is a differential of k(x)/k that has no poles, except possibly for a pole of order 1 at the place x -* oo. Hence cw(Ofx) = 0. But these invariant differentials w have local com- ponents at each point of G that span the entire space of local differentials at the point. Hence if a function u e k (G) is defined at the point ofx e G, we can write du = L HaCa, where each Ha, e k(G) is defined at fix and the wa's are invariant differentials on G that are defined over k. Thus duOx = (du)(0x) = Ea Ha(OfX).a(OifX) = 0. Hence uOfx e k. The point fix is thus rational over. k, proving the constancy of the map Of: D -* G. We are done.

NORTHWESTERN UNIVERSITY

REFERENCES

[1] A. BOREL, Groupes lingaires algebriques, Ann. of Math., vol. 64 (1956), pp. 20-82. [2] M. ROSENLICHT, Equivalence relations on algebraic curves, Ann. of Math., vol. 56 (1952),

pp. 169-191. [3] M. ROSENLICHT, Generalized jacobian varieties, Ann. of Math., vol. 59 (1954), pp. 505-

530. [4] M. ROSENLICHT, Some basic theorems on algebraic groups, Amer. J. Math., vol. 78 (1956),

pp. 401-443. [5] M. ROSENLICHT, Some rationality questions on algebraic groups, to appear in Annali

di Mat. [6] M. ROSENLICHT, A note on derivations and differentials on algebraic varieties, to appear. [7] A. WEIL, Foundations of Algebraic Geometry, Amer. Math. Soc. Colloq. Pub., New

York, 1946. 1si A. WEIL, Varidtds Ab6liennes et Courbes Alg6briques, Paris, 1948.



a universal mapping property of generalized jacobian varieties

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