104 
MATHEMATICS: S. LEFSCHETZ 
An abelian variety of genus Vp, belonging to an 5„, (n > p), is defined 
by equations 
Xj = f{ui, U2, . . . u„), {j = 1,2, . . .n) (1) 
where the (/)'s are 2^-ply periodic meromorphic functions of the (uYs, the 
periods forming a matrix of a type called, after Scorza, a Riemann matrix. 
The rank of Vp is the number of distinct systems of values of the {uYs, 
modulo Q, corresponding to an arbitrary point of Vp. It is easy to show that 
to any 12 corresponds a Vp of rank one. Suppose indeed the (JYs so chosen 
that any other periodic meromorphic function <p belonging to 0 be a rational 
function of them. If Vp were of rank > 1 then to every set of values {ui, 
ti2, ' • . %), would correspond another {ui, U2, . . . u'p), distinct modulo 12, 
such that ip {u) = ip {u') whatever ip. Choosing p such functions with non 
zero jacobian we find that the (mO's are functions of the (w)'s. But 
(p {ui — ai, . . . Up — ap) = (p {ui — ai, . . .Up — ap) whatever the constants 
a, since the function at the right is of same type as (p. Hence at once 
dujt = duh, u'h — Uh = ^h' The (|8)'s are constants which obviously form a 
set of periods of the (/)'s, and therefore Vp is effectively of rank one. — All 
abelian varieties of rank one belonging to 12 are birationally equivalent. 
2. Denoting by m the complex conjugate of any number m, a real variety 
of Sm is defined by the condition that when it contains one of the points, said 
to be conjugate, {%), (x), it contains the other. Let then 12 be a Riemann 
matrix with a corresponding real Vp of rank one. Any simple integral of the 
first kind of Vp is of the form u = Js Rj dxj, where the (R) 's are rational in the 
(x)'s. Replacing in them all coefficients by their conjugates we obtain a 
new integral of the first kind of Vp, say (u), and w is a linear combination of 
the two integrals w + (u), — i(u— (u)), which are of real form. Hence Vp 
possess p independent integrals of real form, Ui, U2, . . . Up. If has a real 
point and we take it for lower limit of integration, our integrals will assume 
conjugate values at conjugate points of Vp. 
Let now 7 be any linear cycle of Vp and 7 its transformed by T, transfor- 
mation of the variety permuting each point with its conjugate. T transforms 
7 + 7 into itself and 7—7 into its opposite. As 2 7, = (7 + 7) + (7 — 7), 
the double of any cycle is the sum of two others transformed by T the one 
into itself and the other into its opposite. The periods of u^ with respect to 
cycles of the first type are real, and with respect to those of the other type 
they are pure complex. If q is the number of cycles of one type 2p — q 
is that of the cycles of the other type. As the real and complex parts of the 
periods of p independent integrals of the first kind with respect to 2p inde- 
pendent cycles form a non zero determinant, we must have q = 2p — q, or 
q = p. Finally we may single out 2p cycles 71, 72, . . . 72^, such that 
T.yh = Jh T.yp+h= — yp+h, h^p while the cycles + ^72 + . . . + 
f^hpyip, (Mk integer), include the double of any cycle of Vp. The corre- 
sponding period matrix for the (w)'s is of the type 
