Real algebraic curves

Abstract

INTRODUCTION THE main object of interest in this paper is a non-singular, -projective, irreducible, complex algebraic curve, which is assumed to be defined over the real numbers, and, until 94 is assumed to have a real point. On the underlying compact, connected Riemann surface X, the complex conjugation and real locus appear, respectively, as a differentiable orientation-reversing involution p and its fixed-point-set Xp, which is a disjoint union of, say, r circles, with t 2 1. The main result (Theorem 1) is a matrix expression, depending only on r and the genus g of X, for the action of p on the homology group H,(X,Z). As a corollary, one has an improvement of Harnack’s inequality r-5 1 + g [ 1,2,4], in the sense that the non-negative number 1 + g r is interpreted (in Corollary 1, § 1) as the dimension of a vector space, namely the quotient of Hr(X, Z/22) by the subspace of cycles fixed by p. In an old manuscript[3], which he has most kindly shared with me, Barry Mazur proved, in the special case where X is the modular curve X0@) classifying elliptic curves together with an isogeny of prime degree p, that Hr(X, Z/Z) is a free module over the group-ring 2/2Z[pl. The present paper implies his result, over Z, using nothing about X,(p) except that its real locus is connected; this property was used in [33, and was actually necessary in view of my Corollary 2, 91. I have taken fairly literally from 131 my use of surgery in 02 (done in [3] only for r = 1, of course) and connected sum in P4. The essence of Mazur’s argument was to reduce by surgery to the case of no real points; for me this case (04) is just an appendix. The twisting operation and shearing cycles of 03 are my adaptations to the general case of Mazur’s constructions on the “fundamental domain” for X0(p); while the former was rather implicit in [3], the latter were quite explicit and even named there.