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Published online by Cambridge University Press: 26 February 2010
Let k be a finite field of q elements. The equation f(x, y) = 0, where f(x, y) is a polynomial with coefficients in k, may be construed to represent a curve, C, in a plane in which x, y are affine coordinates. On the other hand, this equation can be thought of as denning y as an algebraic function of x, where x is transcendental over k. The purpose of this paper is to show that, for a certain class of curves, corresponding in the classical case to curves having n distinct branches at x = ∞, if the degree, n (in y), of the polynomial f is large compared with q, then the genus† of C cannot be too small. We infer this result from a theorem about the genus of a function field; for we can think of C as being a model of such a field.
† In what follows, we shall not distinguish between various definitions of the genus and we assume that they are all equivalent. For the terminology, see: Hasse, H., Zahlentheorie (Berlin 1949), §§21–25; C. Chevalley, Algebraic Functions of One Variable (A.M.S. Surveys, No. 6), Chapters 1-4; and S. Lang, Introduction to Algebraic Geometry (Interscience Tracts, No. 5), Chapters 6 and 10.Google Scholar
† Artin, E., Math. Zeitschrift, 19 (1924), 153–246, especially §2.CrossRefGoogle Scholar
‡ ‡Mahler, K., Annals of Math., 42 (1941), 488–522.CrossRefGoogle Scholar
§ Armitage, J. V., Mathematika, 4 (1957), 132–137. There are two misprints in this paper; the inequality (15) should read |D|> en-2+γ, and in (16) the last term of Li should be 
CrossRef+en-2+γ,+and+in+(16)+the+last+term+of+Li+should+be>Google Scholar.
† Hasse, op. cit., especially p. 297 and p. 349. Our discriminant is the “t-discriminant” of Hasse (so that d1 = deg D2), and we have du = 0.