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The Meromorphisms of an Elliptic Function-Field

Published online by Cambridge University Press:  24 October 2008

H. Davenport
H. Davenport
Affiliation:
Trinity College
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Extract

1. Hasse's second proof of the truth of the analogue of Riemann's hypothesis for the congruence zeta-function of an elliptic function-field over a finite field is based on the consideration of the normalized meromorphisms of such a field. The meromorphisms form a ring of characteristic 0 with a unit element and no zero divisors, and have as a subring the natural multiplications n (n = 0, ± 1, …). Two questions concerning the nature of meromorphisms were left open, first whether they are commutative, and secondly whether every meromorphism μ satisfies an algebraic equation

with rational integers n0, … not all zero. I have proved that except in the case (which is equivalent to |N−q|=2 √q, where N is the number of solutions of the Weierstrassian equation in the given finite field of q elements), both these results are true. This proof, of which I give an account in this paper, suggested to Hasse a simpler treatment of the subject, which throws still more light on the nature of meromorphisms. Consequently I only give my proof in full in the case in which the given finite field is the mod p field, and indicate briefly in § 4 how it generalizes to the more complicated case.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 32 , Issue 2 , May 1936 , pp. 212 - 215
Copyright
Copyright © Cambridge Philosophical Society 1936

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References

* Abh. Math. Seminar Hamburg (1934), 325–45. We use the notation and results of this paper, and refer to it as H.Google Scholar

† Throughout this paper, “meromorphism” means “normalized meromorphism”.

‡ See Hasse, , Göttinger Nachrichten, Neue Folge. 1 (1935). 119–29.Google Scholar

§ The degree of a rational function is the greater of the degrees of its numerator and denominator after removal of common factors.

* The inequality (4) is not given explicitly in H., but is used at the top of p. 346. The new method of He is based on the identity

* deg φ denotes the degree of φ. deg φ>degψ expresses the fact that μ transforms the “infinite” solution into itself, i.e. that μ is a normalized meromorphism.

* H. § 5, simplified by (9).