Counting Points on Curves over Finite Fields

Serguei A. Stepanov
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Serguei A. Stepanov: Bilkent University

Chapter Chapter 6 in Codes on Algebraic Curves, 1999, pp 143-172 from Springer

Abstract: Abstract In this chapter we apply the technique we have worked out earlier to prove the Riemann hypothesis for the zeta-function ζ (X, s) of a curve X defined over a finite field F q .This result was proved for the first time by Hasse (in the case of elliptic curves) and Weil (in the general case) using the correspondence theory on X. Here we give an elementary proof based essentially on using only the Riemann—Roch theorem (see Stepanov [184, 185, 187], Bombieri [17], Schmidt [159] and Stöhr and Voloch [200]).

Keywords: Finite Field; Function Field; Prime Divisor; Algebraic Integer; Finite Extension (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1007/978-1-4615-4785-3_6

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