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• W2396983503 abstract "The Riemann hypothesis, formulated in 1859 by Bernhard Rie- mann, states that the Riemann zeta function �(s) has all its nonreal zeros on the line Res = 1/2. Despite over a hundred years of considerable effort by numerous mathematicians, this conjecture remains one of the most intriguing unsolved problems in mathematics. On the other hand, several analogues of the Riemann hypothesis have been formulated and proved. In this chapter, we discuss Enrico Bombieri's proof of the Riemann hy- pothesis for a curve over a finite field. This problem was formulated as a conjecture by Emil Artin in his thesis of 1924. Reformulated, it states that the number of points on a curve C defined over the finite field with q ele- ments is of the order q + O( p q). The first proof was given by Andre Weil in 1942. This proof uses the intersection of divisors on C × C, making the application to the original Riemann hypothesis so far unsuccessful, because spec Z×spec Z = spec Z is one-dimensional. A new method of proof was found in 1969 by S. A. Stepanov. This method was greatly simplified and generalized by Bombieri in 1973. Bombieri's proof uses functions on C ×C, again precluding a direct transla- tion to a proof of the Riemann hypothesis itself. However, the two coordinates on C × C play different roles, one coordinate playing the geometric role of the variable of a polynomial, and the other coordinate the arithmetic role of the coefficients of this polynomial. The Frobenius automorphismof C acts on the geometric coordinate of C × C. In the last section, we make some suggestions how Nevanlinna theory could provide a model for spec Z × spec Z that is two- dimensional and carries an action of Frobenius on the geometric coordinate. The plan of this chapter is as follows. We first give a historical introduction to the Riemann hypothesis for a curve over a finite field and discuss some of the proofs that have been given. In Section 2, we define the zeta function of the curve. To prove the functional equation, we take a brief excursion to the two- variable zeta function of Pellikaan. In Section 3, we reformulate the Riemann hypothesis forC(s), and give Bombieri's proof. In the last section, we compare �C(s) with the Riemann zeta function �(s), and describe the formalism of Nevanlinna theory that might be a candidate for the framework of a translation of Bombieri's proof to the original Riemann hypothesis." @default.
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• W2396983503 date "2008-01-01" @default.
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• W2396983503 title "FOR FUNCTION FIELDS OVER A FINITE FIELD" @default.
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