Number Theory: An Introduction to Mathematics

5 Generalizations and Analogues 387

This seems quite similar to the number field case, but the function field case is
actually simpler. F.K. Schmidt (1931) deduced from the Riemann–Roch theorem that
there exists a polynomialp(u)of even degree 2g, with integer coefficients and constant
term 1, such that

ζL(s)=p(q−s)/( 1 −q−s)( 1 −q^1 −s),

and that the zeta-function satisfies the functional equation

q(g−^1 )sζL(s)=q(g−^1 )(^1 −s)ζL( 1 −s).

The non-negative integergis thegenusof the field of algebraic functions.
The analogue of the Riemann hypothesis, that all zeros ofζL(s)lie on the line
Rs= 1 /2, is equivalent to the statement that all zeros of the polynomialp(u)have
absolute valueq−^1 /^2 , or that the numberNof prime divisors with normqsatisfies the
inequality

|N−(q+ 1 )|≤ 2 gq^1 /^2.

This analogue has beenprovedby Weil (1948). A simpler proof has been given by
Bombieri (1974), using ideas of Stepanov (1969).
The theory of function fields can also be given a geometric formulation. The prime
divisors of a function fieldLwith field of constantsKcan be regarded as the points
of a non-singular projective curve overK, and vice versa. Weil (1949) conjectured
far-reaching generalizations of the preceding results for curves over a finite field to
algebraic varieties of higher dimension.
LetVbe a nonsingular projective variety of dimensiond, defined by homogeneous
polynomials with coefficients inZ.Foranyprimep,letVpbe the (possibly singular)
variety defined by reducing the coefficients modpand consider the formal power
series

Zp(T):=exp

(∑

n≥ 1

Nn(p)Tn/n

)

,

whereNn(p)denotes the number of points ofVpdefined over the finite fieldFpn.
Weil conjectured that, ifVpis a nonsingular projective variety of dimensiondover
Fp,then

(i) Zp(T)is a rational function ofT,
(ii) Zp( 1 /pdT)=±pde/^2 TeZp(T)for some integere,
(iii)Zp(T)has a factorization of the form

Zp(T)=P 1 (T)···P 2 d− 1 (T)/P 0 (T)P 2 (T)···P 2 d(T),

whereP 0 (T)= 1 −T,P 2 d(T)= 1 −pdTandPj(T)∈Z[T]( 0 <j< 2 d),

(iv) Pj(T)=

∏bj

k= 1 (^1 −αjkT),where|αjk|=p

j/ (^2) for 1≤k≤b
j,(^0 <j<^2 d).