RECIPROCITY LAWS 199

is a polynomial analogue to the ζ -function, and the analogous

statement, called the Riemann Hypothesis for function ﬁelds, was

conjectured and proved in the twentieth century.

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There is yet a third property of conjectures worth mentioning:

They should be interesting, although two mathematicians will

not always agree about what is interesting. The most interesting

conjectures should imply known results, and also should imply

some new and surprising results.

Kinds of Black Boxes

In the reciprocity laws or conjectures of the type we consider in this

book, we always assert the identity of two sequences of numbers.

One of them is always the sequence of traces of the Frobenius

matrix for the unramiﬁed primes of some Galois representation.

The other is a sequence derived as the output of some black box,

which is usually an algebraic, geometric, or topological object. Some

of these black boxes are:

1. The number of solutions to certain systems of Z-equations

modulo various primes.

2. The Fourier coefﬁcients of a modular form corresponding to

the primes.

3. Traces of Hecke operators at the various primes.

We cover the ﬁrst two cases in this book: (1) in chapters 18 and

19, (2) in chapter 21. The third case is crucial for much of the

current research on generalized reciprocity laws, but unfortunately

is beyond the scope of this book. For an expository article on Hecke

operators in cohomology and their use in reciprocity laws, see our

paper (Ash and Gross, 2000).

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An excellent book about the Riemann Hypothesis is (Derbyshire, 2003).

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