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