Reciprocity Laws. From Euler to Eisenstein.

Long ago, after I had learned about the cubic reciprocity law in Prof. Ulrich Felgner's lectures on algebraic number theory, he mentioned that higher reciprocity laws existed and that they would be studied in something called class field theory. Spying a book on "Klassenkörpertheorie" among the used books in the book store at the University of Tübingen, I opened it with great expectations only to find that it seemed to be about everything but number theory: no numbers, no reciprocity laws I could recognize, no nothing (the book was Neukirch's first book on class field theory - the one that starts by building up cohomology and local class field theory).
When it became clear eventually that my notes on reciprocity laws would turn into a book, I made sure that it had lots of numbers in it (not to mention lots of exercises to sink your teeth in). Some of the problems involving the integers that were the first object of study of every number theorist are described below.

1. The Genesis of Quadratic Reciprocity

Euler conjectured the quadratic reciprocity laws after years of work on questions inspired by Fermat. We discuss in detail what exactly is missing in Legendre's attempted proofs and give (crude) description of the eight proofs due to Gauss.

Gauss's second proof of the quadratic reciprocity law was based on genus theory of quadratic forms, and the first proofs of Kummer's reciprocity laws as well as Takagi's class field theory used general versions of genus theory. In this chapter, the genus theory for quadratic number fields is presented, the Lucas-Lehmer test for Mersenne numbers is discussed, and we close with a few remarks on Hilbert symbols and K2Z.

Applications of quadratic reciprocity to integers abound; a few that will be generalized later on are the following:

If p and q = 2p+1 are odd primes, then q | 2p-1 if and only if p = 3 mod 4;
If p and q = 4p+1 are odd primes, then q | 22p+1.
If Un denote the Fibonacci numbers U1 = 1, U2 = 1, Un+1 = Un + Un-1, then primes p different from 5 divide Up-1 if (p/5) = +1, and Up+1 if (p/5) = -1.

Let me also mention a conjecture that I hope to be able to prove in the second volume: Let Fn denote the n-th Fermat number; it is well known that Fn is prime if and only if 322n-1 = -1 mod Fn. The conjecture says that we may save that last n multiplications because Fn is prime if and only if (-3)22n-n-1 = -8 mod Fn.

3. Cyclotomic Number Fields

One of the most beautiful proofs of the quadratic reciprocity laws is the one that compares the decomposition of primes in quadratic and cyclotomic extensions. It became the role model for many proofs of reciprocity laws by comparing the splitting of primes in Kummer extensions and class fields.
We also explain how Pocklington's primality test and the Lucas-Lehmer test fit into the frame of Lenstra's primality test.

4. Power Residues and Gauss Sums

We define power residue symbols in number fields and investigate how symbols in different fields are related. Gauss and Jacobi sums are introduced and studied, and we also present a short discussion of Eisenstein sums.

5. Rational Reciprocity Laws

We discuss the contributions of Dirichlet to this subject and go on demonstrating that his results were the first in a long series of papers that were completely forgotten in our century until they have been rediscovered (often several times). The main focus is on the rational reciprocity laws of Scholz, Burde and Lehmer.
Among the beautiful applications are Reichardt's discovery that the diophantine equation x4 - 17y2 = 2z2 has no nontrivial solutions, or Rédei's result that, for primes p = 1 mod 4 and q = 3 mod 4, the following assertions are equivalent:
qx2 + py2 = z4 has a solution with gcd(x,y) = 1 and odd z;
the class number of the complex quadratic field with discriminant -qp is divisible by 8; (-q/p)_4 = 1: -q is a quartic residue modulo p

6. Quartic Reciprocity

Apart from giving the proofs of rational quartic reciprocity by comparing the splitting of primes in certain extensions, of the full quartic reciprocity law in Z[i] using Gauss sums, and of the quartic reciprocity law in the field of eighth roots of unity, we discuss several applications.

Let q be an odd integer and suppose that p = 4q+1 is prime. Then Sq = 22q+1 is never prime because of the (Aurifeullian) factorization
Sq = Aq Bq, where Aq = 2q - 2(q+1)/2 + 1 and Bq = 2q + 2(q+1)/2 + 1.
Now p = 4q+1 = 5 mod 8, hence the quadratic reciprocity law shows that
Sq = 2(p-1)/2 +1 = (2/p)+ 1 = 0 mod p. Thus p divides AqBq,
and the question (first posed by Brillhart [Concerning the numbers 22p+1, p prime, Math. Comp. 16 (1962), 424-430]) is: which? Since quadratic reciprocity has told us that p | AqBq, we might hope that quartic reciprocity will answer this question. And it does: if we write p = a2 + b2 as a sum of squares with b even, then the result is that
p | Aq if and only if b/2 = 3, 5 mod 8, and
p | Bq if and only if b/2 = 1, 7 mod 8.

7. Cubic Reciprocity

There's a problem related to Brillhart's that can be solved using cubic reciprocity: Let q be an odd integer, u = 1, 5 mod 6, and let p = 6qu+1 be prime. Write p = a2 + ab + b2 with 3 | b and
a+b = 1 mod 4 if 2 | b,
b = 1 mod 4 if 2 | a,
a = 3 mod 4 if ab is odd.
If 3(p-1)/u = 1 mod p, then p | 33q+1 = KqLqMq, where
Kq = 3q+1, Lq = 3q - 3(q+1)/2 + 1, Mq = 3q + 3(q+1)/2 + 1,
and we have
p | Kq iff 9 | b,
p | Lq iff (-1)(p+1)/4 = + (3/u) b/3 mod 3,
p | Mq iff (-1)(p+1)/4 = - (3/u) b/3 mod 3.
Here we need the sextic (i.e. the cubic and the quadratic) reciprocity law in the ring of Eisenstein integers to prove this result.

8. Eisenstein's Analytic Proofs

One of the most surprising proofs of the quadratic reciprocity law is Eisenstein's proof using the sine function. Replacing trigonometric functions (Z-periodic holomorphic functions) by elliptic functions (say Z[i]-periodic meromorphic functions), the quartic reciprocity law follows just as easily. We also discuss proofs of the quadratic reciprocity laws in quadratic number fields and mention in passing a connection to Kronecker's Jugendtraum.

9. Octic Reciprocity

Brillhart's problem for the numbers Sq has an octic analogue: if q and u are odd integers, if p = 8qu+1 = a2+b2 = c2+2d2 is prime (with b even), and if p | S_q = 22q+1, then
p | Aq iff d + b/4u = 1, 7 mod 8,
p | Bq iff d + b/4u = 3, 5 mod 8.
The proof uses the octic reciprocity law, whose proof is based on octic elliptic Gauss sums.

10. Gauss's Last Entry

It is well known that Gauss recorded many of his discoveries in a diary; it ends with the `Last Entry' from July 9, 1814, which reads as follows:
I have made by induction the most important observation that connects the theory of biquadratic residues with the lemniscatic functions. Suppose that a+bi is a prime number, a-1+bi divisible by 2+2i, then the number of all solutions to the congruence 1 = x^2 + y^2 + x^2y^2 mod a+bi including four infinite solutions is (a-1)2+b2.
We discuss its proofs and sketch the development through zeta functions and Artin's thesis up to the Weil conjectures. We also sketch how to count the number of points over Fp of Eisenstein's hyperelliptic curve y2 = x8 - 28x6 + 70 x4 - 28x2 +1.

11. Eisenstein Reciprocity

The historical importance of Eisenstein's Reciprocity Law lies with the fact that there wasn't a single proof of higher (i.e. Kummer's) reciprocity laws that did not use Eisenstein Reciprocity in an essential way before the work of Takagi. We also discuss Stickelberger elements, their generalization, and applications to the class group of abelian extensions of Q.

Appendix

Dramatis Personae

A list of mathematicians who worked on reciprocity and related questions.

Chronology of Proofs

A list of almost 200 published proofs of the quadratic reciprocity law.

Some open problems 