A little bit of number theory

Editorial comments: 

• This is a short note written to illustrate some examples of how the conjectures worked out in very explicit examples.
• (3/16/2017, AVP) At the author's suggestion, Anthony Pulido worked through theorem 2 for $p = 3 \equiv 3 \pmod{4}$ and $p = 5 \equiv 1 \pmod{4}$. This is the note titled Examples. This exercise led to writing a short Haskell program, lbnt.hs, which verifies theorem 2 for the first 100 odd primes.

Author's comments: I am not sure exactly when this text was written. Internal evidence and memory together suggest that it was early in 1973. The internal evidence cannot be interpreted literally, as I was unlikely to be sure even in 1973 exactly when the letter to Weil was written.

The examples are of the type I had in mind when writing that letter. I had not, however, at that time formulated any precise statements. Indeed, not being aware of the Shimura-Taniyama conjecture and not having any more precise concept of what is now known as the Jacquet-Langlands correspondence than that implicit in the letter, I was in no position to provide the examples of the present text, some of which exploit results that had become available in the intervening years. The formulas are as in the original text. I did not repeat the calculations that lead to them.

I have never found anyone else who found the type of theorem provided by the examples persuasive, but, apart from the quadratic reciprocity law over the rationals, explicit reciprocity laws have never had a wide appeal, neither the higher reciprocity laws over cyclotomic fields nor simple reciprocity laws over other number fields (Dedekind: Über die Anzahl der Idealklassen in reinen kubischen Zahlkörpern).

The conjecture referred to in the text as the Weil conjecture is now usually referred to as the Shimura-Taniyama conjecture.