Edward Frenkel on Eichler's reciprocity law

(*Love and Math *, Kindle edition of 2013-10-01,

page 88, location 1812)—

"It seems nearly unbelievable that there

would be a rule generating these numbers.

And yet, German mathematician Martin

Eichler discovered one in 1954.^{11 }"

"11. I follow the presentation of this result

given in Richard Taylor, *Modular arithmetic:
driven by inherent beauty and human
curiosity *, The Letter of the Institute for

Advanced Study [IAS], Summer 2012,

pp. 6– 8. I thank Ken Ribet for useful

comments. According to André Weil’s book

*Dirichlet Series and Automorphic Forms*,

Springer-Verlag, 1971 [pp. 143-144], the

cubic equation we are discussing in this

chapter was introduced by John Tate,

following Robert Fricke."

Actually, the cubic equation discussed
whereas the equation given by Weil,
Whether this is a misprint in Weil's book,
At any rate, the cubic equation discussed by
For further background, see (for instance) |

**Richard Taylor, op. cit. **—

One could ask for a similar method that given any number of polynomials in any number of variables helps one to determine the number of solutions to those equations in arithmetic modulo a variable prime number
Stunningly, in 1954, Martin Eichler (former IAS Member) found a totally new reciprocity law, not included in Artin’s theorem. (Such reciprocity laws are often referred to as non-abelian.) More specifically, he found a reciprocality [
He showed that the number of solutions to this equation in arithmetic modulo a prime number
For example, you see that the coefficient of
should have 5 − 1 = 4 solutions in arithmetic modulo 5. You can check this by checking the twenty-five possibilities for (
( Within less than three years, Yutaka Taniyama and Goro Shimura (former IAS Member) proposed a daring generalization of Eichler’s reciprocity law to all cubic equations in two variables. A decade later, André Weil (former IAS Professor) added precision to this conjecture, and found strong heuristic evidence supporting the Shimura-Taniyama reciprocity law. This conjecture completely changed the development of number theory. |

With this account and its context, Taylor has

perhaps atoned for his ridiculous remarks

quoted at Log24 in The Proof and the Lie.