Log24

Update of Feb. 19: 

Actually, the cubic equation discussed
by Frenkel and by Taylor (see below) is 

Y ² + Y = X ³ – X ² 

whereas the equation given by Weil,
quoting Tate, is

Y ² – Y = X ³ – X ² . 

Whether this is a misprint in Weil's book,
I do not know.

At any rate, the cubic equation discussed by
Frenkel and earlier by Taylor is the same as
the cubic equation discussed in greater detail
by Henri Darmon in "A Proof of the Full
Shimura-Taniyama-Weil Conjecture Is
Announced," AMS Notices , Dec.1999.

For further background, see (for instance)
John T. Tate, "The Arithmetic of Elliptic
Curves," in Inventiones Mathematicae
Volume 23 (1974), pp. 179 – 206, esp. pp.
200-201.

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 p . Such results are referred to as “reciprocity laws.” In the 1920s, Emil Artin gave what was then thought to be the most general reciprocity law possible—his abelian reciprocity law. However, Artin’s reciprocity still only applied to very special equations—equations with only one variable that have “abelian Galois group.”

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 [sic ] law for the two variable equation

Y ² + Y = X ³ – X ².

He showed that the number of solutions to this equation in arithmetic modulo a prime number p  differs from p  [in the negative direction] by the coefficient of q^p in the formal (infinite) product

q (1 – q² )(1 – q¹¹)² (1 – q²)² 
(1 – q ²² )² (1 – q ³)² (1 – q ³³)² 
(1 – q ⁴)² …  =  
q – 2q² – q³ + 2q⁴ + q⁵ + 2q⁶
– 2q⁷ – 2q⁹ – 2q¹⁰ ​+ q¹¹ – 2q¹² + . . .

For example, you see that the coefficient of q⁵ is 1, so Eichler’s theorem tells us that

Y ² + Y = X ³ − X ²

should have 5 − 1 = 4 solutions in arithmetic modulo 5. You can check this by checking the twenty-five possibilities for (X,Y) modulo 5, and indeed you will find exactly four solutions:

(X,Y) ≡ (0,0), (0,4), (1,0), (1,4) mod 5.

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.