1’2s ’1

bm m’s = ’s

1’b

L(g0 , s) = + ,

m=1 primes ∈S

where S is a ¬nite set of bad primes (in our example, S = {5, 17, 37}). Write

1 ’ b X + X 2 = (1 ’ ± X)(1 ’ β X).

The symmetric square L-function is de¬ned to be

’1

L(Sym2 g0 , s) = (1 ’ ±2 ’s

)(1 ’ β 2 ’s ’s

)(1 ’ ± β ) .

∈S

There exists a naturally de¬ned transcendental number „¦ (similar to the pe-

riods considered in Section 9.4), de¬ned by a double integral, such that

L(Sym2 g0 , 2)

= r = a rational number.

„¦

The number n is de¬ned to be the p-part of r (that is, n is a power of p such

that r equals n times a rational number with numerator and denominator

prime to p).

The formula that Wiles proved is therefore that L(Sym2 g0 , 2)/„¦ equals #TA

times a rational number prime to p. This means that the order of an algebraic

object, namely TA , is expressed in terms of the value of an analytic function,

in this case the symmetric square L-function. This formula is therefore of

a nature similar to the analytic class number of algebraic number theory,

which expresses the class number in terms of an L-series, and the conjecture

of Birch and Swinnerton-Dyer (see Section 14.2), which expresses the order

of the Shafarevich-Tate group of an elliptic curve in terms of the value of its

L-series.

© 2008 by Taylor & Francis Group, LLC