<< ًٌٍ. 2(âٌهمî 2)رخؤإذئہحبإ
âˆ‍
1âˆ’2s âˆ’1
bm mâˆ’s = âˆ’s
1âˆ’b
L(g0 , s) = + ,
m=1 primes âˆˆS

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

1 âˆ’ b X + X 2 = (1 âˆ’ خ± X)(1 âˆ’ خ² X).

The symmetric square L-function is deï¬پned to be
âˆ’1
L(Sym2 g0 , s) = (1 âˆ’ خ±2 âˆ’s
)(1 âˆ’ خ² 2 âˆ’s âˆ’s
)(1 âˆ’ خ± خ² ) .
âˆˆS

There exists a naturally deï¬پned transcendental number â„¦ (similar to the pe-
riods considered in Section 9.4), deï¬پned by a double integral, such that

L(Sym2 g0 , 2)
= r = a rational number.
â„¦
The number n is deï¬پned 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)/â„¦ 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.

آ© 2008 by Taylor & Francis Group, LLC

 << ًٌٍ. 2(âٌهمî 2)رخؤإذئہحبإ