Chapter 8 - Functional equations of Hecke L-functions

Summary

In this chapter, after giving a brief summary of the notion of adeles of number fields, we prove the functional equations of Hecke L-functions. For further study of adeles and class field theory, we recommend [W1] and [N].

Adelic interpretation of algebraic number theory

Let us start with the explanation of the adele ring of Q denoted by A. To define the topological ring A, we consider the module Q/Z and its ring End(Q/Z) of additive endomorphisms. Let Q/Z[p^∞] be the subgroup of Q/Z consisting of elements killed by some power of p for a prime p of Z. Then by definition, for any two distinct primes p and q, Q/Z[p^∞]∩Q/Z[q^∞] = {0} and hence ⊗_pQ/Z[p^∞] ⊂ Q/Z. Let us show that this inclusion is in fact surjective. For any given rational number r, we expand r into the standard p-adic expansion ∑^∞_n=-m c_npⁿ defined in §1.3 and put [r]_p = ∑^-1_n=-mc_npⁿ (the p-fraction part). Since r-[r]_p ∈ Z_p∩Q, p does not appear in the denominator of r-[r]_p. Repeating this process of taking out the p-fraction part from r for all prime factors of the denominator of r, we know that r-∑_p[r]_p ∈ Z and hence r = ∑_p[r]_p in Q/Z. Since [r]_p ∈ Q/Z[p^∞], we know that ⊗_PQ/Z[p^∞] = Q/Z. The above process can be applied to p-adic numbers r ∈ Q_p. Then r ↦ [r]_p plainly induces an isomorphism: Q_p/Z_p ≅ Q/Z[p^∞].