The proof of Theorem 1.1 involves many remarkable phenomena, and we wish to explain its structure in this chapter. To this end, we will give the first of a succession of statements, each of which implies the theorem. Passing from each statement to the next is a nontrivial reduction that changes the nature of the problem to be solved. We will outline the ideas of these reductions here and tackle them in detail in subsequent chapters.

**Statement A.** *We have H*_{Γ} = *H*_{Δ}.

This reduction was already mentioned in the first chapter, where Statement A appeared as Theorem 1.2.

The proof that this implies Theorem 1.1 is Theorem 1 of [15]. We review the idea of the proof. To prove the functional equations that *Z*_{Ψ} (** s**;

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