(pp. 206-229)

Let$\phi {\text{ }} = {\text{ }}{\phi _f} \otimes {\phi _\infty } \in {\text{ }}\overline \mathcal{S} {(\mathbb{V}\, \times {\text{ }}\mathbb{A}{\mathbb{Q}^ \times })^{U \times {\text{ }}U}}$be a Schwartz function with standard${\phi _\infty }.$Assume that$ - 1\; \notin \;U$to simply notations. Recall that in §5.1 we have introduced the generating series

$Z{(g,\;\phi )_U} = {Z_0}{(g,\;\phi )_U} + {Z_*}{(g,\;\phi )_U},\quad \;g\; \in \;{\text{G}}{{\text{L}}_2}(\mathbb{A})$.

Here the non-constant part

${Z_*}{(g,\;\phi )_U} = \sum\limits_{a \in {F^ \times }} {\sum\limits_{x \in K\backslash \mathbb{B}_f^ \times } {r(g)\phi {{(x)}_a}} } \;Z{(x)_U}$.

We further have the height series as follows:

$Z{(g,\;({h_1},\;{h_2}),\;\phi )_U} = \;{\langle Z{(g,\;\phi )_U}\;[{h_1}]_U^ \circ ,\;[{h_2}]_U^ \circ \rangle _{{\text{NT}}}},{\text{ }}{h_1},\;{h_2}\; \in \;{\mathbb{B}^ \times }$;

$Z{(g,\;\chi ,\;\phi )_U} = \;\int_{T(F)\backslash T(\mathbb{A})/Z(\mathbb{A})}^* {Z{{(g,\;(t,\;1),\;\phi )}_U}\;\chi (t)\,dt} $.

By Lemma 3.19,$Z{(g,\;({h_1},\;{h_2}),\;\phi )_U}$is cuspidal in*g*. So we can replace$Z{(g,\;\phi )_U}$by${Z_*}{(g,\;\phi )_U}$in the definition of$Z{(g,\;({h_1},\;{h_2}),\;\phi )_U}$. The constant term${Z_0}{(g,\;\phi )_U}$will be ignored in the rest of this book.

The goal of this chapter is to decompose the height series

$Z{(g,\;({t_1},\;{t_2}),\;\phi )_U} = \;{\langle {Z_*}{(g,\;\phi )_U}\;[{t_1}]_U^ \circ ,\;\;[{t_2}]_U^ \circ \rangle _{{\text{NT}}}},\quad \;{t_1},\;{t_2}\; \in \;{\mathbb{B}^ \times }$.

We presume the assumptions in §5.2. Then there is not horizontal self intersection in...