Pole bounds for the Jacobi quotient at a finite divisor

For real $r$ let $G_{r}T$ be the span of $s^{-p}b\mathbf{v}$ with $p\ge 0$, $b\in B_s$, and homogeneous tensor $\mathbf{v}$ of weight $\tau$ satisfying $p+\tau \le r$. We check the reduction in F1 preserves this filtration. For a relation obtained from a positive integral weight state $u\in V_d$ and a base tuple of weight $\tau$, its designated $u_{-1}$ term has weight $\tau +d$. Ordinary-mode terms have weight $\tau +d-k-1$; their coefficient has pole order at most $k+1$. Opposite-mode terms $u_{-1-k}^{\ast }$ have weight $\tau -d-k$ and their coefficients are regular at every finite divisor. These bounds cover every term of $A_i$ and $C$, including $z_a=0$ and $z_a=z_b$. Thus each has pole order plus tensor weight at most $\tau +d$.

In $D$, the last opposite-mode term has weight $\tau -d$. A term $L(1{)}^a{u}_{k}L(1{)}^b$ in $K_i(u_k)$ has weight shift $d-k-1-a-b$ and scalar factor a constant times $z_i^{2(d-k-1)-a-b}$. After multiplication by $z_i^{k+1}$ in $D$, its scalar power is $z_i^{2d-k-1-a-b}$. At a collision divisor, or a zero divisor other than $z_i=0$, this factor is regular. At $z_i=0$ its pole order is $\max (0,k+1+a+b-2d)$; adding its weight shift gives either $d-k-1-a-b\le d$ or $-d\le d$. All terms of $D$ therefore satisfy the same bound. Multiplication by an existing $s^{-p}b$ preserves the bound additively. Every non-designated term also has strictly smaller tensor weight, exactly as in F1.

Apply the terminating reductions of F1, keeping the scalar factors rather than discarding their poles. Steps 1.1–2.1 show that every element of $G_{r}T$ has a representative in $(R_n\otimes E)\cap G_{r}T$ modulo $J\cap G_{r}T$. Let $B$ be a common lower bound for the total tensor weights. Choose $S$ such that $\sigma +S$ is a nonnegative integer and $S+B\ge 0$. Set $b_k=w_0\otimes \cdots \otimes L(-1{)}^k{w}_l\otimes \cdots \otimes w_{n+1}$ and choose a reduced representative $v_k\in (R_n\otimes E)\cap G_{\sigma +k}T$ with $b_k-v_k\in J$. In each term $s^{-p}b\mathbf{v}$ of $v_k$, the bound $p+\mathrm{w}\mathrm{t}(\mathbf{v})\le \sigma +k$ implies $p\le \sigma +k+S$. Consequently $q_k=s^{\sigma +k+S}{v}_k$ belongs to the finite $B_s$-module $B_s\otimes E$. The exponent is an integer for every $k$.

The submodule generated by $q_0,q_1,\dots$ is finite by F2. Express a finite generating set as finite combinations of these vectors; for some $m\ge 1$ the initial segment $q_0,\dots ,q_{m-1}$ therefore generates the entire submodule. There are $c_j\in B_s$ such that $q_m+{\sum }_{j<m}{c}_j{q}_j=0$. Divide in $R_n\otimes E$ by $s^{\sigma +m+S}$ to get $v_m+{\sum }_{j<m}{c}_j{s}^{j-m}{v}_j=0$. Subtracting the relations $b_k-v_k\in J$ gives the same identity for the classes $[b_k]\in T\mathrm{/}J$. Applying F1's annihilating coefficient map and its derivative identity proves the displayed equation. No differentiation of the chosen representatives $v_k$ is used.

Write $ϵ={\mathrm{\partial }}_{z_l}s\in \{1,-1\}$ and keep the other coordinates fixed, so ${\mathrm{\partial }}_{z_l}=ϵ{\mathrm{\partial }}_s$. After division by ${ϵ}^m$ the equation has the same form in ${\mathrm{\partial }}_s$, with coefficients ${\tilde{c}}_j={ϵ}^{j-m}{c}_j$ holomorphic at the chosen smooth divisor point. Put $U_j=s^j{\mathrm{\partial }}_s^{j}f$, $0\le j<m$. Then $s{\mathrm{\partial }}_s{U}_j=j{U}_j+U_{j+1}$ for $j<m-1$, and $s{\mathrm{\partial }}_s{U}_{m-1}=(m-1)U_{m-1}-{\sum }_{j<m}{\tilde{c}}_j{U}_j$. Hence $s{\mathrm{\partial }}_{s}U=A(s)U$ with $A$ holomorphic there, the explicit Fuchsian first-order system. This proves precisely the asserted pole bound and its Fuchsian interpretation.