Convergence of finite logarithmic Euler systems with parameters

Work with one alpha. Multiplication by the matrices $A_i$, whose q-powers are nonnegative integers, preserves its exponent coset. Distinct chosen cosets are linearly independent as formal generalized-power series, so the system holds on each alpha-summand. Let B be the finite-dimensional vector space of $C^d$-valued polynomials in $L_1,\mathrm{.}\mathrm{.}\mathrm{.},L_R$ with degree bounds $K_i$. Give it the sum norm on its coefficients, using a fixed vector norm on $C^d$. On B, ${\mathrm{\partial }}_{L_i}$ has norm at most $K_i$.

For the module assertion, F2 supplies finitely many generators. Replace them by their finitely many joint generalized-weight components, still a finite generating set. On a joint weight lambda define $N_i=L_i(0)-{\lambda }_i$. F3 shows that $N_i$ commutes with every component mode as an operator between the appropriate generalized-weight spaces: the scalar weight shift exactly cancels the conformal commutator. Independently of the fixed formal-solution bounds $K_i$ in step 1.1, choose nonnegative integers $J_i$ for $1\le i\le k$ so that $J_i+1$ is at least the nilpotence exponent of $N_i$ on every chosen generator. Then $N_i^{J_i+1}$ annihilates every mode word on those generators, and hence $N_i^{J_i+1}=0$ on the whole module. No boundedness of dimensions of all weight spaces is being inferred from this argument.

Expand $A_i(q,b)={\sum }_{m\ge 0}{A}_{i,m}(b)q^m$. The coefficient equation at n is $(({\alpha }_i+n_i)I-A_{i,0}+{\mathrm{\partial }}_{L_i})P_{\alpha ,n}={\sum }_{0<m\le n}{A}_{i,m}{P}_{\alpha ,n-m}\mathrm{.}$ Here $m\le n$ is componentwise and $m>0$ means m is not the zero vector. Fix a compact parameter subset H. The norms of all $A_{i,0}$ on H and the finitely many alpha_i and $K_i$ have one bound. For $\mathrm{\mid }n\mathrm{\mid }=\sum n_i$ sufficiently large, choose i with $n_i\ge \mathrm{\mid }n\mathrm{\mid }\mathrm{/}R$. The operator on the left is $n_i$ I plus a uniformly bounded operator on B. Its geometric-series inverse has norm at most $2\mathrm{/}{n}_i\le 2R\mathrm{/}\mathrm{\mid }n\mathrm{\mid }$. This choice uses no relations between the $A_i$. Resonances at the finitely many smaller n cause no difficulty because their formal coefficients have already been given.

Choose radii $0<s_i<{\rho }_i<r_i$. F1 gives a constant $M_H$ such that $\mathrm{\parallel }{A}_{i,m}(b)\mathrm{\parallel }\le M_H{\prod }_j{\rho }_j^{-m_j}$ for every i,m,b in H. Put $d_n={\sup }_{b\in H}\mathrm{\parallel }{P}_{\alpha ,n}(\cdot ,b)\mathrm{\parallel },e_n=d_n{\prod }_j{s}_j^{n_j},t_j=s_j\mathrm{/}{\rho }_j<1.$ For large |n|, step 2.2 gives $e_n\le \frac{2R{M}_H}{\mathrm{\mid }n\mathrm{\mid }}{\sum }_{0<m\le n}\left({\prod }_j{t}_j^{m_j}\right)e_{n-m}\mathrm{.}$ The full positive sum $S={\sum }_{m>0}{\prod }_j{t}_j^{m_j}={\prod }_j(1-t_j{)}^{-1}-1$ is finite. Choose N larger than the inverse threshold in step 2.2 and larger than 2RM_H S. Choose $C_H$ bounding all $e_n$ with $\mathrm{\mid }n\mathrm{\mid }\le N$. Induction on |n| then yields $e_n\le C_H$ for every n, since every n-m on the right has smaller total degree and the displayed bound is at most $C_H$ times 2RM_H S/|n|.

To obtain convergence on a prescribed compact q-subset choose smaller radii $u_i<s_i$ containing that subset. Step 3.1 gives ${\sum }_n{d}_n{\prod }_i{u}_i^{n_i}\le C_H{\prod }_i(1-u_i\mathrm{/}{s}_i{)}^{-1}<\mathrm{\infty }\mathrm{.}$ There are only finitely many alpha and logarithmic monomials. Their factors $q^{\alpha }$ and powers of the chosen logarithms are bounded on every compact subset of the covering domain. Thus the entire series is absolutely locally uniformly convergent with the stated coefficient bounds. The same argument on slightly larger compact subsets permits termwise q and parameter differentiation by Cauchy's formula; its sum is holomorphic and solves the system. Since $s_i,{\rho }_i$ can approach $r_i$ arbitrarily closely, this is convergence on the entire allowed punctured polydisc, not merely some unspecified neighborhood of zero.

F2 gives finitely many joint integer cosets and coordinatewise lower bounds. In each occurring coset choose a representative alpha and subtract its coordinatewise minimum integer shifts; the resulting representative need not occur, but every occurring weight has the form alpha+n with n in $N^R$. Keep one representative per distinct coset. Step 2.1 bounds every logarithmic degree in $q^{L(0)}$ by the separately chosen module bound $J_i$. Consequently, when the generalized-weight sewing series satisfies an actual finite matrix Euler system, apply the already proved formal-series assertion afresh with $R=k$ and log-degree bounds $K_i:=J_i$. That assertion holds for every choice of nonnegative finite bounds, so this application needs no relation between the module bounds $J_i$ and the bounds of the initially given formal solution $F$. It proves convergence of this module series. The existence of that geometric matrix system and its holomorphic coefficient domain are separate obligations.