Explicit algebraic KZB-type connections and the maximal metabelian de Rham period map for punctured curves of genus at least two

For a once-punctured elliptic curve, Moore showed that the algebraic Knizhnik–Zamolodchikov–Bernard (KZB) connection of Levin–Racinet and Luo represents Kim's universal unipotent connection, and used it to write the maximal metabelian quotient of Kim's de Rham period map explicitly in terms of elliptic polylogarithms — which the author states is, as far as he is aware, the first such formula on any curve of positive genus. The paper states that no explicit algebraic connection of the required local form is known on any curve of genus at least 2, and that it is unknown whether the analytic gauge transformations of Enriquez and Enriquez–Zerbini on Schottky covers descend to algebraic connections. This problem asks for an explicit algebraic KZB-type connection (or any connection of the specified form) on punctured curves $X = C-P$ of genus $g\ge2$ over number fields, valued in the maximal metabelian quotient of the free pronilpotent Lie algebra and satisfying Moore's conditions on its local form, and for the resulting closed formula for the maximal metabelian quotient of the de Rham period map, generalizing Moore's genus-one formulas. All higher-genus constructions of such connections are analytic over the complex numbers, and all algebraic constructions of them are confined to genus one; no algebraic connection of the required local form is known on any curve of genus $g \ge 2$.

Background

A hyperbolic curve is a smooth algebraic curve, defined over the complex numbers or over a number field $K$, obtained from a smooth projective curve $C$ of genus $g$ by removing a nonempty finite set $P$ of points; its topological fundamental group is free, and one studies its enriched linear-algebraic analogues. The unipotent (or pronilpotent) fundamental group $U$ is the inverse limit of the nilpotent quotients of the topological fundamental group, reorganized by the Tannakian formalism (the framework that reconstructs a generalized symmetry group from the category of its representations equipped with a fiber functor): to each embedding of the ground field into $\mathbb C$ (or $\mathbb Q_p$) corresponds a realization, and the de Rham realization $U^{dR}$ is a prounipotent group over $K$ dual to iterated integrals of regular algebraic differential forms. The maximal metabelian quotient $W = U/[[U,U],[U,U]]$ is the largest quotient whose Lie algebra $\mathfrak w$ is metabelian, meaning its commutator subalgebra is abelian; writing $X_1,\dots,X_n$ for the $n=2g$ generators $A_1,B_1,\dots,A_g,B_g$, its Lie algebra has an explicit basis built from the generators and the adjoint-iterated commutators $\sigma_{\mathbf u}|_{i,j} = \mathrm{ad}_{X_n}^{u_n}\cdots \mathrm{ad}_{X_1}^{u_1}[X_i,X_j]$ indexed by multi-indices $\mathbf u$, and it is in a precise sense the largest algebraically comprehensible quotient of $U$. The Knizhnik–Zamolodchikov–Bernard (KZB) connection — named after the physicists Knizhnik, Zamolodchikov, and Bernard — is a flat connection generalizing the genus-zero Knizhnik–Zamolodchikov connection to curves of positive genus: on a punctured elliptic curve it is the explicit algebraic connection constructed by Levin–Racinet from the Kronecker function (the elliptic analogue of the kernel out of which classical polylogarithms are built by iterated integration) and developed by Luo, which Moore showed represents the universal unipotent connection of the curve (the Tannakian group object classifying unipotent local systems on the curve, recoverable from iterated integrals). In the arithmetic context of Kim's Selmer varieties (nonabelian generalizations of Selmer groups used to bound rational points on curves, e.g. in nonabelian Chabauty), the de Rham period map $j^{dR}$ assigns to a $K$-point of the curve the de Rham realization of the motivic path from a fixed basepoint; computing $j^{dR}$ explicitly on a positive-genus curve requires an explicit connection representing the universal unipotent connection, and Moore carried this out for genus one, expressing the metabelian quotient of $j^{dR}$ via elliptic polylogarithms (iterated integrals of the elliptic curve's regular differentials). For genus $g\ge2$, Enriquez constructed a KZB-type flat connection on Schottky covers of configuration spaces of complex curves (Schottky uniformization expresses a Riemann surface as a quotient of the complex plane by a free group of Möbius transformations), and Enriquez–Zerbini produced explicit Maurer–Cartan elements (one-forms satisfying the Maurer–Cartan flatness equation $d\omega+\tfrac12[\omega,\omega]=0$, the algebraic condition for a connection to be flat) and, in principle, gauge transformations (invertible changes of trivialization $g$ acting on connection one-forms by $\omega \mapsto -dg\,g^{-1} + g\,\omega\,g^{-1}$) under which the connection descends to the curve itself; however, these are analytic constructions over $\mathbb C$, and it is not known whether they produce algebraic connections on the compact algebraic curve. Moore's introduction pins down exactly what would suffice for the period-map application: a connection on $X = C-P$ built from $\mathfrak w$-valued one-forms $\omega_X, \omega_Y$ on an open cover $X, Y \ni P$ and a gauge transformation $g$ on $X \cap Y$, satisfying (1) $\omega_X = \sum_{i=1}^{g}(\alpha_i A_i + \beta_i B_i)$ modulo higher-degree words, where $\{\alpha_1,\dots,\alpha_g,\beta_1,\dots,\beta_g\}$ is a basis of $H^1_{dR}(X/K)$ and the $\alpha_i$ extend to global one-forms on $C$; (2) $\omega_X$ and $\omega_Y$ take values in the Lie subalgebra of words of degree at most 1 in the $A_i$ (not necessarily Galois-equivariant); (3) $g$ acts by left multiplication by words in the $B_i$. The relation $[A_1,B_1]+\cdots+[A_g,B_g]=0$ is quotiented out when passing to the completion $C$. Since the genus-one constructions rely on elliptic-specific inputs — the Kronecker function in Levin–Racinet, and the universal vector extension (the canonical commutative algebraic group through which every extension of an elliptic curve by a vector group factors) in Fonseca–Matthes — none of which has a direct analogue for a general curve, the higher-genus problem is genuinely open.

Problem Statement

Let $K$ be a number field, let $C$ be a smooth projective curve over $K$ of genus $g \ge 2$, let $P \in C(K)$, and set $X = C - P$. Let $\mathfrak u$ denote the free pronilpotent Lie algebra over $K$ on generators $A_1, B_1, \dots, A_g, B_g$, let $\mathfrak w$ be its maximal metabelian quotient (the largest quotient whose commutator Lie subalgebra is abelian), and let $W$ be the corresponding prounipotent group. Construct explicitly an algebraic connection on $X$ with values in $\mathfrak w$ as follows: exhibit one-forms $\omega_X$ on $X$ and $\omega_Y$ on an open neighbourhood $Y$ of $P$, with algebraic coefficients over $K$, $\omega_X$ regular on $X$ and $\omega_Y$ logarithmic on $Y$ with at most a simple pole at $P$, flat (each satisfying the Maurer–Cartan equation $d\omega+\tfrac12[\omega,\omega]=0$) and such that the glued connection represents Kim's universal unipotent pointed connection of $X$ at a basepoint $b$ (either a rational basepoint $b \in X(K)$ or a tangential basepoint at $P$), together with a gauge transformation $g$ on $X \cap Y$ satisfying $\omega_Y = -dg\,g^{-1} + g\,\omega_X\,g^{-1},$ such that: (1) $\omega_X = \sum_{i=1}^{g}(\alpha_i A_i + \beta_i B_i)$ modulo words of higher degree, where $\{\alpha_1,\dots,\alpha_g,\beta_1,\dots,\beta_g\}$ is a basis of $H^1_{dR}(X/K)$ and the $\alpha_i$ extend to global one-forms on $C$; (2) $\omega_X$ and $\omega_Y$ take values in the image in $\mathfrak w$ of the Lie subalgebra of $\mathfrak u$ consisting of words of degree at most 1 in the $A_i$ (not necessarily Galois-equivariant; the connection has the required logarithmic local form at the puncture); and (3) $g$ acts by left multiplication by words in the $B_i$. Then, using this connection, write down explicitly the maximal metabelian quotient of Kim's de Rham period map $j^{dR}(X,b)$ for the punctured curve $X$ (the analogues of Moore's Propositions 5–8 and Theorems 1 and 9, for which the metabelian Lie algebra is free and no relation is imposed), expressing its coordinates as algebraic iterated integrals of the regular differentials of $X$, thereby generalizing Moore's genus-one metabelian formula to genus $g \ge 2$; the period map of the completion $C$ is then obtained by further quotienting by the relation $[A_1,B_1] + \cdots + [A_g,B_g] = 0$.

References

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