# The de Rham period map for punctured elliptic curves and the KZB equation

Ben Moore  Mathematics Institute, University of Warwick, CV4 7AL, UK, and
School of Mathematics, University of Edinburgh, King’s Buildings, Edinburgh,
UK

(Date: May 2023)

###### Abstract.

We demonstrate that the algebraic KZB connection of Levin–Racinet and Luo on a
once-punctured elliptic curve represents Kim’s universal unipotent connection,
and we observe that the Hodge filtration on the KZB connection has a
particularly simple form. This allows us to generalise previous work of Beacom
by writing down explicitly the maximal metabelian quotient of Kim’s de Rham
period map in terms of elliptic polylogarithms. As far as we are aware this is
the first time that the de Rham period map has been written out for an
infinite dimensional quotient of the de Rham fundamental group on any curve of
positive genus.

##  1\. Introduction

Let $C/K$ be a geometrically smooth complete curve of genus $g$ over a local
field $K$ with good reduction over the residue field $k$. Write $X$ for the
hyperbolic curve $C-D$, where $D$ is a finite subset of cardinality $r\geq
3-2g$. As part of his program [16, 15, 3, 4] for understanding hyperbolic
curves over number fields via _Selmer schemes_ , Kim has defined a _de Rham
period map_ which associates to each $K$-valued point of $X$ an element inside
a certain moduli space of torsors over $X$ (here, $K$ is the completion of the
number field at some finite place of good reduction). The entire Selmer scheme
apparatus is controlled by the $K$-prounipotent completion of the motivic
fundamental group $U$ of $X$, which is an infinite dimensional
$K$-prounipotent group, and the de Rham period map is controlled by the de
Rham realisation $U^{dR}$ of $U$.

Previous work [1, 2, 5] has generally focussed on the Selmer schemes induced
by certain small finite-dimensional quotients of $U$, such as its descending
central series quotients $U_{n}$ for $n\leq 3$, but such quotients contain
nontrivial information about $X$ only when $n$ exceeds some (increasing)
function of the rank of the Jacobian of $X$. It is therefore necessary (and
natural) to try to understand Selmer schemes for $U$ itself — or more
realistically, for algebraically well-behaved infinite-dimensional quotients
of $U$, such as its maximal metabelianisation $W=U/[[U,U],[U,U]]$. In this
article we give a complete description of the de Rham period map for $W^{dR}$
in the simplest nontrivial case, namely for $X$ a once-punctured elliptic
curve.

The target of the de Rham period map is the affine space
$F^{0}U^{dR}\backslash U^{dR}$, where $F^{\bullet}$ is the _Hodge filtration_
on $U^{dR}$. In general, we have
$F^{0}U^{dR}=\exp\left(F^{0}\mathfrak{u}\right)$, where $\mathfrak{u}$ is the
Lie algebra of $U^{dR}$ and $F^{0}\mathfrak{u}$ is a subspace. For the
purposes of applications to Selmer schemes, it is necessary to understand the
composition of the period map with a coordinate $\iota:F^{0}U\backslash
U\rightarrow\mathfrak{u}$. In general, one obtains such a coordinate $\iota$
from a splitting $\mathfrak{u}=F^{0}\mathfrak{u}\oplus M$ of $K$-vector
spaces, with $M$ a Lie ideal, simply by observing that the composite

$M\xrightarrow{\mathrm{exp}}U\twoheadrightarrow F^{0}U\backslash U$

is bijective, and defining $\iota$ to be its inverse.

When $X$ is the thrice-punctured line, $F^{0}$ is trivial, and so $\iota$ is
simply the logarithm. It is easy to compute the image in $W$ of the logarithm
of the de Rham period map; however, most authors choose to compute its image
in Deligne’s polylogarithmic quotient [10], which is even smaller than $W$.
For a once-punctured elliptic curve $X=E-\\{\infty\\}$, $\mathfrak{u}$ is the
free Lie algebra on $A$ and $B$ ($A$ corresponds to an algebraic $1$-form on
$E$ and $B$ to an algebraic $1$-form on $X$). Furthermore $F^{0}\mathfrak{u}$
is one-dimensional. Choosing a Lie ideal $M$ allows us to define the
projection $\pi_{F^{0}\mathfrak{u}}$ of $\mathfrak{u}$ onto
$F^{0}\mathfrak{u}$, and we can check that the map

$U\ni
x\mapsto\log\left(\exp\left(-\pi_{F^{0}\mathfrak{u}}(\log(x))\right)x)\right)\in
M$

descends to $F^{0}U\backslash U$ and is equal to $\iota$. It is natural to
wonder whether our results are contained in the earlier article of Beacom [6],
which is dedicated to the de Rham period map for elliptic and hyperelliptic
curves. The answer is that Beacom describes a recursive algorithm for
computing the de Rham period map attached to $U_{n}$ from the one attached to
$U_{n-1}$, but the algorithm is sufficiently complicated that it is infeasible
to use it to directly write down the period map for every $U_{n}$.

We circumvent this difficulty by exploiting the fact, which seems to be
generally under-appreciated, that the presentation of $F^{0}U^{dR}$, and hence
$\iota$, depends on a choice of representative for the _universal connection_
on $X$. Beacom, along with all other writers on the subject, chooses the so-
called _naive connection_ $\nabla_{0}$, for which the _crystalline path
torsor_ appearing in the period map assumes a particularly simple form. But
the drawback to this approach is that the Hodge filtration is harder to
describe: for example, at level four, we have

$F_{\nabla_{0}}^{0}\mathfrak{u}_{4}=\mathrm{Span}_{K}\left\\{B-\lambda[B,[A,B]]+(\mu+\kappa/3)[B,[B,[A,B]]]\right\\},\quad\lambda,\mu,\kappa\in
K.$

However, we choose to employ (a variant of) the algebraic KZB connection
$\nabla_{KZB}$ of Levin–Racinet and Luo. There is a unique (pointed)
isomorphism between these (pointed) connections, but this variant of the KZB
connection enjoys a number of highly desirable properties. Among them is the
particularly simple form taken by its the Hodge filtration:

$F_{\nabla_{KZB}}^{0}\mathfrak{u}=\mathrm{Span}_{K}\\{B\\}.$

Now there is a canonical choice for $M$: namely, the Lie ideal spanned by Lie
words of degree at least $1$ in $A$. Then $\iota$ takes the form

(1)
$\iota_{KZB}(x)=\log\left(\exp\left(-\pi_{\mathrm{Span}_{K}\\{B\\}}(\log(x))\right)x)\right).$

Finally, we note that as for the thrice-punctured line, it is very convenient
to pass to $W$, in order to make the calculations algebraically tractable. We
require descriptions of the flat section of the (adjoint) of the KZB equation,
of the logarithm of this flat section, and of the Baker–Campbell–Hausdorff
formula, and all of these are rather difficult to obtain over $U$.

Before we state the main result of the article, we need to introduce some
algebraic functions on our punctured elliptic curve. Let $E/K$ be an elliptic
curve over a local field $K$ of characteristic zero, with good reduction over
the residue field $K$. Let $X=E-\\{\infty\\}$ denote the hyperbolic curve,
with model

$y^{2}=4x^{3}-60e_{4}x-140e_{6}$

obtained by removing the point at infinity and let $b\in E(K)$ be a basepoint.
The Lie algebra $\mathfrak{w}$ of the maximal metabelian quotient of the de
Rham fundamental group of $X$ based at $b$ is spanned by $A$, $B$ and
$\sigma_{r,s}=\mathrm{ad}^{r}_{B}\mathrm{ad}^{s}_{A}[A,B]$ for $r,s\geq 0$ and
we can choose these so that the isomorphism $\mathfrak{w}^{\mathrm{ab}}\cong
H^{dR}_{1}(X/K)$ identifies the dual of $A$ with the global regular
differential $\alpha=dx/y$ and the dual of $B$ with the differential form
$\beta=xdx/y$ which is regular away from $\infty$. Let $e_{n}$ be $1/(2n+1)$
times the $(n-2)$th Laurent series coefficient of the Weierstrass function for
$E$: alternatively, the $e_{n}$ are characterised by $e_{n}=0$ for $n\geq 3$
odd and

$\frac{1}{3}(m-3)(4m^{2}-1)e_{2m}=\sum_{r=2}^{m-2}(2r-1)(2m-r-1)e_{2r}e_{2m-2r}$

for $m\geq 4$. Then let $P_{1}=-2x^{2}/y$, $P_{2}=x$, $P_{3}=-y/2$ and for
$m,n\geq 2$, define $P_{m}$ by

$P_{m}P_{n}-P_{m+n}=(-1)^{n}\sum_{k=1}^{m-2}\binom{n+k-1}{k}e_{n+k}P_{m-k}+(-1)^{m}\sum_{k=1}^{n-2}\binom{m+k-1}{k}e_{m+k}P_{n-k}+(-1)^{m}\binom{m+n}{m}e_{m+n}.$

We recall the Bernoulli numbers $B_{m}$ with the convention

(2) $B_{0}=-1,B_{1}=\frac{1}{2},B_{2}=\frac{1}{6},\dots.$

######  Theorem 1.

We have

(3) $\iota_{KZB}\circ j^{dR}(X,b)(x)=\int_{b}^{x}\alpha A+\sum_{r,s\geq
0}g_{r,s}\sigma_{r,s}-\int_{b}^{x}\beta
S\left(\int_{b}^{x}\alpha\mathrm{ad}_{A},\int_{b}^{x}\beta\mathrm{ad}_{B}\right)\left(-\int_{b}^{x}\alpha\sigma_{0,0}+\sum_{r,s\geq
0}g_{r,s}\sigma_{r+1,s}\right)$

where $j^{dR}(X,b)(x)$ is the de Rham period map, $\iota_{KZB}$ is the image
inside $\mathfrak{w}$ of the coordinate described above, and

$S(U,V)=\frac{1}{U+V}\left(1-\frac{\exp(-V)-1}{V}\frac{U+V}{\exp(U+V)-1}\right).$

The $g_{r,s}$ are iterated integrals of certain algebraic functions on $E/K$,
characterised by the equality of formal power series

$\sum_{r,s\geq 0}g_{r,s}X^{s}Y^{r}=\sum_{r,s\geq
0}\left(\sum_{\begin{subarray}{c}u+n=r\\\
v+m=s\end{subarray}}\frac{B_{u+v}\left(\int_{b}^{x}\beta\right)^{u}\left(\int_{b}^{x}\alpha\right)^{v}G_{n,m}}{u!v!}\right)X^{s}Y^{r}-\left(\int_{b}^{x}\alpha\right)\left(\int_{b}^{x}\beta\right)T\left(\int_{b}^{x}\alpha
X,\int_{b}^{x}\beta Y\right),$

where

$T(U,V)=\frac{1}{U}\left(1-\frac{U+V}{V}\frac{\exp(U)-1}{\exp(U+V)-1}\right)$

and $B_{m}$ is the $m$th Bernoulli number from (2). The algebraic iterated
integrals $G_{r,s}$ are in turn characterised as the iterated integral of the
coefficient of $X^{r}Y^{s}$ in

$-\left(1-\alpha Y-\beta X\right)^{-1}\left(1+(1-\beta X)\sum_{i\geq
0}\sum_{k=2}^{i}\beta^{i-k+1}\alpha p_{k}X^{i}\right),$

where we define the algebraic functions $p_{k}$ by

$p_{n}=\sum_{2a_{2}+3a_{3}+\dots+na_{n}=n}\frac{1}{a_{2}!a_{3}!\dots
a_{n}!}\prod_{k=2}^{n}\left(\frac{(-1)^{k+1}P_{k}}{k}\right)^{a_{k}}.$

In Theorem 9, we prove a similar result in the case that $b$ is a tangential
basepoint at $\infty$.

We note, in passing, that our KZB connection is a connection on a bundle of
cocommutative Hopf algebras, whereas the KZB connections used by Levin–Racinet
[19], Hain [14], Luo [20], Brown–Levin [9] and others are closely related
connections on a bundle of Lie algebras. This latter KZB connection is
important primarily because it is the specialisation to $E$ of the unique
logarithmic connection over the moduli stack of elliptic curves satisfying
certain properties [14, Corollary 14.4]. With respect to higher genus curves,
Enriquez has defined a KZB connection on Schottky covers of configuration
spaces of complex curves [11], and Enriquez–Zerbini have given (in principle)
gauge transformations under which this connection descends to the curve [12],
but at the time of writing these constructions haven’t been worked out
explicitly for any curve of genus at least $2$. In fact it is not known if
their gauge transformations yield an algebraic connection.

However, for the purposes of writing down the de Rham period map on hyperbolic
curves $X=C-P$, where $C$ is a complete curve of genus $g\geq 2$ and $P\in C$,
with respect to the maximal metabelian quotient $\mathfrak{w}$ of
$\mathfrak{u}$, it suffices to construct any connection of a certain form,
which ensures that generalisations of Propositions 5 and 6 hold. Then,
straightforward generalisations of Propositions 7 and 8 to the maximal
metabelian Lie algebra on $n=2g$ generators $A_{1},B_{1}\dots,A_{g},B_{g}$,

$\mathfrak{w}_{n}=\mathrm{Span}_{K}\left\\{X_{1},\dots
X_{n},\sigma_{\mathbf{u}}\rvert_{i,j}\right\\}_{1\leq i<j\leq
n},\quad\mathbf{u}=(0,\dots,0,u_{i},\dots,u_{n}),\quad\sigma_{\mathbf{u}}\rvert_{i,j}=\mathrm{ad}_{X_{n}}^{u_{n}}\dots\mathrm{ad}_{X_{1}}^{u_{1}}[X_{i},X_{j}],$

would yield analogous formulae for the period map. Specifically, cover $C$ by
two open subsets $X=C-P$ and $Y$: then we may construct our connection by
specifying two $\mathfrak{w}$-valued one-forms $\omega_{X}$ and $\omega_{Y}$
on $X$ and $Y$ together with a gauge transformation $g$ such that
$\omega_{Y}=-dg.g^{-1}+g\omega g^{-1}$ on $X\cap Y$. Then the obvious
analogues of Propositions 5 and 6 hold if:

  1. (1)

$\omega_{X}=\sum_{i=1}^{2g}\left(\alpha_{i}A_{i}+\beta_{i}B_{i}\right)$ modulo
higher degree words (the $\alpha_{i}$ and $\beta_{i}$ are a basis for
$H^{1}_{dR}(X/K)$ such that the $\alpha_{i}$ extend to global $1$-forms on
$X$);

  2. (2)

$\omega_{X}$ and $\omega_{Y}$ take values in the (not necessarily Galois-
equivariant) Lie subalgebra $\mathfrak{g}$ on $\mathfrak{u}$ consisting of Lie
words of degree at most $1$ in $A_{1},\dots,A_{g}$; and

  3. (3)

$g$ acts as left multiplication by words in the $B_{i}$s.

The period map for the completion of $C$ is obtained by quotienting by the
relation $[A_{1},B_{1}]+\dots+[A_{g},B_{g}]=0$.

##  2\. Definitions

This section is devoted to recalling the definition of the de Rham period map.
We make definitions for general smooth $K$-schemes as far as possible,
restricting to hyperbolic curves only when necessary. We will use the word
“connection” throughout to refer to a vector bundle equipped with a
connection.

Let $Z/K$ be a scheme over a field of characteristic zero. Let $D$ be a
divisor of $Z$ with normal crossing singularities, and set $Z^{\prime}=Z-D$.
Let $\mathrm{UnFC}(Z^{\prime})$ denote the Tannakian category of unipotent
flat connections over $Z^{\prime}$: that is, the objects are connections that
may be constructed as successive extensions of the trivial connection. Let
$\mathrm{UnFC}(Z)(\log D)$ denote the category of flat unipotent logarithmic
connections on $Z$. These are the flat meromorphic connections on $Z$ that are
holomorphic outside $D$ and have at worst logarithmic singularities within
$D$. Deligne has shown that every connection $\nabla$ over in
$\mathrm{UnFC}(Z^{\prime})$ with a meromorphic extension to $Z$ has a
canonical extension $\overline{\nabla}$ to a connection in
$\mathrm{UnFC}(Z)(\log D)$. Furthermore, for $P\in D$, any such logarithmic
connection on $C$ induces a connection, called the _residue connection_ , on
$T_{P}Z\setminus\\{0\\}$.

For any $K$-schemes $S$, let $Z_{S}$ denote the base-change of $Z$ to $S$, and
write $\mathrm{Vect}_{S}$ for the category of vector bundles on $S$. Let $b$
be a $K$-rational basepoint of $Z^{\prime}$, meaning that $b\in
Z^{\prime}(K)$, or a tangential basepoint at $P\in D$, in which case
$b\in(T_{P}Z)(K)$. When $b$ is a rational basepoint, there is an obvious fibre
functor $F_{b}(S):\mathrm{UnFC}(Z^{\prime}_{S})\rightarrow\mathrm{Vect}_{S}$
defined by sending a connection $\mathscr{E}$ to its fibre over $b$. When $b$
is a tangential basepoint at $P$, we define the fibre functor
$F_{b}(S):\mathrm{UnFC}(Z_{S})(\log(D))\rightarrow\mathrm{Vect}_{S}$ by
sending a connection on $Z_{S}$ to the fibre of the residue connection
$\mathrm{Res}\left(\mathscr{E}\right)$ on $T_{P}Z_{S}-\\{0\\}$ over $b$.

When $b$ is a rational basepoint, a _pointed_ unipotent flat connection on
$Z^{\prime}$ (based at $b$) is a connection $(\mathscr{E},\nabla)$ in
$\mathrm{UnFC}(Z^{\prime})$ together with an element $v\in\mathscr{E}_{b}$.
When $b$ is a tangential basepoint, a pointed unipotent flat connection is a
connection $(\mathscr{E},\nabla)$ in $\mathrm{UnFC}(Z)(\log(D))$ together with
an element $v\in\mathrm{Res}\left(\mathscr{E}\right)_{b}$. In either case
there exists a _universal_ unipotent pointed flat connection
$(\mathscr{E}_{0},\nabla_{0},v_{0})$: that is, for any other unipotent pointed
flat connection $(\mathscr{E},\nabla,v)$, there exists a unique isomorphism of
connections $\phi$ such that the underlying isomorphism
$\mathscr{E}_{0}\rightarrow\mathscr{E}$ of $K$-vector bundles on $X$ satisfies
$\phi_{b}(v_{0})=v$. The universal pointed unipotent flat connection is unique
up to a unique isomorphism.

The _de Rham fundamental group_ of $Z^{\prime}$ based at $b$, denoted
$U^{dR}(Z^{\prime},b)$ (or simply $U$, though the dependence on $b$ is
crucial), is defined to be the $K$-group scheme representing the functor
$S\mapsto\mathrm{Aut}^{\otimes}F_{b}(S)$ of tensor-compatible automorphisms.
Similarly, for $z^{\prime}\in Z^{\prime}$, the functor
$S\mapsto\mathrm{Isom}^{\otimes}(F_{b}(S),F_{z^{\prime}}(S))$ of tensor-
compatible isomorphisms is represented by a right
$U^{dR}(Z^{\prime},b)$-torsor, denoted by $P^{dR}(Z^{\prime},b)$, which we
call the _de Rham path torsor_. On general grounds, $U^{dR}$ and $P^{dR}$
admit a certain filtration, called the _Hodge filtration_ : but we will see
below that a lemma of Hadian provides a characterisation of the Hodge
filtration which is rather more useful in practice. It is well-known that $U$
is a prounipotent group, so it is representable as the inverse limit of its
descending central series quotients $U_{n}$, defined by

$U^{1}=U,\quad U^{n}=[U,U^{n-1}],\quad U_{n}=U/U^{n+1}.$

The construction extends to produce descending central series quotients of its
Lie algebra $\mathfrak{u}$, the universal enveloping algebra
$\mathscr{U}(\mathfrak{u})$ of its Lie algebra, the de Rham path torsor, and
unipotent connections on $Z^{\prime}$. We even have descending central series
quotients for all of the quotients of these objects induced by quotients of
$U$. The $n$th level quotient by the descending central series of any of these
objects is always denoted with a subscript index $n$.

A logarithmic filtration $F^{\bullet}$ on a logarithmic connection
$(\mathscr{E},\nabla)\in\mathrm{UnFC}(Z)(\log D)$ with poles along $D$ is a
filtration of $\mathscr{E}$ by subbundles $F^{i}\mathscr{E}$ such that

$\mathscr{E}=F^{m}\mathscr{E}\supseteq
F^{m+1}\mathscr{E}\supseteq\dots\supseteq F^{n}\mathscr{E}=0$

for some $m\leq n$ which satisfy Griffiths transversality:

$\nabla\left(F^{i}\mathscr{E}\right)\subseteq
F^{i-1}\mathscr{E}\otimes\Omega_{Z/K}^{1}\left(\log(D)\right).$

When the connection in question is pointed, with a marked element
$v\in\mathscr{E}_{b}$, we additionally require that
$v\in\left(F^{0}\mathscr{E}\right)_{b}$. The logarithmic extension of the
universal connection inherits a logarithmic filtration, also called the _Hodge
filtration_ , from the one on $U^{dR}$ mentioned earlier.

From now on, suppose that $K$ is a nonarchimedean field with residue field
$k$, and that $Z^{\prime}$ has good reduction over $k$. Let $Z^{\prime}_{k}$
be the reduction of $Z^{\prime}$ to $k$ and write
$\mathrm{UnIsoc}(Z^{\prime}_{k})$ for the category of overconvergent unipotent
isocrystals on $Z^{\prime}_{k}$. For $c\in Z^{\prime}_{k}(k)$, let $]c[$
denote the residue disk of $c$. The fibre functor
$e_{c}:(\mathscr{E},\nabla)\mapsto\mathrm{ker}(\nabla)_{]c[}$, taking an
element of $\mathrm{UnIsoc}(Z^{\prime}_{k})$ to the $k$-vector space of its
flat sections over $]c[$, provides $\mathrm{UnIsoc}(Z^{\prime}_{k})$ with a
Tannakian structure. The crystalline fundamental group based at $c$, denoted
$U^{\mathrm{cr}}(Z^{\prime}_{k},c)$, is the group scheme over the fraction
field of the ring of Witt vectors of $k$ representing the tensor-compatible
automorphisms, and the crystalline path torsor is the right
$U^{\mathrm{cr}}(Z^{\prime}_{k},c)$-torsor representing the tensor-compatible
isomorphisms. If $c$ is instead a tangential basepoint, then there are
suitable parallel definitions due to Besser–Furusho [8]. By a comparison
theorem of Chiarellotto between crystalline and de Rham fundamental groups,
the Frobenius automorphism on $U^{cr}$ induces a Frobenius automorphism on
$P^{dR}(Z^{\prime},z^{\prime})$ for any $z^{\prime}\in Z^{\prime}$.

When $z^{\prime},b\in Z^{\prime}(K)$, Besser [7] shows that there is a unique
Frobenius-invariant de Rham path $p^{cr}(z^{\prime})$ from $b$ to
$z^{\prime}$. When $z^{\prime}$ and $b$ are tangential basepoints,
Besser–Furusho [8] prove that the existence of a unique Frobenius invariant de
Rham path generally depends on a choice of a branch of the $p$-adic logarithm.
However, if $z^{\prime}$ is an integral point, then a choice of branch of the
$p$-adic logarithm is not required.

Kim defines [17] an _admissible_ right $U^{dR}$-torsor $T$ over $Z^{\prime}$
to be a right $U^{dR}$-torsor $T$ over $Z^{\prime}$ with a number of extra
structures: an Eilbenberg–Maclane filtration on $\mathcal{O}(T)$, a Hodge
filtration $F^{\bullet}$ on $\mathcal{O}(T)$ and a Frobenius automorphism
$\phi:T\rightarrow T$ of $Z^{\prime}$-schemes. These structures are required
to be compatible in various ways (with each other and with the Hodge
filtration on $U^{dR}$), and there must be a $\phi$-invariant element
$p^{cr}_{T}\in T(Z^{\prime})$ and a Hodge element $p^{H}_{T}\in
F^{0}T(Z^{\prime})$. When $Z^{\prime}$ is affine, there is a unique
$p^{cr}_{T}$: otherwise, we may remove a point $P_{0}$ and make use of the
functoriality of the Hodge filtration. Thus, once a point $p^{H}_{T}$ is
chosen, there is a unique $u_{T}\in U^{dR}(Z^{\prime})$ such that
$p^{cr}_{T}=p^{H}_{T}u_{T}$. We write $[T]$ for the image of $u_{T}$ inside
$F^{0}U^{dR}(Z^{\prime})\backslash U^{dR}(Z^{\prime})$: it is independent of
the choice of $p^{H}_{T}$.

######  Proposition 2.

For a $K$-scheme $Z^{\prime}$, the map $T\mapsto[T]$ defines a bijection from
the space of isomorphism classes of admissible right $U^{dR}$-torsors over
$Z^{\prime}$ to the affine $K$-space $F^{0}U^{dR}\backslash U^{dR}$.

For each $z^{\prime}\in Z^{\prime}$, Kim has shown (in loc. cit.) that
$P^{dR}(z^{\prime})$ is an admissible right $U^{dR}$-torsor over $Z^{\prime}$.
Finally we may define the de Rham period map.

######  Definition 3.

Let $b$ be a rational or tangential basepoint for $(Z,D,Z^{\prime})$. Then the
de Rham period map is given by

$j^{dR}(Z^{\prime},b):x\mapsto[P^{dR}(Z^{\prime},b)]\in F^{0}U^{dR}\backslash
U^{dR},$

associating a point $z^{\prime}\in Z^{\prime}(K)$ to the class of the de Rham
path torsor inside the moduli space of admissible $U^{dR}$-torsors.

In practice (for example, when working with Selmer schemes), we work with the
composition of the de Rham period map with a Lie algebra-valued coordinate
$\iota:F^{0}U^{dR}\backslash
U^{dR}=\exp\left(F^{0}\mathfrak{u}\right)\backslash\exp\left(\mathfrak{u}\right)\rightarrow\mathfrak{u}$.
At this point, we replace $(Z,D,Z^{\prime})$ with the data $(C,D,X)$ of a
hyperbolic curve from the introduction, and explain how to work out
$\iota\circ j^{dR}(X,x)$ for each $x\in X(K)$.

First we need to construct unipotent connections on $X$, and for this it is
necessary to choose some bases. Fix some $P_{0}\in C$ such that $P_{0}\in D$
if $D\neq\varnothing$. Select a basis $\alpha_{1},\dots,\alpha_{g}$ of global
algebraic differential forms for $H^{1}_{dR}(C/K)$ and a basis
$\beta_{1},\dots,\beta_{g}$ of meromorphic differential forms for
$H^{1}_{dR}(C/K)$ which are algebraic outside $P_{0}$. Choose meromorphic
differential forms $\gamma_{1},\dots,\gamma_{r-1}$ with poles contained in $D$
such that the $\alpha$s, $\beta$s and $\gamma$s together form a basis of
$H^{1}_{dR}(X/K)$. Then if $X$ is affine, $\mathscr{U}(\mathfrak{u})$ is
isomorphic to

$H=K\langle\langle A_{1},B_{1}\dots,A_{g},B_{g},C_{1},\dots
C_{r-1}\rangle\rangle,$

the free cocommutative Hopf algebra on the symbols $A_{i}$, $B_{j}$, $C_{k}$,
and one can arrange things so that the canonical isomorphisms
$H_{1}^{dR}(X/K)\cong\log(U_{1})\cong U_{1}$ identify $\alpha_{i}$ with
$\log(A_{i})$, $\beta_{j}$ with $\log(B_{j})$ and $\gamma_{k}$ with
$\log(C_{k})$. In fact, we can identify $U$ with the grouplike elements of $H$
and $\mathfrak{u}$ with the primitive elements. If $D=\varnothing$, so that
$X$ is a complete projective curve of genus at least $2$, then one obtains
results analogous to the affine case by quotienting by the relation
$\sum_{i}[A_{i},B_{i}]=0$. Returning to the affine case, Kim has proved (loc.
cit.) that the connection

$\nabla_{0}=d-\sum_{i=1}^{g}\left(A_{i}\alpha_{i}+B_{i}\beta_{i}\right)-\sum_{k=1}^{r-1}\gamma_{k}C_{k}$

on $\mathcal{O}_{X}\otimes H$, which we call the _naive connection_ ,
represents the universal pointed unipotent connection on $X$. Fix any
representative $(\mathcal{O}_{X}\otimes H,\nabla)$ of the universal unipotent
pointed connection with marked element $1\in H$ (which we interpret as the
fibre over $b$). We choose a primitive $p^{H}_{\nabla}(x)$ in the $K$-algebra
generated by $\left(F^{0}\nabla\right)_{x}$ and define
$\iota_{\nabla}(m)=\log\left(\exp\left(-p_{\nabla}^{H}(x)\right)m\right)$, so
that

(4) $\iota_{\nabla}\circ
j^{dR}_{\nabla}(x)=\log\left[\exp\left(-p_{\nabla}^{H}(x)\right)p^{cr}_{\nabla}(x)\right].$

If $\nabla$ is any trivialised representative of the universal unipotent
pointed connection on $X$ with marked element $1\in H$ in the fibre over $b$,
and $b\in X(K)$ is a rational basepoint, then $p_{\nabla}^{cr}(x)$ is the
value at $x$ of the unique flat section $G$ of $\nabla$ which begins $G=1+$
terms of degree $\geq 1$ in $A_{i}$, $B_{j}$ and $C_{k}$. If $b\in(T_{P}C)(K)$
is a tangential basepoint, let $Y$ be an open set containing $P$ over which
the logarithmic extension of $\nabla$ is trivialised, and choose some $y\in
X(K)\cap Y(K)$: then for $x\in X(K)-Y(K)$,

$p_{\nabla}^{cr}(x)=g_{\nabla}^{-1}(y)p_{\nabla^{\prime}}^{cr}(y)+p_{\nabla}^{cr}(x),$

where $g_{\nabla}$ is the gauge transformation over $X\cap Y$ gluing $\nabla$
to the logarithmic connection $\nabla^{\prime}$ on $Y$.

We turn now to the problem of determining $p^{H}_{\nabla}$. A lemma of Hadian
[13, Lemma 3.6] states that there is a filtration $F^{\bullet}$ of logarithmic
subbundles (with poles contained in $D$) on the logarithmic extension
$\overline{\nabla_{0}}$ of
$(\mathcal{O}_{X}\otimes\mathscr{U}(\mathfrak{u}),\nabla_{0})$ to $C$ such
that the exact sequence

$0\rightarrow V^{\otimes
n}_{dR}\otimes\mathcal{O}_{C}\rightarrow\left(\overline{\nabla_{0}}\right)_{n}\rightarrow\left(\overline{\nabla_{0}}\right)_{n-1}\rightarrow
0$

of pointed logarithmic connections becomes an exact sequence of filtered
pointed logarithmic connections. Here, we write $V_{dR}$ for $H_{1}^{dR}(X/K)$
and freely identify it with $U_{1}$.

The Hodge filtration on $V^{\otimes n}_{dR}\otimes\mathcal{O}_{C}$ is the
tensor product filtration of the filtration on $V_{dR}$ inducing by the de
Rham complex of $X/K$ and the trivial filtration on $\mathcal{O}_{C}$.
Specifically,

(5) $F^{-j}\left(V^{\otimes
n}_{dR}\otimes\mathcal{O}_{C}\right)=\begin{cases}\mathrm{Span}_{\mathcal{O}_{C}}\left\\{\text{words
}w\text{ in }A_{i}\text{s, }B_{j}\text{s, s.t.
}\sum_{i=1}^{g}\mathrm{deg}_{A_{i}}w\leq|j|\right\\}&j\leq 0\\\
0&j>0.\end{cases}$

Hadian’s pointed filtration is unique up to a unique isomorphism, and is
completely determined by a choice of representative for the isomorphism class
of the universal pointed connection. Thus, the Hodge filtration on any
representative of the universal unipotent pointed connection is the
restriction of this filtration from the logarithmic extension. Based on this
observation, Beacom [6, Section 4.2] provides an algorithm for recursively
computing the Hodge filtration on $\left(\nabla_{0}\right)_{n}$ from the Hodge
filtration on $\left(\nabla_{0}\right)_{n-1}$ for affine elliptic and
hyperelliptic curves.

Thus, given a representative $\nabla$ for the universal unipotent pointed
connection, the data of the logarithm of its fundamental solution together
with its Hodge filtration $F^{\bullet}$ is sufficient to define Kim’s de Rham
period map. For the naive connection $\nabla_{0}$ on $X$, the fundamental
solution is given in terms of iterated integrals as

$G(z)=1+\sum_{\begin{subarray}{c}\text{words }w\\\ \text{ in
}A_{i},B_{j},C_{k}\end{subarray}}G_{w}(z)w,\quad
G_{A_{i}B_{k}^{2}C_{j}}=\int_{b}^{z}\alpha_{i}\beta_{k}\beta_{k}\gamma_{j}\text{
etc},$

so the computation of the de Rham period map with respect to $\nabla_{0}$
reduces to the computation of the Hodge filtration, at least when $b$ is a
rational basepoint. When $b$ is a tangential basepoint, more work is
necessary, but when the endpoint $z$ is an integral point contained in the
open set containing $b$, the period map is obtained simply by replacing
$\nabla_{0}$ with its logarithmic extension over that open set. In particular
there is no need to choose a branch of the $p$-adic logarithm. Beacom’s
recursive algorithm for computing the Hodge filtration on descending central
series quotients gives rise to a recursive algorithm [6, Sections 5 and 6] for
computing $\left(j^{dR}_{\nabla_{0}}\right)_{n}$ from
$\left(j^{dR}_{\nabla_{0}}\right)_{n-1}$. Unfortunately, it it appears to be
very difficult to use his algorithm to directly write down the full Hodge
filtration on $U$, or indeed on any infinite dimensional quotient of $U$, so a
straightforward description of the de Rham period map has proved elusive.

However, any choice of representative for the universal unipotent connection,
giving rise to a coordinate $\iota_{\nabla}$ on the period space, is
sufficient for applications to Selmer schemes. As we shall demonstrate later,
it is quite possible to compute explicitly the logarithm of the crystalline
parallel transport operator $p^{cr}_{\nabla}$ (or more truthfully, its image
in a certain large quotient of $U$) for any connection $\nabla$. Thus, it is
natural to seek representatives of the universal connection whose Hodge
filtration is as simple as possible. We will demonstrate that when $C$ is an
elliptic curve and $D=\\{\infty\\}$, the elliptic KZB connection, introduced
by Levin–Racinet [19] and developed further by Luo [20] meets this
requirement.

##  3\. The KZB connection and the universal connection

From now on, $C=E$ is an elliptic curve over a nonarchimedean field $K$, with
good reduction over the residue field $k$ of $K$, and $X$ is the affine curve
$E-\\{\infty\\}$. Fix an embedding $K\hookrightarrow\mathbb{C}$.

Without loss of generality, $X$ is given by the equation

$y^{2}=4x^{3}-60e_{4}x-140e_{6},$

for some $e_{4},e_{6}\in K$. Let $\alpha=dx/y$ and $\beta=xdx/y$ so that
$\\{\alpha,\beta\\}$ is a $K$-basis for $H^{1}_{dR}(X/K)$ and $\alpha$ is the
restriction to $X$ of a global algebraic $1$-form on $E$. Let $\wp(z)$ denote
the Weierstrass elliptic function associated to $E$, so that the Abel–Jacobi
map is $AJ:P\mapsto z=\int_{\infty}^{P}\alpha$ and $x=AJ^{*}\wp(z)$. We have

$\wp(z)=z^{-2}+\sum_{n=1}^{\infty}(2n+1)e_{2n+2}z^{2n},$

for some $e_{k}\in K$ and we set

$\wp_{2}(z)=\wp(z),\quad\wp_{k}(z)=\frac{(-1)^{k}}{(k-1)!}\frac{d^{k-2}\wp}{dz^{k-2}}$

for $k\geq 3$. The principal part of each $\wp_{k}(z)$ near $z=0$ is given by

$\wp_{k}(z)=z^{-k}+e_{k}+O(z),$

and the pullback functions

$P_{k}=AJ^{*}\left(\wp_{k}(z)-e_{k}\right)$

are algebraic on $X$. Following Weil [21, Chapter II, Sections 7 and 8, and
Chapter VI, Sections 1, 2 and 6], we can characterise these by recurrence
relations. Set $e_{3}=0$ and for $m\geq 5$, define $e_{m}$ by

$\frac{1}{3}(m-3)(4m^{2}-1)e_{m}=\sum_{r=2}^{m-2}(2r-1)(2m-r-1)e_{2r}e_{2m-2r}.$

Then $P_{2}=x$, $P_{3}=-\frac{1}{2}y$ and for $m,n\geq 2$ we have

$P_{m}P_{n}-P_{m+n}=(-1)^{n}\sum_{k=1}^{m-2}\binom{n+k-1}{k}e_{n+k}P_{m-k}+(-1)^{m}\sum_{k=1}^{n-2}\binom{m+k-1}{k}e_{m+k}P_{n-k}+(-1)^{m}\binom{m+n}{m}e_{m+n}.$

Observe that the undefined $P_{1}$ never actually contributes to the right
hand side since it is always multiplied by an $e_{k}$ for $k$ odd, which is
zero.

Following Luo [20, Sections 7 and 8], we define the (algebraic) KZB connection
by constructing suitable logarithmic connections on an open cover of $E$, then
gluing the connections along a gauge transformation.

######  Proposition 4.

Fix some meromorphic function $f$ on $E$ such that $df-\beta$ is holomorphic
at $\infty$, with $f=z^{-1}+O(z)$ with respect to the parameter $z$ above. Let
$Y$ denote the open subset of $E$ obtained by removing the points at which $f$
has poles, except $\infty$. Define a connection

(6) $\nabla_{KZB}=d-\omega_{KZB},\quad\omega_{KZB}=\beta
B+\alpha\exp\left(-\sum_{k=2}^{\infty}\frac{(-1)^{k}P_{k}\mathrm{ad}_{B}^{k}}{k}\right)A$

on $X$ and a connection

$\displaystyle\nabla_{KZB}^{\prime}=d-\omega_{KZB}^{\prime},\quad\omega_{KZB}^{\prime}=(\beta-
df)B+\alpha\exp\left(-f-\sum_{k=2}^{\infty}\frac{(-1)^{k}P_{k}\mathrm{ad}_{B}^{k}}{k}\right)A$

on $Y$. The gauge transformation

$g=\exp(-fB),$

acting by $g(d+\omega)=d-dg.g^{-1}-g\omega g^{-1}$, satisfies
$g(\nabla_{KZB})=\nabla_{KZB}^{\prime}$. Observe that $\nabla_{KZB}$ is
nonsingular on $X$, and $\nabla_{KZB}^{\prime}$ is nonsingular on $Y$, apart
from at $\infty$, where we claim it has a simple pole of residue $[B,A]$. It
follows that $\nabla_{KZB}$, $\nabla_{KZB}^{\prime}$ and $g$ descend to an
algebraic logarithmic $\mathscr{U}(\mathfrak{u})$-bundle with connection on
$E$ with a simple pole at $\infty$. We denote this connection
$(\nabla_{KZB},\nabla_{KZB}^{\prime},g)$.

###### Proof.

It is straightforward to verify that the gauge transformation takes
$\nabla_{KZB}$ to $\nabla_{KZB}^{\prime}$ using the identity
$\mathrm{Ad}_{\exp(X)}=\exp\left(\mathrm{ad}(X)\right)$. To see that
$\nabla_{KZB}^{\prime}$ has a regular singularity at $z=0$, recall that
$P_{k}(z)$ has Laurent tail $z^{-k}+O(z)$ by construction, so

$\displaystyle\exp\left(-f-\sum_{k=2}^{\infty}\frac{(-1)^{k}P_{k}\mathrm{ad}_{B}^{k}}{k}\right)$
$\displaystyle=\exp\left(-\sum_{k=1}^{\infty}\frac{(-1)^{k}\mathrm{ad}_{B}^{k}}{kz^{k}}+H(z)\right)=\left(\frac{\mathrm{ad}_{B}}{z}+1\right)\exp(H(z)),$

where $H$ is holomorphic at $z=0$. The claim concerning the algebraicity
follows from the definition of the $P_{k}$. ∎

It will be convenient to write the KZB connections in the form

(7) $\displaystyle\nabla_{KZB}=d-\beta B-\alpha
A-\alpha\sum_{k=2}^{\infty}p_{k}\mathrm{ad}_{B}^{k}(A),\quad\nabla_{KZB}^{\prime}=d-\left(\beta-
df\right)B-\alpha A-\alpha\sum_{k=1}^{\infty}q_{k}\mathrm{ad}_{B}^{k}(A),$

where, following Luo [20, Sections 7 and 8],

$\displaystyle
p_{n}=\sum_{2a_{2}+3a_{3}+\dots+na_{n}=n}\frac{1}{a_{2}!a_{3}!\dots
a_{n}!}\prod_{k=2}^{n}\left(\frac{(-1)^{k+1}P_{k}}{k}\right)^{a_{k}}$
$\displaystyle
q_{n}=\sum_{a_{1}+2a_{2}+\dots+na_{n}=n}\frac{1}{a_{1}!a_{2}!\dots
a_{n}!}\prod_{k=1}^{n}\left(\frac{(-1)^{k+1}P_{k}}{k}\right)^{a_{k}},$

with $P_{1}$ newly defined to be $-f$.

######  Proposition 5.

The connection $\nabla_{KZB}$ on $X$ with the marked element
$1\in\mathcal{O}_{X}\langle\langle A,B\rangle\rangle$ represents the universal
unipotent pointed connection on $X$.

###### Proof.

It suffices to show that $\nabla_{KZB}$ and the naive connection are
isomorphic as pointed connections on $X$. In fact, we will prove a stronger
result: any connection on $\mathcal{O}_{X}\langle\langle A,B\rangle\rangle$
with marked element $1$ over $b$ of the form

$\nabla=d+\omega,\quad\omega=\alpha A+\beta B+\sum_{|w|>1}r_{w}w,\quad
r_{w}\in\mathcal{O}_{X},$

where the sum is over words in $A$ and $B$ of total degree greater than one,
is isomorphic to the naive connection. That is, we will show that it is
possible to solve the equation

(8) $dg=g\omega_{0}-\omega g,$

where $\omega_{0}=\alpha A+\beta B$, for some
$g\in\mathrm{Aut}\left(\mathcal{O}_{X}\langle\langle A,B\rangle\rangle\right)$
which satisfies $g(1)(b)=1$. To begin, we demonstrate that there exists such a
gauge transformation which replaces the $r_{w}$ with $1$-forms belonging to
$\mathrm{Span}_{K}\\{\alpha,\beta\\}$. Indeed, each $r_{w}$ is of the form
$df_{w}+a_{w}\alpha+b_{w}\beta$ for some $f_{w}\in\mathcal{O}_{X}$ and
$a_{w},b_{w}\in K$. So we may rewrite $\omega$ as

(9)
$\omega=\alpha(A+\sum_{w}a_{w}w)+\beta(B+\sum_{w}b_{w}w)+\sum_{|w|=n}r_{w}w+\sum_{|w|>n}r_{w}w.$

Then the invertible gauge transformation $g=1+\sum_{w}(f_{w}-f_{w}(b))w$,
acting by left multiplication, transforms $\omega$ into

$\omega^{\prime}=\alpha(A+\sum_{w}a_{w}w)+\beta(B+\sum_{w}b_{w}w)+\sum_{|w|=n}\left(a_{w}\alpha+b_{w}\beta\right)+\sum_{|w|>n}r^{\prime}_{w}w,$

and the demonstration is complete after induction. To finish the proof, it
suffices to observe that the gauge transformation characterised by

$g(1)=1,\quad g(A)=A+\sum_{w}a_{w}w,\quad g(B)=B+\sum_{w}b_{w}w,$

and extended to $\mathscr{U}(\mathfrak{u})$ by multiplicativity and linearity,
takes the naive connection with $\omega_{0}=\alpha A+\beta B$ to the
connection with one-form

$\omega^{\prime}=\alpha(A+\sum_{w}a_{w}w)+\beta(B+\sum_{w}b_{w}w).$

To check that $g$ is invertible, fix the lexicographical ordering

$1<A<B<A^{2}<AB<BA<B^{2}<A^{3}<\dots$

on the standard basis $\mathscr{B}$ of $K\langle\langle A,B\rangle\rangle$ and
observe that since $g(w)=w+\sum_{v>w\in\mathscr{B}}k_{v}v$ for some $k_{v}\in
K$, the matrix for $g$ with respect to $\mathscr{B}$ is unipotent. ∎

We have shown that the universal unipotent connection on $X$ is represented by
$\nabla_{KZB}$, so on general grounds, there is a Hodge filtration for
$\nabla_{KZB}$. We demonstrate that the Hodge filtration has a particularly
simple presentation.

######  Proposition 6.

The Hodge filtration $F^{\bullet}$ on $\nabla_{KZB}$ is

$F^{j}\mathcal{O}_{X}\langle\langle
A,B\rangle\rangle=\mathrm{Span}_{\mathcal{O}_{X}}\left\\{\text{words }w\text{
in }A,B\text{ s.t. }\mathrm{deg}_{A}w\leq-j\right\\}.$

###### Proof.

Define a filtration

$\left(F^{\prime}\right)^{j}\mathcal{O}_{Y}\langle\langle
A,B\rangle\rangle=\mathrm{Span}_{\mathcal{O}_{Y}}\left\\{\text{words }w\text{
in }A,B\text{ s.t. }\mathrm{deg}_{A}w\leq-j\right\\}$

on $\nabla_{KZB}^{\prime}$. It is clear that the gauge transformation $g$
induces an isomorphism of filtrations $F^{j}\cong\left(F^{\prime}\right)^{j}$,
so the data $(F^{\bullet},\left(F^{\prime}\right)^{\bullet},g)$ is a candidate
for the Hodge filtration on the logarithmic extension
$\left(\nabla_{KZB},\nabla_{KZB}^{\prime},g\right)$ of $\nabla_{KZB}$. By
Hadian’s lemma, it suffices to check that
$(F^{\bullet},\left(F^{\prime}\right)^{\bullet})$ descends to a filtration
satisfying Griffiths transversality, and upgrades the exact sequence of
logarithmic connections

$0\rightarrow V_{dR}^{\otimes
n}\otimes\mathcal{O}_{E}\rightarrow\left(\left(\nabla_{KZB}\right)_{n},\left(\nabla_{KZB}^{\prime}\right)_{n},g_{n}\right)\rightarrow\left(\left(\nabla_{KZB}\right)_{n-1},\left(\nabla_{KZB}^{\prime}\right)_{n-1},g_{n-1}\right)\rightarrow
0$

to an exact sequence of filtered logarithmic connections. The latter is clear
upon comparing the Hodge filtration on the logarithmic extension to the
earlier definition (5) of Hodge filtration on $V^{\otimes
b}_{dR}\otimes\mathcal{O}_{E}$. The condition concerning Griffiths
transversality amounts to the commutativity of the diagram

$F^{i}\mathcal{O}_{X}\langle\langle
A,B\rangle\rangle$$F^{i}\mathcal{O}_{Y}\langle\langle
A,B\rangle\rangle(\log(\infty))$$F^{i-1}\mathcal{O}_{X}\langle\langle
A,B\rangle\rangle\otimes\Omega_{X/K}^{1}$$F^{i-1}\mathcal{O}_{Y}\langle\langle
A,B\rangle\rangle\otimes\Omega_{Y/K}^{1}\left(\log(\infty)\right)$$g$$g$$\nabla_{KZB}^{\prime}$$\nabla_{KZB}$

which follows from the fact that $g$ only increases the $B$-degree of elements
of $K\langle\langle A,B\rangle\rangle$. ∎

##  4\. The de Rham period map

At this stage, the only remaining obstacle to writing down the de Rham period
map is the computation of the logarithm of a flat section of $\nabla_{KZB}$.
Such a computation would require a basis for $\mathfrak{u}$ as a $K$-vector
space, but recall that this is quite complicated: a basis for the free Lie
algebra on a finite set of generators is given by Lyndon words. In order to
obtain a reasonable description of the de Rham period map, we therefore
concentrate on working out its image inside the maximal metabelian quotient
$W=U/[[U,U],[U,U]]$ of $U$. This is in some sense the largest (Galois-
equivariant) quotient of $U$ which is algebraically comprehensible. In fact,
the set

$\left\\{A,B,\sigma_{r,s}\right\\}_{r,s\geq
0},\quad\sigma_{r,s}=\tau_{r,s}[A,B],\quad\tau_{r,s}=\mathrm{ad}_{B}^{r}\mathrm{ad}_{A}^{s}$

is a $K$-basis for the Lie algebra $\mathfrak{w}$ of $W$. The adjoint map on
$\mathfrak{w}$ is injective, so there is an isomorphism
$\mathfrak{w}\cong\mathrm{ad}(\mathfrak{w})\subseteq\mathrm{End}_{K}(\mathfrak{w})$,
which induces an isomorphism $W\cong\mathrm{Ad}(W)\subseteq GL(W)$. If $G$ is
a flat section of some connection $\nabla$ on
$\mathcal{O}_{X}\otimes\mathscr{U}(\mathfrak{w})$ and $\nabla=d+\omega$ where
$\omega$ acts as left multiplication by a primitive element, then $G$ is
primitive. Furthermore,
$\mathrm{Ad}(G)=\exp\left(\mathrm{ad}_{\log(G)}\right)$ is a flat section of
the induced connection $\mathrm{Ad}(\nabla)=d+\mathrm{ad}_{\omega}$ on
$\mathcal{O}_{X}\otimes\mathscr{U}(\mathrm{ad}(\mathfrak{w}))$. To describe
such a section, we require a basis for a $K$-vector space containing
$\exp\left(\mathrm{ad}(\mathfrak{w})\right)$ (not for all of
$\mathscr{U}(\mathrm{ad}(\mathfrak{w}))$):

$\exp\left(\mathrm{ad}(\mathfrak{w})\right)\subset\mathrm{Span}_{K}\\{\tau_{u,v},\mathrm{ad}_{\sigma_{r,s}}\\}_{\begin{subarray}{c}u,v\geq
0\\\ r,s\geq 0\end{subarray}}.$

Thus, in order to understand $\log(G)$, it suffices to understand (part of)
the flat section $\mathrm{Ad}(G)$ of the adjoint KZB connection
$\mathrm{Ad}\left(\nabla_{KZB}\right)$ together with the exponential map on
$\mathrm{ad}(\mathfrak{w})$. We prove a slightly more general result (the
functions $p_{1}$ is not assumed to be zero) so that we can recycle it for the
case where $b$ is a tangential basepoint.

######  Proposition 7.

Let $\mathrm{Ad}\left(\nabla_{KZB}\right)=d-\mathrm{ad}_{\omega}$, where

$\mathrm{ad}_{\omega}=\beta\tau_{1,0}+\alpha\tau_{0,1}-\alpha\sum_{k=1}p_{k}\mathrm{ad}_{\sigma_{k-1,0}}.$

Then the unique flat section of $\mathrm{Ad}\left(\nabla_{KZB}\right)$ of the
form

$G=1+\sum_{\begin{subarray}{c}u,v\geq 0\\\
u+v>0\end{subarray}}G^{*}_{u,v}\tau_{u,v}+\sum_{r,s\geq
0}G_{r,s}\mathrm{ad}_{\sigma_{r,s}},\quad
G\in\mathscr{U}(\mathrm{ad}(\mathfrak{w}))$

satisfies

(10) $G^{*}_{1,0}=\int\beta,\quad G^{*}_{0,1}=\int\alpha,$

and $G_{r,s}$ is the iterated integral of the coefficient of $X^{r}Y^{s}$ in

(11) $-\left(1-\alpha Y-\beta X\right)^{-1}\left(1+(1-\beta X)\sum_{i\geq
0}\sum_{k=1}^{i}\beta^{i-k+1}\alpha p_{k}X^{i}\right),$

where the “iterated integral” of an element of
$\mathcal{O}_{X}\langle\langle\alpha,\beta\rangle\rangle$ is defined in the
obvious way.

###### Proof.

Comparing coefficients of $\tau_{1,0}$ and $\tau_{0,1}$ in
$dG=\mathrm{ad}_{\omega}G$ yields (10), and an inspection of the coefficient
of $\mathrm{ad}_{\sigma_{i,j}}$ in $\mathrm{ad}_{\omega}G$ yields

$dG_{i,j}=\begin{cases}-\alpha\left(p_{1}+\int\beta\right)&i=0,j=0\\\ \beta
G_{i-1,0}-\alpha p_{i+1}&i\geq 1,j=0\\\ \alpha G_{0,j-1}&i=0,j\geq 1\\\ \beta
G_{i-1,j}+\alpha G_{i,j-1}&i\geq 1,j\geq 1,\end{cases}$

where we have used the identity

$\tau_{0,1}\tau_{u,v}=\begin{cases}\tau_{u,v+1}&(u,v)\neq(1,0)\\\
-\mathrm{ad}_{\sigma_{0,0}}-\tau_{1,1}&(u,v)=(1,0).\end{cases}$

We obtain $G_{0,0}=-\int\alpha p_{1}-\int\alpha\beta$, then by induction we
have

$G_{i,0}=\sum_{k=1}^{i}\int\beta^{i-k+1}\alpha
p_{k}-\int\beta^{i}\alpha\beta,\quad
G_{0,j}=-\int\alpha^{j+1}p_{1}-\int\alpha^{j+1}\beta.$

If we let the $F_{i,j}$ be elements of the noncommutative ring
$\mathcal{O}_{X}\langle\langle\alpha,\beta\rangle\rangle$ satisfying the
recurrence relations

$F_{i,j}=\begin{cases}-\alpha\left(p_{1}+\beta\right)&i=0,j=0\\\ -\alpha
p_{i+1}+\beta F_{i-1,0}&i\geq 1,j=0\\\ \alpha F_{0,j-1}&i=0,j\geq 1\\\ \beta
F_{i-1,j}+\alpha F_{i,j-1}&i\geq 1,j\geq 1,\end{cases}$

and the boundary conditions

$F_{i,0}=-\sum_{k=1}^{i}\beta^{i-k+1}\alpha p_{k}-\beta^{i}\alpha\beta,\quad
F_{0,j}=-\alpha^{j+1}p_{1}-\alpha^{j+1}\beta,$

then we find the following relation among generating series

$\sum_{i,j\geq 0}F_{i,j}X^{i}Y^{j}=-\left(1-\alpha Y-\beta
X\right)^{-1}\left(1+(1-\beta X)\sum_{i\geq
0}\sum_{k=1}^{i}\beta^{i-k+1}\alpha p_{k}X^{i}\right),$

which completes the proof. ∎

Next, we compute the logarithm of any grouplike element in
$\mathscr{U}(\mathrm{ad}(\mathfrak{w}))$ (and we note that it is possible to
do this for the maximal metabelian group on any finite number of generators).

######  Proposition 8.

Suppose that

$H=1+\sum_{u,v\geq 0}H^{*}_{u,v}\tau_{u,v}+\sum_{r,s\geq
0}H_{r,s}\mathrm{ad}_{\sigma_{r,s}}$

is grouplike in $\mathscr{U}(\mathrm{ad}(\mathfrak{w}))$, with

$\log(H)=h_{B}\mathrm{ad}_{B}+h_{A}\mathrm{ad}_{A}+\sum_{r,s\geq
0}h_{r,s}\mathrm{ad}_{\sigma_{r,s}}.$

Then we have $h_{A}=H^{*}_{0,1}$, $h_{B}=H^{*}_{1,0}$, and an equality of
formal power series

$\sum_{r,s\geq 0}h_{r,s}X^{s}Y^{r}=\sum_{r,s\geq
0}\left(\sum_{\begin{subarray}{c}u+n=r\\\
v+m=s\end{subarray}}\frac{B_{u+v}h_{B}^{u}h_{A}^{v}H_{r,s}}{u!v!}\right)X^{s}Y^{r}-h_{A}h_{B}T(h_{A}X,h_{B}Y),$

where

$T(U,V)=\frac{1}{U}\left(1-\frac{U+V}{V}\frac{\exp(U)-1}{\exp(U+V)-1}\right)$

and $B_{m}$ is the $m$th Bernoulli number from (2).

###### Proof.

An inspection of the coefficients of $\tau_{1,0}$ and $\tau_{0,1}$ in
$\exp(\log(H))$ reveals the identities for $h_{A}$ and $h_{B}$. The rest of
the claim is proved by studying the coefficient of
$\mathrm{ad}_{\sigma_{r,s}}$. To begin, this yields

$\displaystyle H_{r,s}$
$\displaystyle=\sum_{\ell=0}^{\infty}\frac{1}{\ell!}\left\\{\text{coeff. of
}\mathrm{ad}_{\sigma_{r,s}}\text{ in
}\left(h_{A}\mathrm{ad}_{A}+h_{B}\mathrm{ad}_{B}+\sum_{r,s\geq
0}h_{r,s}\mathrm{ad}_{\sigma_{r,s}}\right)^{\ell}\right\\}$
$\displaystyle=\sum_{\ell=0}^{\infty}\frac{1}{\ell!}\left\\{\text{coeff. of
}\mathrm{ad}_{\sigma_{r,s}}\text{ in }\left(\sum_{u,v\geq
0}h_{u,v}\mathrm{ad}_{\sigma_{u,v}}\right)^{\ell}+\sum_{\begin{subarray}{c}u\leq
r\\\ v\leq s\end{subarray}}h_{u,v}\times\text{ coeff. of }\tau_{r-u,s-v}\text{
in }\left(h_{A}\mathrm{ad}_{A}+h_{B}\mathrm{ad}_{B}\right)^{\ell-1}\right\\}.$

But

$\left(h_{A}\mathrm{ad}_{A}+h_{B}\mathrm{ad}_{B}\right)^{\ell}=\sum_{i+j=\ell}h_{A}^{j}h_{B}^{i}\times\sum_{\begin{subarray}{c}\text{words
}w\\\ \text{in }A,B\text{ with }\\\ jA\text{s,
}iB\text{s}\end{subarray}}\mathrm{ad}_{w},$

where

(12) $\sum_{\begin{subarray}{c}\text{words }w\\\ \text{in }A,B\text{ with }\\\
jAs,iBs\end{subarray}}\mathrm{ad}_{w}=\binom{i+j}{i,j}\tau_{i,j}+\binom{i+j-1}{i-1,j}\mathrm{ad}_{\sigma_{i-1,j-1}}.$

We prove this via induction by writing

$\displaystyle\sum_{\begin{subarray}{c}\text{words }w\\\ \text{in }A,B\text{
with }\\\ jAs,iBs\end{subarray}}\mathrm{ad}_{w}$
$\displaystyle=\sum_{\begin{subarray}{c}\text{words }w\\\ \text{in }A,B\text{
with }\\\
j-1As,iBs\end{subarray}}\mathrm{ad}_{w}\mathrm{ad}_{A}+\sum_{\begin{subarray}{c}\text{words
}w\\\ \text{in }A,B\text{ with }\\\
jAs,i-1Bs\end{subarray}}\mathrm{ad}_{w}\mathrm{ad}_{B},$

then substituting in (12) and using relation
$\tau_{i-1,j}\mathrm{ad}_{B}=\tau_{i,j}+\sigma_{i-1,j-1}$ for $i\geq 1$, with
the convention that $\sigma_{i-1,j-1}=0$ when $j=0$.

Then the coefficient of $\mathrm{ad}_{\sigma_{r,s}}$ in
$\left(h_{A}\mathrm{ad}_{A}+h_{B}\mathrm{ad}_{B}\right)^{\ell}$ is

$\binom{r+s+1}{r,s+1}h_{A}^{s+1}h_{B}^{r+1}$

if $r+s+2=\ell$, and is otherwise zero. Similarly, the coefficient of
$\tau_{r-u,s-v}$ in
$\left(h_{A}\mathrm{ad}_{A}+h_{B}\mathrm{ad}_{B}\right)^{\ell-1}$ is

$\binom{r-u+s-v}{r-u,s-v}h_{A}^{r-u}h_{B}^{s-v}$

if $r-u+s-v=\ell-1$ and is otherwise zero. Therefore

(13)
$H_{r,s}=\frac{h_{B}^{r+1}h_{A}^{s+1}}{(r+s+2)r!(s+1)!}+\sum_{\begin{subarray}{c}0\leq
u\leq r\\\ 0\leq v\leq
s\end{subarray}}\frac{h_{u,v}h_{B}^{r-u}h_{A}^{s-v}}{(r-u+s-v+1)(r-u)!(s-v)!}.$

We note that

$\sum_{r,s\geq
0}\frac{X^{s}Y^{r}}{(r+s+2)r!(s+1)!}=\frac{1}{X}\left(\frac{\exp(X+Y)-1}{X+Y}-\frac{\exp(X)-1}{X}\right)$

because both sides are holomorphic functions near the origin which satisfy the
differential equation

$\left(X\partial_{X}+Y\partial_{Y}+2\right)F=\frac{\exp(Y)(\exp(X)-1)}{X}.$

The same reasoning applies to prove

$\sum_{r,s\geq 0}\frac{X^{s}Y^{r}}{(r+s+1)r!s!}=\frac{\exp(X+Y)-1}{X+Y}$

where the differential equation is

$\left(X\partial_{X}+Y\partial Y+1\right)F=\exp(X+Y)$

Then the proposition is proved upon passing to generating series in (13)), and
using the fact that

$\frac{\exp(X+Y)-1}{X+Y}=\sum_{r,s\geq 0}\frac{B_{r,s}}{r!s!}X^{r}Y^{s}.\qed$

So we have proved that the function $\log(p^{cr}_{\nabla})(x)$ appearing in
(4) is of the form

$\log(p^{cr}_{\nabla})(x)=\int_{b}^{x}\alpha A+\int_{b}^{x}\beta
B+\sum_{r,s\geq 0}g_{r,s}\sigma_{r,s},$

where the $g_{r,s}$ are given in terms of the $G_{u,v}$ by Proposition 8 and
the $G_{u,v}$ are given in terms of the functions $p_{n}$ by (11).

Now the proof of Theorem 1 comes down to calculating the composition of
$\iota_{KZB}$ with the period map.

###### Proof of Theorem 1.

We begin by recalling a theorem of Kurlin [18], which shows how the
Baker–Campbell–Hausdorff formula simplifies for the maximal metabelian Lie
algebra on two generators:

(14)
$\log\left(e^{X}e^{Y}\right)=X+Y+\frac{1}{\mathrm{ad}_{Y}}\left(1-\frac{\exp\left(\mathrm{ad}_{X}\right)-1}{\mathrm{ad}_{X}}\frac{\mathrm{ad}_{X}+\mathrm{ad}_{Y}}{\exp\left(\mathrm{ad}_{X}+\mathrm{ad}_{Y}\right)-1}\right)[X,Y].$

But $[\sigma_{r,s},[X,Y]]=0$ so when we set $X=-\int_{b}^{x}\beta B$ and
$Y=\log(p_{\nabla}^{cr})(x)$, we can replace the $\mathrm{ad}_{Y}$s in (14)
with $\int_{b}^{x}\alpha\mathrm{ad}_{A}+\int_{b}^{x}\beta\mathrm{ad}_{B}$.
This yields the formula (3). ∎

We conclude by describing the period map for a tangential basepoint $b$ at
$\infty$. We use the notations $\alpha$, $\beta$, $e_{n}$ and $P_{n}$ from the
introduction, and write $Y$ for the open subset of $E$ obtained by removing
the points with $y=0$. Let $\beta^{{}^{\prime}}$ be the one-form
$\beta+dP_{1}$ and recall that at (1) we constructed from the KZB connection a
coordinate $\iota_{KZB}$ on the de Rham period space. We will assume that the
endpoint of the de Rham period map is an integral point on $X$ so that we do
not need to choose a branch of the $p$-adic logarithm. With respect to
applications to Selmer schemes, this is by no means a severe restriction.

######  Theorem 9.

For $x\in X(K)\cap Y(K)$, we have

$\iota_{KZB}\circ j^{dR}(X,b)(x)=\int_{b}^{x}\alpha A+\sum_{r,s\geq
0}g_{r,s}\sigma_{r,s}-\int_{b}^{x}\beta^{\prime}S\left(\int_{b}^{x}\alpha\mathrm{ad}_{A},\int_{b}^{x}\beta^{\prime}\mathrm{ad}_{B}\right)\left(-\int_{b}^{x}\alpha\sigma_{0,0}+\sum_{r,s\geq
0}g_{r,s}\sigma_{r+1,s}\right)$

where $j^{dR}(X,b)(x)$ is the de Rham period map and

$S(U,V)=\frac{1}{U+V}\left(1-\frac{\exp(-V)-1}{V}\frac{U+V}{\exp(U+V)-1}\right).$

The $g_{r,s}$ are iterated integrals of certain algebraic functions on $E/K$,
characterised by the equality of formal power series

$\sum_{r,s\geq 0}g_{r,s}X^{s}Y^{r}=\sum_{r,s\geq
0}\left(\sum_{\begin{subarray}{c}u+n=r\\\
v+m=s\end{subarray}}\frac{B_{u+v}\left(\int_{b}^{x}\beta^{{}^{\prime}}\right)^{u}\left(\int_{b}^{x}\alpha\right)^{v}G_{n,m}}{u!v!}\right)X^{s}Y^{r}-\left(\int_{b}^{x}\alpha\right)\left(\int_{b}^{x}\beta^{{}^{\prime}}\right)T\left(\int_{b}^{x}\alpha
X,\int_{b}^{x}\beta^{{}^{\prime}}Y\right),$

where

$T(U,V)=\frac{1}{U}\left(1-\frac{U+V}{V}\frac{\exp(U)-1}{\exp(U+V)-1}\right)$

and $B_{m}$ is the $m$th Bernoulli number from (2). The algebraic iterated
integrals $G_{r,s}$ are in turn characterised as the iterated integral of the
coefficient of $X^{r}Y^{s}$ in

$-\left(1-\alpha
Y-\beta^{{}^{\prime}}X\right)^{-1}\left(1+(1-\beta^{{}^{\prime}}X)\sum_{i\geq
0}\sum_{k=1}^{i}\left(\beta^{{}^{\prime}}\right)^{i-k+1}\alpha
q_{k}X^{i}\right),$

where we define the algebraic functions $q_{k}$ by

$q_{n}=\sum_{a_{1}+2a_{2}+\dots+na_{n}=n}\frac{1}{a_{1}!a_{2}!\dots
a_{n}!}\prod_{k=1}^{n}\left(\frac{(-1)^{k+1}P_{k}}{k}\right)^{a_{k}}.$

###### Proof.

The proof is exactly the same as that of Theorem 1, except that we replace the
KZB connection over $X$ throughout by the KZB connection over $Y$, as it
appears at (7). ∎

## Acknowledgements

It is a great pleasure to express my gratitude to Minhyong Kim for over two
years’ worth of guidance on Selmer schemes and related topics, as well as his
continued patience as the fruits of these discussions slowly coalesce into
definitive results.

I have also benefited greatly from numerous discussions with Rob de Jeu and
Herbert Gangl on the subject of de Rham period maps and polylogarithms.
Special thanks are also due to Richard Hain and Federico Zerbini for
clarifying some points about the KZB connection.

The author would like to thank the Isaac Newton Institute for Mathematical
Sciences, Cambridge, for support and hospitality during the programme _K-
theory, algebraic cycles and motivic homotopy theory_ where work on this paper
was undertaken. This work was supported by EPSRC grant no EP/R014604/1.

The author would also to acknowledge the support of the University of Warwick
through its Chancellor’s International Scholarship. In addition, the author is
grateful for the visiting studentship provided by the University of Edinburgh.

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