Computation of the unipotent Albanese map on elliptic and hyperelliptic curves

We study the unipotent Albanese map appearing in the non-abelian Chabauty method of Minhyong Kim. In particular we explore the explicit computation of the p-adic de Rham period map jndr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j^{dr}_n$$\end{document} on elliptic and hyperelliptic curves over number fields via their universal unipotent connections U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {U}}$$\end{document}. Several algorithms forming part of the computation of finite level versions jndr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j^{dr}_n$$\end{document} of the unipotent Albanese maps are presented. The computation of the logarithmic extension of U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {U}}$$\end{document} in general requires a description in terms of an open covering, and can be regarded as a simple example of computational descent theory. We also demonstrate a constructive version of a lemma of Hadian used in the computation of the Hodge filtration on U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {U}}$$\end{document} over affine elliptic and odd hyperelliptic curves. We use these algorithms to present some new examples describing the co-ordinates of some of these period maps. This description will be given in terms iterated p-adic Coleman integrals. We also consider the computation of the co-ordinates if we replace the rational basepoint with a tangential basepoint, and present some new examples here as well.


Introduction
This paper will examine some explicit aspects of a method introduced by Minhyong Kim in [20]. This method, known as the "Kim-Chabauty method" or "non-abelian Chabauty" is an analogue to and extension of an earlier method developed by Chabauty in [11] and made more explicit by Coleman in [14].
Suppose X is a curve over a number field K. Let v be a non-archimedean place of K, and K v it's completion at v. The main restriction in the method of Chabauty-Coleman is the rank-genus condition, which requires our curve to have genus strictly greater than the rank of its Jacobian. Kim in [20] provides a possible way around this by replacing the Jacobian with a certain quotient of the de Rham fundamental group U dr of X, relative to some fixed basepoint b ∈ X(K). The quotient in question comes from the Hodge filtration F • on U dr defined in [22]. In [20] Kim shows that the quotient group F 0 U dr \ U dr naturally classifies the de Rham path spaces.. This property of the quotient group allows us to define a higher de Rham unipotent Albanese map j dr . Taking quotients of U dr by its lower central series yields a chain of sub-quotients U dr n , each of which also carries a Hodge filtration. Composition of the map j dr with the natural projection U dr ։ U dr n give us a family of maps j dr n : X(K v ) → F 0 U dr n \ U dr n which are compatible in the following sense: . . .
The maps j dr n have a Zariski dense image in the quotient F 0 U dr n \U dr n . It is this property which allows us to carry out a Chabauty type argument. Kim constructs a variety, the Selmer variety, with suitable finite level versions denoted here by V n , along with finite levelétale unipotent Albanese maps jé t n : X(K v ) → V n . There is also an algebraic map log v,n : V n → F 0 U dr n \ U dr n such that the the image of X(K) in F 0 U dr n \ U dr n (K v ) will be contained in the image of V n . The connection to Chabauty then is that if the following dimension hypothesis holds the algebraicity of log v,n implies the existence of an algebraic function on X(K v ) such that X(K) is contained in it's zero locus. The density of X(K v ) in F 0 U dr n \ U dr n implies that this algebraic function is non-zero on X(K v ), and hence it's zero locus is finite. The finiteness of X(K) then follows.
In several cases it is already known that the dimension hypothesis is satisfied for some n: if X = P 1 Q \ {0, 1, ∞} [19]; if X is a once punctured rank 1 CM elliptic curve over Q then n = 3 suffices [20] and if all the Tamagawa numbers are 1 then n = 2 is sufficient [21]; if X is a complete hyperbolic curve with CM Jacobian [13]; if X is a complete hyperbolic curve a solvable cover of P 1 Q , so in particular any smooth superelliptic curve over Q with genus at least 2 [17]. Also of interest is the fact that several conjectures -Bloch-Kato, Fontaine-Mazur, Janssen -imply the existence of an n ≫ 0 such that the dimension hypothesis is satisfied. In [7], [8] Balakrishnan and Dogra have made an explicit application of non-abelian Chabauty when n = 2 -what they refer to as Quadratic Chabauty -to p-adic heights on elliptic and hyperelliptic curves.
The algebraic function whose existence is implied by the dimension hypothesis will be described locally as a p-adic analytic power series, and in fact will be defined by p-adic iterated integrals. These p-adic iterated integrals will come from the parallel transport associated to the n-th finite level quotient of the universal unipotent connection U associated to X.
The universal unipotent connection U is a universal object among pointed unipotent connections on X and is in fact a pro-unipotent connection with finite level quotients U n . It transpires that the dual of U is the co-ordinate ring of the canonical U dr torsor P dr . As outlined in [20] the universal connection comes with an associated Hodge filtration inducing the Hodge filtration on U dr . As shall be shown in Section 5, in order to determine the co-ordinates of j dr n (x) it will be necessary to determine this Hodge filtration. Lemma 3.6 in [18] will be used to demonstrate how this Hodge filtration may be computed when X is an affine elliptic or hyperelliptic curve. The practical computation of the Hodge filtration is inspired by the approach of Dogra in [16] and Balakrishnan and Dogra in [8].
As part of this computation the logarithmic extension U of the universal connection U on X will need to be computed. In Section 3 a general algorithm for computation of U is outlined (Algorithm 3.5), which is an example of computational descent theory. That is, the logarithmic extension of the universal connection on the complement X = C − D of a complete curve C is computed as a collection of logarithmic connections on an open cover (U i ) of X together with isomorphism satisfying certain descent conditions. This will be necessary because unlike the case of P 1 \{0, 1, ∞}, for curves of positive genus the extension U will in general have non-trivial bundle, or it may not be possible to express the connection on just one affine piece as having logarithmic poles at all the missing points. This is what necessitates taking the approach outlined above. The existence of suitable logarithmic extensions of the U n compatible with projection is proven (Theorem 3.15). Conditions are imposed on the extensions, and using these two algorithms are provided, Algorithm 3.18 and Algorithm 3.22, allowing for the iterative computation of U n for elliptic and odd hyperelliptic curves respectively. We then demonstrate how the computation of the Hodge filtration is contained in the computation of these extensions. In Theorem 4.9 we present a constructive version of Lemma 3.6 in [18] when X is an elliptic curve. From this we develop Algorithm 4.10 for the explicit computation of the Hodge filtration on the U n in this case. We also consider some examples when X is a hyperelliptic curve.
In Section 5 we apply the previous algorithms to the computation of j dr n for several new n. Previously, j dr n has been determined for elliptic curves only when n = 1, 2 and for hyperelliptic curves it has only been computed for n = 1, 2 in specific cases. We compute the co-ordinates of j dr 3 (Proposition 5.1) and j dr 4 (Proposition 5.2) for elliptic curves, and j dr 2 (Proposition 5.3) for general odd hyperelliptic curves. In Section 6 we consider the scenario where our basepoint is tangential: this is useful in those cases where a rational basepoint is lacking, and to provide a greater wealth of examples. We provide new explicit descriptions of the coordinates of the maps j dr 2 (Proposition 6.2) and j dr 3 (Proposition 6.3) for elliptic curves with a tangential basepoint at infinity, and j dr 2 (Proposition 6.4) for odd hyperelliptic curves with a tangential basepoint at infinity.

The Universal Connection
Let K be a field of characteristic 0, let X be a K-scheme, and suppose we have a fixed basepoint b ∈ X(K). Let V be a vector bundle on X. Remark 2.2. We will often refer to a vector bundle V with connection ∇ simply as a connection and write such objects either as (V, ∇) or simply as V. Definition 2.3. We say that V is a flat or integrable connection if the induced morphism is the zero map. Note that if X is a curve, then any connection V is automatically flat.
Given a connection (V, ∇) with V of rank n, there is a matrix Ω ∈ gl n ⊗ Ω 1 X/K called the connection matrix which determines ∇: suppose that we have a local basis e i : O X ֒→ V (1 ≤ i ≤ n). Let U ⊂ X be some trivialising neighbourhood in X. Then ∇(e i ) ∈ V ⊗ Ω 1 X/K (U ), and so there are ω ij ∈ Ω 1 X/K (U ) such that ∇(e i ) = j e j ⊗ ω ij We let Ω := (ω ij ). We may show that in matrix notation ∇(e · f ) = e · (df + Ω · f ), and so ∇ acts locally as d + Ω. Definition 2.5. A connection V is unipotent with index of unipotency less than or equal to n if there is a decreasing sequence of sub-connections such that the quotients V i+1 /V i are isomorphic to a direct sum of copies of (O X , d) i.e. they are trivial.
We obtain the following category on X: Definition 2.6. Let Un n (X) be defined to be category whose objects are unipotent vector bundles on X with flat connection having index of unipotency less than or equal to n with morphisms being morphisms of connections.
Given some b ∈ X(K) we can define a functor e b : Un(X) → Vect K from Un(X) to the category of vector spaces over K, sending V → V b := b * V. We may show that e b is a fibre functor (Un(X), e b ) is a neutral Tannakian category. In [1] they make the following definition: Definition 2.7. Given a neutral Tannakian category (C, ω) over a filed k, define the pointed category C * to be the category whose objects are pairs (V, v), where v ∈ ω(V ) and morphisms f : With this in mind we may define the universal connection as a universal projective system of connections: That is, there is a morphism φ : U n → V of connections such that It is shown in [23, I, Chapter 2] that such a universal projective system {U n , u n } n≥0 exists in Un(X) * . There it is the object referred to as the "generic pro-sheaf", G b,dr . Another construction is contained in Theorem A in [24]. A nice reference for the construction on a general scheme X over a field K is contained in [20, §1] where we see that we can take (U 0 , u 0 ) = (O X , 1) and u n = 1 for all n.
In loc.cit., Kim shows how the universal connection together with its Hodge filtration defines a map -the unipotent Albanese map j dr -may be used to implement higher order Chabauty-Coleman arguments. In order to compute the co-ordinates of this map on X, we need to compute the parallel transport associated to U as well as its Hodge filtration. We turn our attention to the case when X is an affine curve, since U has an explicit description in this case. Definition 2.12. Let C be a smooth projective curve of genus g over a field K of characteristic 0. Let D be a non-empty divisor of size r and let X := C − D. Let α 0 , ..., α 2g+r−1 be a K-basis of H 1 dr (X). We will assume that this basis is chosen so that α 0 , ..., α g−1 form a basis of H 0 (C, Ω 1 ). Let V dr := H 1 dr (X) ∨ with basis elements A i dual to α i . Let R be the tensor algebra of V dr i.e.
Let I be the two-sided ideal generated by A 0 , ..., A 2g+r−1 and define R n to be the quotient of R by words of length ≥ n + 1. Then define and let ∇ n be the connection such that 13. Note that although we have only defined ∇ n on f ∈ R n , this defines the connection on U n via the Leibniz identity: if f ∈ R n and s is a section of O X , then Remark 2.14. For convenience we will often write ∇ instead of ∇ n . Remark 2.16. The connections U n each carry a Hodge filtration, and it is this we initially wish to compute. The Hodge filtration on the U n induces the Hodge filtration on U and hence on U ∨ . As noted in loc.cit., this dual connection turns out to be the co-ordinate ring of the canonical de Rham torsor, P dr , on X, and so induces a Hodge filtration on the de Rham fundamental group with basepoint b, U dr . We shall see that the computation of the Hodge filtration is contained in the computation of the logarithmic extension of U on X to the complete curve C.

Logarithmic Extensions of the Universal Connection
Let C, D and X be as in Definition 2.12 and let Ω 1 C (D) be the sheaf of logarithmic differentials on C along D. This sheaf will consist of differentials on C regular on X and with at worst logarithmic (i.e. simple) poles along D.
Definition 3.1. A logarithmic connection on C with logarithmic poles along D is a vector bundle V equipped with a linear sheaf morphism ∇ : V → V ⊗ Ω 1 C (D) satisfying the Leibniz property. A morphism of logarithmic connections is defined analogously to a morphism of connections.
Remark 3.2. It is important to note here that a logarithmic connection is not a connection -it is simply a sheaf morphism satisfying similar properties. In particular, there is no diretly analagous notion of parallel transport. This shall be explored further in Section 6. Definition 3.3. Let V be a connection on X = C − D. A logarithmic extension of V to C along D is a logarithmic connection V on C with logarithmic poles along D such that V| X ∼ = V.
In [18] Hadian defines unipotent logarithmic connections, which are iterated extensions of trivial connections (P ⊗ K O C , id P ⊗ K d). Here P is a finitely dimensional K-vector space and d : is the canonical differential. The logarithmic extension of the universal connection on the affine curve X is then used as a model for the universal connection, replacing the connection on the affine curve with the logarithmic connection on the complete curve. In loc.cit. Lemma 3.6 gives us a practical way to compute the Hodge filtration of this extension on the complete curve, where it will be the unique filtration satisfying several properties. It is important to note that the same lemma fails if applied to the connection on the affine curve (see Remark 4.5). That such extensions exist is a consequence of a theorem of Deligne ( [15, Proposition 5.2]).
To compute logarithmic extensions U n of the U n we utilise a description in terms of an open covering of C. The idea here is that objects and morphisms over schemes should be described locally. That is, an object is described on open subsets of some cover together with gluing morphisms called descent datum which satisfy some cocycle condition.
For example, take an open cover Uij such that φ ii = id and which satisfy the coycle condition: then there is a vector bundle V of rank r on Z, with transition functions defined by local trivialisations e i : V| Ui → O r Uij in the obvious way. So descent provides us with a way of deciding when an object on a cover Y of Z descends to an object on Z, by checking conditions on the proposed descent datum.
Descent for logarithmic connections on C along D is described by the following descent datum: an open cover (U i ) i of C; logarithmic connections (O r Ui , ∇ i ) along U i ∩ D; and isomorphisms of connections ,∇i) and which satisfy the cocycle condition Thus to explicitly construct a logarithmic extension U n of U n we need to determine an open cover of C with logarithmic connections on each patch satisfying the descent conditions. This is because in general, the extension of the connection will not have a trivial bundle. We shall also see later that it is necessary for us to construct the U n so that they form a projective system of logarithmic connections. That is, for each n we want a projection map U n → U n−1 . However, we note that the construction of U n is functorial in the descent datum in the following sense: if we have a family of morphisms ρ i : (O r Ui , ∇ i ) → U n−1 | Ui compatible with G ij over U ij and satisfying the cocycle condition, then there is a morphism ρ : U n → U n−1 . Thus it is enough to construct descent datum compatible with projection to U n−1 . To construct the descent datum we will need the following lemma, which follows from an easy computation: Lemma 3.4. Given a (logarithmic) connection V on a curve Z, suppose that with respect to the local basis (e i ) it has connection matrix Ω. If G is an automorphism of V, the transformation is called a gauge transformation of Ω. This is the connection matrix of V with respect to the local basis (G −1 e i ).
Suppose now that U is the universal connection on X and let U n be the level n quotient of U. We want an iterative algorithm by which we may compute the logarithmic extensions U n of the U n successively. Observe that U 0 = (O C , d) is a logarithmic extension of U 0 = (O X , d), and we use this as our base case. The algorithm below iteratively computes descent datum compatible with projections. It determines gauge transformations G which map the connection matrix of U n over X to a connection matrix with at worst logarithmic poles at points of D, and then determining a suitable open cover of C such that this defines logarithmic connections U i n on the patches Y i . This is done subject to the condition that on each candidate open patch Y i we should have a commutative diagram The construction ensures that the descent datum satisfies the descent conditions, and thus defines a logarithmic connection on C along D compatible with projections. Note that it will be convenient for the computations that follow to describe the gauge transformations G as elements of K(C) ⊗ K gl N for some N .
n , G ij n with:

Output
• the logarithmic extension U n+1 of U n+1 , defined by defined by logarithmic connections n+1 is compatible with projection to level n • the image C i n+1 of C n+1 has at worst logarithmic poles at d i (3) Define: where the map is defined by the matrices defining G 0j n+1 (4) Glue the logarithmic connections U i n+1 , U n+1 together via the isomorphisms G ij n+1 to obtain a logarithmic connection U n+1 with log poles along D.
Remark 3.6. In step (2) above we let U i n+1 := R n+1 ⊗ O Yi . This is because the extension of the finite level quotients U n to U n should fit into an exact sequence 0 → V ⊗n dr ⊗ O C → U n → U n−1 → 0 of connections as in [18, §2-3]. There it is shown that the logarithmic connection constructed in this way will define a universal projective system in the category of unipotent logarithmic connections on C with logarithmic poles along D. Therefore, the universal connection on C will be a maximal quotient of this logarithmic connection.
Remark 3.7. Strictly speaking, on curves we will only need to use two open subsets of C in our construction, one of which contains all of the missing points of C missing from X. The above algorithm is included since it will be useful for higher dimensions.

Computing the extension on elliptic curves
In what follows, we describe this process explicitly for an arbitrary elliptic curve. Let C be an elliptic curve over a field K of characteristic 0 with point at infinity P ∞ . Let X := C − {P ∞ } be the punctured elliptic curve. Recall that C is a genus 1 curve. We specialise the construction of Definition 2.12 to X: Let α 0 , α 1 ∈ H 0 (X, Ω 1 X ) be 1-forms on X with α 0 regular on C and α 1 having a pole of order 2 at P ∞ such that α 0 , α 1 are a K-basis for H 1 dr (X/K). Let V dr := H 1 dr (X/K) ∨ have K-basis A 0 , A 1 dual to α 0 , α 1 . Let R be the tensor algebra of V dr , I := (A 0 , A 1 ), and R n := R/I n+1 . We write the basis element A ⊗i1 Then U n := R n ⊗ O X and ∇ n is the connection on U n defined by f ∈ R n → −A 0 f α 0 − A 1 f α 1 ∈ R n ⊗ Ω 1 X Remark 3.8. It will be convenient at this stage to fix a choice of ordered basis B n for R n . We take as a K-basis the words of length less than or equal to n with a graded lexicographic ordering such that A 0 > A 1 > 1. Note that there are 2 l words of length l and so at level n there are 2 n + 2 n−1 + .. + 1 = 2 n+1 − 1 basis elements. Thus if we have a word w l k , then A 0 w l k = w l+1 k and A 1 w l k = w l+1 k+2 l . We can describe the action of ∇ on a basis for R n : Lemma 3.9. The connection matrix of U 0 is the zero matrix. If C n is the connection matrix of U n with respect to the basis B n , then is the connection matrix of U n+1 with respect to the basis B n+1 where Proof. This is just a straightforward calculation given Remark 3.8.
We now explicitly compute the extension of the U n to logarithmic connections on C as outlined above. At level 0 we set U 0 = (O C , d). Next we compute as an explicit example the extension at level 1.
The connection matrix of ∇ with respect to B 1 is We have a projection map π 1 : U 1 ։ O X . We wish to find an open Y ⊂ C containing P ∞ , a connection U ′ 1 on Y , and a gauge transformation G 1 such that over Y ∞ = X ∩ Y we have a commuting diagram The above condition means that G 1 must be of the form The gauge transformation of C 1 by G 1 is Choose g = 0 and f a function on C with a pole of order 1 at P ∞ such that −α 1 + df is regular at P ∞ . Let Y be the open set {P ∈ C : α 1 + df is regular at P } ⊂ C. Then we define U ′ 1 to be the connection on Y defined with connection matrix C ′ 1 and bundle R 1 ⊗ O Y . Thus we construct the logarithmic extension U 1 of U 1 by gluing U 1 and U ′ 1 together along Y ∞ via the isomorphism G 1 .
Remark 3.11. Note that with Y as in Example 3.10, X ∩ Y = Y − {P ∞ }, and we will write Y ∞ for this open subset of C.
Remark 3.12. Suppose that y 2 = g(x) is an affine model for an elliptic curve C over K, where g(x) ∈ K[x] is a degree 3 polynomial. Suppose that e 1 , e 2 , e 3 are the roots of g(x) in an algebraic If we let f be a multiple of y/x ∈ K(C) in the above, then we may let Y : We now describe a process for iteratively computing G n+1 given G n . As in the above example, we find that G n+1 should be of the form Lemma 3.13. Let n ≥ 1 and let C n (resp. C n+1 ) be the connection matrix of U n (resp. U n+1 ) with respect to the basis B n (resp. B n+1 ). Suppose that there is a gauge transformation G n at level n and a connection U ′ n over Y extending the connection at level n to a logarithmic connection U n on C with connection matrix C ′ n . Suppose that this extension is compatible with the projection to level n − 1 and that U ′ n has connection matrix C ′ n . If G n+1 is a gauge transformation of C n+1 of the form Proof. This follows easily from the definition of the gauge transformation.
Remark 3.14. Note that in the above, C ′ n will be a matrix of 1-forms with at worst logarithmic poles at P ∞ by inductive hypothesis. Therefore, in order to compute a suitable gauge transformation G n+1 we need to find a matrix of functions H n+1 such that dH n+1 + D n+1 G n − H n+1 C ′ n has entries with at worst logarithmic poles at P ∞ . In the following theorem, we will show that we can do this in such a way that the matrix H n+1 can be easily computed. Before coming to the theorem, we define the following two maps for r, i, j, k ∈ Z with r, k ≥ 1 and j ≥ i + 1.
These will allow us to later describe the computation of U n for both elliptic curves and odd hyperelliptic curves with one general algorithm.
Theorem 3.15. Let C, X, U n and C n be as above. Then there is an open Y ⊂ C containing P ∞ such that for all n there is an isomorphism G n of U n with a connection Proof. In Example 3.10 we have already shown that this is true in the case that n = 1, so now we proceed by an induction argument, where our induction hypothesis will be that a) and b) and hold for all n. Let Y be as in Example 3.10. Suppose that we have computed the extension up to level n satisfying the conditions of the theorem. Recall that the matrix C ′ n+1 is , by inductive hypothesis we need to show that can H n+1 can be chosen so that D ′ n+1 has entries with at worst logarithmic poles at P ∞ whilst satisfying a) and show that this implies C ′ n+1 satisfies b). For i between 0 and n consider the matrix D i n+1 formed by the columns 2 n+1 − 2 i+1 + 1 to Consider the r-th block matrix V r,i n+1 in this matrix. We want to express this matrix V r,i n+1 in terms of the block matrices in H n , C ′ n and H n+1 . We will say that W r,i n+1 in H n+1 stems from the block matrix W s,j n+1 for i + 1 ≤ j ≤ n if the rows of H n+1 containing W r,i n+1 are a subset of the rows of H n+1 containing W s,j n+1 . It is not difficult then to see that W r,i n+1 in H n+1 stems from the block matrix W A(r,i,j,2),j n+1 . A simple calculation shows that and by the inductive hypothesis the contribution of dH n+1 + D n+1 G n to V r,i n+1 is The contribution coming from H n+1 C ′ n is more complicated to work out. We introduce a second inductive step to the argument, with our inductive hypothesis being that a) is satisfied by H n+1 for j = n, n − 1, ..., i + 1. We then show that a) is satisfied by H n+1 for j = i.
For our base case, we need to show that H n+1 satisfies a) for j = n. It's straightforward, however, to see that and so D r,n n+1 = dW r,n n+1 − α r−1 I 2 n for r = 1, 2. We choose W r,n n+1 = h r,n n+1 I 2 n where h r,n n+1 is a function on C such that dh r,n n+1 − α r−1 is regular at P ∞ . So we have that a) holds for j = n. Now let i < n and assume then that a) holds for j = n, ..., i + 1. Now, rows (r − 1)2 i + 1 to (r − 1)2 i + 2 i are the rows of H n+1 containing W r,i n+1 . They have the form where each H r,i,j n+1 is a 2 i × 2 j matrix. Then by the inductive hypothesis on H n n+1 , ..., H i+1 n+1 we have that for j = n, .., i + 1 and so we can conclude that the contribution of H n+1 C ′ n to V r,i n+1 is given by This expression can be simplified further by noting the following: since we are assuming that i < n, then the terms c Applying this repeatedly to the expression above The bracketed expression is a 1-form on C, and so we can choose some h r,i n+1 a function of C over k such that has at worst logarithmic poles at P ∞ and which has no poles on Y (for example by choosing By induction we conclude that for each n there is an isomorphism G n from U n to U ′ n over Y ∞ such that the connection matrix of U ′ n is logarithmic on Y along P ∞ . Gluing U ′ n , U n along Y ∞ by the isomorphism G n+1 we obtain a logarithmic connection U n on C extending U n .
The following lemma shows how we simplify the computations in the above theorem: Lemma 3.16. Let C, D and X be as above. Let n ≥ 1 and let U n be the n-th level finite quotient of the universal connection U on X. Let U ′ n be the connection and G n the gluing morphism defined in Theorem 3.15. Then the h r,i n can be chosen such that for Proof. This is a simple induction argument. The base case n = 1 is simple: for r = 1, 2 we have c r,1 2 = dh r,1 2 − α r−1 , c r,0 1 = dh r,0 1 − α r−1 and so we can choose h r,0 1 = h r,1 2 and thus c r,0 1 = c r,1 2 . In fact it is clear that we can choose h r,n n+1 = h r,n−1 n for r = 1, 2 and all n. We now proceed by induction. Suppose that the hypothesis is true at level n, and that for n ≥ j > i > 0 we can choose h r,j n+1 = h r,j−1 n , c r,j n+1 = c r,j−1 n . We have by Theorem 3.15 that for Note that A(r, i, j, 2) = A(r, i−1, j −1, 2) and B(r, i, j, 2) = 2B(r, i−1, j −1, 2). The inductive hypothesis above then implies that : . Hence for i = 1, .., n we can choose the h r,i n+1 in this way and hence the original inductive hypothesis is true at level n + 1.
Remark 3.17. The previous two results combined allow us to compute gauge transformations G n+1 with relatively greater ease than otherwise. We simply need to determine the image of 1 at each level, and by Lemma 3.16 this immediately determines a suitable gauge transformation by previous knowledge of G n . That is, at level n + 1 we need to compute h r,0 n+1 for r = 1, .., 2 n such that has at worst logarithmic poles at P ∞ . By expanding (3.3) in a local parameter t at P ∞ , we can compute h r,0 n+1 locally as the formal integral of  • elliptic curve C over a characteristic 0 field K with affine model X of the form • universal connection U on X with respect to the basis α 0 , α 1 of H 1 dr (X) as at the beginning of Section 3.1 • the connection matrix C ′ n over Y ⊂ C of the logarithmic extension U n of U n • the gauge transformation G n defining the extension U n with respect to the basis B n Output: • the connection matrix C ′ n+1 over Y of the logarithmic extension U n+1 of U n+1 • the gauge transformation G n+1 defining the extension U n+1 with respect to the basis B n+1 Algorithm: has at worst logarithmic poles at P ∞ . Put c r,0 n+1 = (3.4).
. This gives us an efficient way to compute the gauge transformations G n , since at each level we only need to compute 2 n entries of the matrix rather than all 2 n × 2 n − 1 entries of H n . = −α 1 + df . Note that since α 1 has a pole of order 2 at P ∞ we must have that f has a pole of order 1. Following Algorithm 3.18, we set h 1,1 2 = h 1,0 1 = 0, h 2,1 2 = h 2,0 1 = f . We need to compute h r,0 2 for r = 1, 2, 3, 4 such that 1 2 has at worst logarithmic poles at P ∞ . Here we find that Since α 0 is regular at P ∞ and f has a simple pole there we can take h 1,0 2 = h 2,0 2 = h 3,0 2 = 0 and also take h 4,0 2 = 1 2 f 2 . Therefore, we find that we can take the matrix G 2 to be

Computing the extension on hyperelliptic curves
Let C be an odd hyperelliptic curve of genus g ≥ 2 over a field K of characteristic 0. Suppose that we have an affine model of C of the form with deg(f ) = 2g + 1. There is a single K-rational point P ∞ at infinity. Let X be the affine curve y 2 = f (x) over K so that X = C − {P ∞ }. The de Rham cohomology of X has a basis of size 2g. We now specialise Definition 2.12 to X: Let α 0 , α 1 , ...α 2g−1 ∈ H 0 (X, Ω 1 X ) be 1-forms on X such that these form a K-basis for H 1 dr (X/K). Since C is of genus g, we may further assume that α 0 , .., α g−1 is a K-basis of H 0 (C, Ω 1 C ). Let V dr := H 1 dr (X/K) ∨ have basis A 0 , ..., A 2g−1 dual to α 0 , ..., α 2g−1 . Define R to be the tensor algebra of V dr , I := (A 0 , ..., A 2g−1 ). and R n := R/I n+1 . Then U n := R n ⊗ O X and ∇ n is the connection on U n defined by

Give the basis a graded lexicographic order such that
So we can describe the action of ∇ on a basis for R n : The connection matrix of U 0 is 0. If C n is the connection matrix of U n with respect to the basis B n , then is the connection matrix of U n+1 with respect to the basis B n+1 where It is straightforward to prove the analogous versions of Theorem 3.15 and Lemma 3.16. Thus, we iteratively compute the logarithmic extensions of U n using the following algorithm: • genus g hyperelliptic curve C over a characteristic 0 field K with affine model X of the form y 2 = f (x) with deg f = 2g + 1 • universal connection U on X with respect to the basis α 0 , ..., α 2g−1 of H 1 dr (X) as at the start of Section 3.2 • the connection matrix C ′ n over Y ⊂ C of the logarithmic extension U n of U n • the gauge transformation G n defining the extension U n with respect to the basis B n Output: • the connection matrix C ′ n+1 over Y of the logarithmic extension U n+1 of U n+1 • the gauge transformation G n+1 defining the extension U n+1 with respect to the basis B n+1 Algorithm: (1) For 0 < i ≤ n, has at worst logarithmic poles at P ∞ . For 1 ≤ r ≤ (2g) n+1 put c r,0 n+1 = (3.5). ( Remark 3.23. If we let g = 1 then the steps above are the same as those in Algorithm 3.18, and so this provides a general algorithm to compute the logarithmic extensions of U n for elliptic curves or odd hyperelliptic curves. Remark 3.24. If we consider the above computations for an even model X of a genus g hyperelliptic curve C, we have entirely analogous results. The only difference is that we will need to consider an extension of U n around each point at infinity. However, the computations will be exactly the same, and the results above will hold over each point at infinity with 2g replaced by 2g + 1 since H 1 dr (X) has dimension 2g + 1. We thus obtain at each level two logarithmic connections: U + n over Y + open containing P ∞ + and U − n over Y − open containing P ∞ − , as well as gauge transformations G + n , G − n such that Therefore, we can glue the connections U n , U + n , U − n together via the G + n , G − n and obtain a logarithmic connection U n on C. By construction this will be compatible with the projections to level n − 1.

The Hodge Filtration on the Universal Connection
Here we utilise the work of Hadian in [18] in characterising the Hodge filtration on the universal connection U of X := C − D, where C is once more a general smooth projective curve over a field K of characteristic 0 and D is a non-empty divisor. Fix a basepoint b ∈ X(K).
for m < n ∈ Z satisfying the Griffiths transversality property: Note that the dual space V dr in Definition 2.12 has a Hodge filtration induced by the natural filtration on the de Rham complex of X, and this in turn induces a Hodge filtration on V ⊗n dr : F 1 V ⊗n dr = 0; and for p ≥ 0 the filtered subspace F −p V ⊗n dr is generated by those words of containing at most p occurrences of the A i dual to a regular differentials on C. With this we can state the following lemma of Hadian which uniquely characterises the Hodge filtration on U. Lemma 4.4 ( [18], Lemma 3.6). Let C be a smooth projective curve over a field K of characteristic 0, D a non-empty divisor and X := C − D, b ∈ X(K). Let V dr := H 1 dr (X) ∨ and U n be the n-th finite level quotient of the universal connection on X with respect to the basepoint b. Let U n be the canonical extension of this to a logarithmic connection on C. Then there exists for all n a unique filtration (F • U n ) of vector bundles such that i) for all n the filtration F • on U n satisfies Griffiths transversality giving U n the structure of a filtered logarithmic connection ii) for all n the exact sequence of logarithmic connections becomes an exact sequence of filtered logarithmic connections, where V ⊗n dr ⊗ O C has the Hodge filtration induced by the filtration on V ⊗n dr iii) the element 1 such that {(U n , 1)} a universal projective system lies and 1 in b * F 0 U n . Remark 4.5. That the analogous lemma fails for U n on X is precisely why we needed to consider the logarithmic extensions in Section 3. Over X we have an exact sequence of trivial bundles 0 → V ⊗n dr ⊗ O X → U n → U n−1 → 0 If we try to lift generators of F 0 U n−1 as an O X -module up to F 0 U n , we will find that there are multiple ways of doing this whilst still satisfying Griffith's transversality.
For example, taking n = 1 in the above, F −1 U 1 is necessarily U 1 . So any lift of the generators of U 0 as an O X -module together with F 0 V dr will together generate an O X -submodule of U 1 satisfying Griffiths transversality. It can be easily arranged that 1 ∈ b * F 0 U 1 by replacing the generator 1 + i g i A i (where g i ∈ H 0 (X, O X )) with 1 + i (g i − g i (b))A i . So we do not get a unique lift of the Hodge filtration and this approach will not work for affine curves. Thus we can view the computation of the Hodge filtration as being somehow contained in the computation of the extension to a logarithmic connection on C.
We now describe the iterative method of computing the Hodge filtration at level n. This method is based on the application presented by Dogra in [16, §4]. There the Hodge filtration at level 2 for a family of hyperellptic curves of genus 2 with affine model is computed. The idea is as follows: say we have computed the extension U n of U n to a logarithmic connection on C. We compute this as in Section 3 by computing G i n gauge transformations about Y i ⊂ C containing P i ∈ D. These G i n give isomorphisms of U n with a connection U i n over Y i ∩ X. What we want to compute are sub-bundles F • U n over X, F • U i n over Y i satisfying Griffiths transversality and such that they glue together along X ∩ Y i via the G i n . The computation of this is contained in the following algorithm: Input: • Smooth projective curve C over field K of characteristic 0, non-empty divisor D, X = C−D, basepoint b ∈ X(K) • Hodge filtration F • on U n−1 the logarithmic extension of the n-th finite level quotient U n−1 of universal connection on X • Logarithmic extension U n of U n with gauge transformations G i n Output: • Hodge filtration F • on U n Algorithm: 1. For p > 0 put F p U n = 0.
2. For p ≤ 0: (a) Take arbitrary lifts of generators of F p U n−1 to U n over X, F p U i n−1 to U i n over Y i , and, include generators of F p V ⊗n dr (b) For each i: i. For each candidate generator of F p U n restrict to X ∩ Y i and compute the image by G i n ii. Express the images computed in 2.b.i) using restrictions to X ∩Y i of the candidate generators of F p U i n iii. Use expressions determined in 2.b.ii) to determine conditions over X ∩ Y i on lifts of generators of F p U n , F p U i n .
3. Compute generators of F 0 U n . For p = 0 use conditions determined in (2) and 1 ∈ b * F 0 U n to determine the generators of F 0 U n .

For p < 0:
(a) For each local generator of F p+1 U n compute the image under ∇ (b) Use conditions imposed by Griffiths Transversality, (2) and 1 ∈ b * F 0 U n to determine the generators of F p U n .
Remark 4.7. Hadian's lemma ensures that this algorithm will compute the unique Hodge filtration on U n . Note that it is easy to see that this algorithm will terminate with the computation of F −n U n . Observe that F −n V ⊗n dr = V ⊗n dr , and then it is straightforward to see by induction that F −n U n = U n . Example 4.8. In this example we compute the Hodge filtration on U 1 for X := C − D as in Definition 2.12. So U 1 = R 1 ⊗ O X with R 1 = A 0 , ..., A 2g+r−2 K , where C is of genus g and |D| = r > 0 and the A i are dual to the basis differentials α i H 1 dr (X). Note that we have ordered these so that α 0 , .., α g−1 are regular on C. We know that F 0 V dr = A g , ..., A 2g+r−2 K .
Then consider the logarithmic extension of U 1 to a logarithmic connection U 1 on C. Let G i 1 be the gauge transformations over Y i ⊂ C as in Section 3. Then we find that over X ∩ Y i Using the proposed exact sequence over X the generators of F 0 U 1 as an O X -module will be for some a k ∈ H 0 (X, The image of these generators over X ∩ Y i by G i 1 are and hence the generators of F 0 U i 1 | X∩Yi as an O X∩Yi -module are then the above conditions imply that The h i k will be regular on Y i k since they are chosen so that dh i k − α k are regular on Y i k , and α k is regular on C for k < g. So, therefore, we conclude that the sections a k , b i k − h i k glue to give a global section in H 0 (C, O C ) and are, therefore, constant.
Since we require that 1 ∈ b * F 0 U 1 then we must have that a k (b) = 0 and hence that a k = 0 for all k.
Therefore, the Hodge filtration at level 1 on F 0 U 1 is generated over X by 1, A g , ..., A 2g+r−2 and over Y i is generated by In particular, if we choose all h i k = 0 then we find that the Hodge filtration is generated over C by 1, A g , ..., A 2g+r−2 Finally, we note that F 0 U 1 will not be a trivial bundle. This is because the gluing morphisms G i 1 are in general non-trivial.

Computing the Hodge filtration when C is an elliptic curve.
We again return to elliptic curves, and demonstrate that Algorithm 4.6 will output a unique Hodge filtration on the logarithmic extension U we computed in Section 3. We do this by explicitly determining the conditions identified by step (2) in Algorithm 4.6, and showing that these determine a unique solution. Suppose that C is now an elliptic curve over K, and we computed the logarithmic extension of the universal connection on the affine curve X = C − {P ∞ } using Algorithm 3.18. We wish to use this to determine the Hodge filtration on U . We have the following constructive version of Hadian's lemma to compute F 0 U : Theorem 4.9. Let C, X be as above. For all n there is a unique filtration of U n by sub-bundles F i U n such that U n has the structure of a filtered logarithmic connection fitting into an exact sequence of filtered connections 0 → V ⊗n dr ⊗ O C → U n → U n−1 → 0 and such that 1 ∈ b * F 0 U n for all n. Over X we have generators of F 0 U n as an O X -module where a t,s k ∈ H 0 (X, O X ) and w t s is the s-th word of length t in A 0 , A 1 . Suppose also that over Y we have generators of F 0 U ′ n−1 as an O U -module . Then the following conditions are satisfied on restricting sections to Y ∞ = X ∩ Y : i) b n,s n−1 = a n,s n−1 + δ s,2 n−1 h 1,n−1 where j(s, t) = A(s, 0, t, 2).
Proof. We wish to find a filtration F • of U n such that we have, for each i, an exact sequence it is easy to see by induction that F 1 U n = 0. So suppose that we have determined F 0 U n−1 . Over X, we have generators of F 0 U n−1 as an O X -module a t,s k w t s for k = 0, ..., n − 2 (4.1) for some a t,s k ∈ H 0 (X, O X ). Note that in the above, we assume that the a t,2 t k = 0, since the word w t 2 t = A t 1 and we suppose the above are generators of the O X -module.
Over Y , F 0 U ′ n−1 is generated as an O Y -module by . We want to show that there is a unique sub-bundle F 0 U n of U n such that we have an exact sequence We consider this over X and Y separately; over X we find We note then that A n 1 is contained inside F 0 (U n ), and is one of the generators of this O Xmodule. Then, we consider arbitrary lifts of the generators in (4.1): We have assumed that there are no w n 2 n = A n 1 terms appearing in the lifts above since we assume that these lifts are generators of F 0 U X and this also has A n 1 as a generator.
We consider the extension U n as computed in Algorithm 3.18, and the gauge transformation G n used to determine it. With this choice of G n we find that on Y ∞ for t < n and all s and G n : w n s → w n s for all s. We need to determine the image of the generators in (4.3) under G n above. We find that: So, therefore, the image of this generator under G n is We want to describe the generators of the O Y∞ -module F 0 (U ′ n )| Y∞ in terms of these T k n . First of all, we note that the coefficient of the w i 2 i for i = k + 1, ..., n are simply h 2 i−k ,k i . Now, let S n n := T n n = A n 1 , and define S k n recursively by So, therefore, the coefficient of w t 2 t = A t 1 in S k n is δ tk . Now lift the generators in (4.2) to U ′ n and suppose that these lifts are generators of the O Y -module F 0 (U n ). As before, for some b n,s k ∈ H 0 (Y, O Y ) these will be Then we must have that S k n | Y∞ = S k n | Y∞ by comparing co-efficients of the w t 2 t = A t 1 . Hence, (4.6) is true over Y ∞ if we replace the S k n with S k n . Using this we can give a recursive description of the b t,s k .
First of all, note that a n,s n−1 w n s + 2 j=1 w n j2 n−1 h j,n−1 n (4.7) Hence, S n−1 n | Y∞ = T n−1 n − h 2,n−1 n S n n = A n−1 1 + 2 n −1 s=1 a n,s k w n s + w n 2 n−1 h 1,n−1 n | Y∞ . Therefore we obtain the condition b n,s n−1 | Y∞ = a n,s n−1 + δ s,2 n−1 h 1,n−1 n | Y∞ (4.8) Now we consider the S k n for k < n − 1. We first split up the sum by taking the terms with words of length n outside the sum and find that Also, we have S k n = S k n−1 + 2 n −1 s=1 b n,s k w n s . So then restricting to Y ∞ and putting (4.6) and (4.9) together we get: We need only consider coefficients of words of length n. Take the sum n−1 Then if we fix t, note that for each p in {1, ..., 2 n − 1} there are unique s, j such that s + (j − 1)2 t = p. Here, j will be the unique integer lying between 1 and 2 n−t such that 1 ≤ p−(j −1)2 t ≤ 2 t . Then j = A(p, 0, t, 2), where A is the function defined in (3.1). Since there will be no terms in the sum with s = 2 t , we find that there are no terms with p = j ′ 2 t . Let j(p, t) := A(p, 0, t, 2). Then we find that Hence the coefficient of w n s which will be a n,s k + where δ uv is the Kronecker delta. The restriction of (4.10) to Y ∞ is exactly the restriction of b n,s k to Y ∞ . To summarise, we find that the sections b n,s k , a n,s k should satisfy the following two conditions: i) b n,s n−1 |Y ∞ = a n,s n−1 + δ s,2 n−1 h 1,n−1 We now show how these conditions imply uniqueness of the lifts. We know that when n = 0, all the a n,s k are uniquely determined -this is a trivial. Fix s and suppose that for all m < n, that the a m,s k and b m,s k are uniquely determined. Then suppose that we have a n,s k , b n,s k and a n,s k , b n,s k defining lifts of the generators at level n − 1 to level n. We claim that that a n,s k = a n,s k and b n,s k = b n,s k for all k, s. Since the a n,s k , b n,s k also satisfy conditions i) and ii) we find that over Y ∞ b n,s n−1 = a n,s n−1 + δ s,2 n−1 h 1,n−1 n b n,s k = a n,s k + where the final equality is by inductive hypothesis. The above immediately show that b n,s n−1 − b n,s n−1 = a n,s n−1 − a n,s n−1 . Since a n,s n−1 , a n,s n−1 are constant we find that b n,s n−1 − b n,s n−1 is constant.We now consider the difference b n,s n−2 − b n,s n−2 . Our inductive hypothesis implies that b n,s n−2 − b n,s n−2 = a n,s n−2 − a n,s n−2 − (b n,s n−1 − b n,s n−1 )h 2,n−2 n−1 Notice that h 2,n−2 n−1 is a function on C such that dh 2,n−2 n−1 − α 1 is regular at P ∞ . That is, h 2,n−2 n−1 has a pole of order 1 at P ∞ . Furthermore, b n,s n−1 − b n,s n−1 is constant. Since b n,s n−2 − b n,s n−2 ∈ H 0 (Y, O Y ) and since there are no regular functions on X with a simple pole at P ∞ , we must have that b n,s n−1 = b n,s n−1 , and hence that a n,s n−1 = a n,s n−1 . This then implies that b n,s n−2 − b n,s n−2 = a n,s n−2 − a n,s n−2 on Y ∞ . Therefore, the sections b n,s n−2 − b n,s n−2 ∈ H 0 (Y, O Y ) and a n,s n−2 − a n,s n−2 ∈ H 0 (X, O X ) glue to give a global section of O C and hence are constant. Now we suppose the following: for some k < n − 2 that b n,s k+1 − b n,s k+1 = a n,s k+1 − a n,s k+1 is constant and that for all l > k + 1, b n,s l = b n,s l and a n,s l = a n,s l . Then, repeating the argument above we will conclude that b n,s k − b n,s k = a n,s k − a n,s k − (b n,s k+1 − b n,s k+1 )h 2,k k+1 and as before we find that b n,s k+1 = b n,s k+1 ,a n,s k+1 = a n,s k+1 and that the difference b n,s k − b n,s k = a n,s k − a n,s k is constant. By induction then we conclude the following: for all k > 0, b n,s k = b n,s k , a n,s k = a n,s k and when l = 0 we have b n,s 0 − b n,s 0 = a n,s 0 − a n,s 0 is constant. To conclude that this last constant is zero, we recall that we have that we are fixing the Hodge filtration so that 1 ∈ b * F 0 U n . Therefore, a n,s 1 (b) = a n,s 1 (b) = 0, and hence we conclude that the equality b n,s k = b n,s k , a n,s k = a n,s k holds for all l and F 0 U n is unique.
It remains to show the existence of suitable lifts (4.3), (4.6) satisfying conditions i) and ii). Condition i) implies that for each s we have a n,s n−1 = λ n,s n−1 ∈ k since h 1,n−1 n = h 1,0 1 is constant. We now use this in condition ii) with k = n − 2: b n,s n−2 = a n,s n−2 + over Y ∞ . Note that ( * ) is known, since we assume we already know the Hodge filtration at level n − 1 and have computed the gauge transformations G n−1 , G n . Since b n,s n−2 must be regular at P ∞ , by expanding ( * ) in a local parameter t. we determine a n,s n−2 uniquely up to addition of a constant.That is, there is a unique polynomial f n,s n−2 (x, y) ∈ k[x, y] with no constant term and some constant λ n,s n−2 ∈ k such that a n,s n−2 = f n,s n−2 (x, y) + λ n,s n−2 . Since no sections in H 0 (X, O X ) can have simple pole at P ∞ , and f has simple pole there, we can then determine λ n,s n−1 as the unique constant such that b n,s n−2 (t) ≡ 0 mod k[[t]]. By iterating this process,we find for each s and k ≥ 2 polynomials f n,s k (x, y) ∈ k[x, y] with no constant term and constants λ n,s k such that a n,s k = f n,s k (x, y) + λ n,s k . We also find polynomials f n,s 1 (x, y) ∈ k[x, y] such that there is a constant λ n,s 1 with a n,s 1 = f n,s 1 (x, y) + λ n,s 1 . Finally, we use the fact that a n,s 1 (b) = 0 to determine the λ n,s 1 and find that a n,s . These a n,s k then define b n,s k using conditions i) and ii).
The construction at the end of the proof above motivates the following algorithm to determine the Hodge filtration at level n. • elliptic curve C over a characteristic 0 field K with affine modelX of the form y 2 = f (x) with deg f = 3 • logarithmic extension U n of n-th finite level quotient U n of universal connection on C computed using Algorithm 3.18 • the gauge transformation G n defining the extension U n over an open Y ⊂ C containing P ∞ • generators of F 0 U n−1 over X, Y given by a t,s k w t s for k = 0, ..., n − 2 over X and b t,s k w t s for k = 0, ..., n − 2 over Y . Output: • generators of F 0 U n over X, Y given by a t,s k w t s for k = 0, ..., n − 1 over X and Algorithm: (1) For 1 ≤ s ≤ 2 n − 1 put a n,s n−1 = λ n,s n−1 ∈ k some constant to be determined, let k := n − 2.
(2) While k > 1 do (a) for 1 ≤ s ≤ 2 n − 1 do i) Compute the Laurent expansion p n,s k (t) in the parameter t = x y of ii) Compute a polynomial f n,s k (x, y) ∈ k[x, y] with no constant term such that f n,s k (x, y)(t) + p n,s . iii) If f n,s k (x, y)(t)+ p n,s k (t) has expansion µt −1 + o(1) and f has expansion νt −1 + o(1) in t for constants µ, ν ∈ k, then put λ n,s k+1 = µ ν . iv) Put a n,s k = f n,s k (x, y) + λ n,s k for λ n,s k ∈ k some constant to be determined. v) Put k := k − 1.
(4) For 1 ≤ s ≤ 2 n − 1 put a n,s 1 = f n,s 1 (x, y) − f n,s 1 (x, y)(b). Remark 4.11. We could attempt to produce some similar conditions on the Hodge filtration of a hyperelliptic curve -and indeed, we do this in the case that n = 2 for odd curves of genus g below -but in general these conditions will be even more complicated than those described above.

Example: elliptic curves
We now use the results of Sections 3 and 4 to compute F 0 U 2 , F 0 U 3 and F 0 U 4 . In [21, Lemma 3.2] Kim shows that F 0 U 2 is generated by 1, A 1 , A 2 1 , so we expect to determine the same description of the Hodge filtration here.
Proposition 4.12. The generators of F 0 U 2 are 1, A 1 , A 2 1 , and in particular, these generate F 0 U 2 as an O X -module.
Proof. In Example 3.20 we computed that the gauge transformation G 2 is of the form 1}, where f has a pole of order 1 at P ∞ . First considering our bundles over X, we must have that F 0 U 2 is generated by A 2 1 and some lifts of 1, A 1 . We let these lifts be 1 + for k = 0, 1 are some sections giving lifts of the generators of F 0 U ′ 1 . Note that h 1,1 2 = h 1,0 1 = 0 and that h 2,1 2 = h 2,0 1 = f . Using Theorem 4.9 we conclude for s = 1, 2, 3 that So, therefore, the sections b 2,s 1 , a 2,s 1 glue to give a global section of C and hence are constant. Then since f has a pole of order 1 at P ∞ but b 2,s 0 does not we must conclude that the b 2,s 1 = a 2,s 1 = 0. Otherwise, a 2,s 0 ∈ H 0 (X, O X ) would have a pole of order 1 at P ∞ which is not possible. Then by a similar argument, we conclude that the b 2,s 0 = a 2,s 0 are constants. Finally, since a 2,s 0 (b) = 0, we conclude that the a 2,s 0 = 0. Hence, the filtration at level 2 is generated by 1, A 1 , A 2 1 .
Proposition 4.13. The extension U 3 of the connection U 3 on X can be computed via the gauge transformation G 3 with matrix where λ ∈ K is such that λdf − 1 2 f 2 α 0 has a logarithmic pole at P ∞ . Proof. This is a straightforward application of Algorithm 3.18.
Remark 4.14. By expanding α 0 , α 1 , f in the local parameter t = x y around P ∞ , we can easily verify that λ depends only on the choice of α 0 , α 1 .
and in particular, these generate F 0 U 3 as an O X -module.
Proof. As per the previous example, this is a straightforward application using the gauge transformation G 3 computed in Proposition 4.13, and the conditions in Theorem 4.9 when n = 3.
Remark 4.16. It is at this stage that we see some dependency in the Hodge filtration on the choice of basis of H 1 dr (X). If we choose the standard basis with α 0 = dx y , α 1 = xdx y then we can take f = y x . In this case, we need to take λ = −2 in the generators above. Remark 4.17. Note that we may rewrite the generator Proposition 4.18. There are constants µ, ν, κ ∈ K such that Proof. Using Algorithm 3.18, we calculate that the gauge transformation G 4 at level 4 is such that the image of 1 is modulo words of length ≤ 3 where µ ∈ K is a constant such that 1 3 λf df + µdf − 1 6 f 3 α 0 has at worst a logarithmic pole at P ∞ . By Theorem 4.9, we need to determine sections a 4,s k ∈ H 0 (X, O X ), b 4,s k ∈ H 0 (Y, O Y ) for which the following conditions hold over X ∩ Y = Y ∞ : The first condition immediately implies that the a 4,s 3 = b 4,s 3 are all constants. We iteratively compute the a 4,s k using Algorithm 4.10: where ν, κ ∈ K are such that νx+ λf 2 has at worst a logarithmic pole at P ∞ and νx+ κf + λf is regular on Y . These constants appear when computing the co-efficients at k = 0. For example we find that This forces us to take the constants as described above. Note that the computations show also that for k = 0, a 4,s k = b 4,s k for s = 1, .., 15.
Remark 4.19. If we again take α 0 = dx y , α 1 = xdx y and f = y x then we will find that µ = κ = 0 and ν = 2. Hence the generators will be given by

Example: hyperelliptic curves
As a simple example, say we have a hyperelliptic curve C of genus 2 over K a field of characteristic 0 with odd affine model X : y 2 = x 5 + ax 4 + bx 3 + cx 2 + dx + e with one point at infinity, P ∞ . Fix a basepoint b ∈ X(K). Let α i := x i dx y for i = 0, 1, 2, 3. The classes of these in H 1 dr (X) for a basis. Note that α 0 , α 1 are regular on C while α 2 , α 3 have poles of order 2 and 4 at P ∞ respectively. With this choice of basis for H 1 dr (X), define the universal connection U on X as at the start of Section 3.2.
Then from Example 4.8 we know that F 0 U 1 is generated by 1, A 2 , A 3 . Using Algorithm 3.22 we compute the level 2 gauge transformation G 2 , and using Algorithm 4.6 we compute the following: Proposition 4.20. The generators of F 0 U 2 as an O X -module by For more general hyperelliptic curves C of genus g with odd affine model X over K, suppose that we take a basis α 0 , ..., α 2g−1 for H 1 dr (X) where α g , α g+1 , ..., α 2g−1 have poles of order 2, 4, .., 2g at P ∞ . Then with the A i as previous, using Algorithm 3.22 and Example 4.8 we find that F 0 U 1 is generated by A g , ...A 2g−1 , whilst F 0 U 2 is generated by where k, r, s ∈ {g, ..., 2g − 1} and in the sums i, j are not both greater than g − 1 with the following conditions on sections a ij , b ijk ∈ H 0 (X, O X ): An argument similar to that in Theorem 4.9 demonstrates that the a ij , b ijk are unique. Now, using Algorithm 3.22, it is clear that the functions h 2gi+j+1,0 2 satisfy the following condition: that has at worst a logarithmic pole at P ∞ . Now, if i, j < g then using the algorithm we can assume that h i+1,0 Adding these two conditions together shows that the sum h 2gi+j+1,0 + h 2gj+i+1,0 2 must be constant. When j ≥ g and i < g the same statement must also hold. Therefore, noting that will be regular at P ∞ it follows that b ijk + b jik = 0, and hence b ijk = −b jik . In particular b iik=0 . It also follows that a ij + a ji will be constant. However, evaluation of a ij , a ji at b must be 0, and hence we conclude that a ij = −a ji and a ii = 0. This is summarised in the following proposition: are regular at P ∞ and a ij (b) = 0.

The Unipotent Albanese Map
The p-adic unipotent Albanese map j dr was introduced by Kim in [20], in an attempt to widen the scope of Chabauty-Coleman type finiteness arguments for integral points on curves. This map comes equipped with finite-level versions, the maps j dr n . In what follows, we describe in some explicit detail how we determine j dr n (x). From there, we use this to determine j dr 3 , j dr 4 on an affine elliptic curve and j dr 2 on an affine hyperelliptic curve. From now on, we will assume the following: K is a number field, v is a non-Archimedean valuation on K and K v is the completion of K with respect to v. Let R v be the ring of integers of K v , and k its residue field. Let C be a smooth curve over K with some non-zero divisor D and let X := C − D and let b ∈ X(K). Let X v := X ⊗ K v denote the basechange of X. Below we outline how j dr n is defined. We let Un n (X) be the category of unipotent connections on X v over K v with unipotency index less than or equal to n. We have a fibre functor e n b : Un n (X) → Vect Kv ; (V, ∇) → b * V, and Tannka duality gives a group scheme U dr n which is a quotient of the de Rham fundamental group U dr of X v . We also have a right U dr n -torsor P dr n (x) over X v for each x ∈ X v (K v ) coming from the right U dr -torsor of de Rham paths P dr (x).
It will also be important to recall from loc.cit. that in our construction of U n = (R n ⊗O X , ∇), R n is a Hopf algebra, where the coproduct ∆ comes from the universal property of U n sending 1 ∈ b * U n → 1 ⊗ 1 ∈ b * (U n ⊗ U n ). In particular, this coproduct maps A i to A i ⊗ 1 + 1 ⊗ A i , and so the A i are primitive for this coproduct. Additionally, R n is the Lie algebra of U dr n , with the group-like elements of R n being identified with K v -points of U dr n . The exponential map exp and logarithm map log on b * U n = R n define bijections between the group-like elements (i.e. U dr n ) and the primitive elements. Note that linear combinations and commutators of primitive elements are also primitive. Thus, for example, we may identify U dr 1 with V dr = H 1 dr (X) ∨ . In a more general sense, x * U n ∼ = Hom(e n b , e n x ) functorially. Then as P dr n (x) consists of the tensor compatible elements in Hom(e n b , e n x ), we may identify P dr n (x) with the group-like elements with respect to the coproduct ∆. Now let Y be the reduction of X v over k. Let Un(Y ) be the category of overconvergent unipotent isocrystals on Y . If we basechange to K v , this will be identified with unipotent connections convergent on every residue disk on X, and overconvergent near points of D v = D ⊗ K v . For c ∈ Y (k), let ]c[ denote the residue disk of c. Then there is a fibre functor e c : (V, ∇) → V(]c[) ∇=0 , the horizontal sections of V on the residue disk of c. Tannka duality then gives us a crystalline fundamental group U cr , and a right-torsor of crystalline paths P cr (y) for y ∈ Y (k). We similarly obtain U cr n , P cr n (y) with overconvergent isocrystals of unipotency index less than or equal to n.
The q = |k|-power map on O Y induces a Frobenius automorphism φ : P cr n (y) ≃ P cr n (y). Due to the comparison theorem of Chiarletto ( [12]), this gives a Frobenius automorphism φ on P dr n (x) also, for x ∈]y[. Corollary 3.2 in [9] shows that the Frobenius map φ will have a fixed point in P cr n (y) -that is, a Frobenius-invariant crystalline path from c to y. If b ∈]c[, then this gives us a Frobenius-invariant de Rham path p cr (x) from b to x.
In [20], Kim defines admissible torsors T for U dr n for a K v -scheme Z. Such torsors will be required to possess a Frobenius map φ, and the existence of a Frobenius-invariant point p cr T ∈ T (Z). It should also have a Hodge structure for which it is trivialisable, so that there is a p H T ∈ F 0 T (Z). Let T be an admissible torsor over a K v -algebra L. Then as it is a right torsor, there is a u T ∈ U dr n such that p cr T = p H T u T . The u T will be unique up to multiplication on the left by F 0 U dr n , and so we have a [u T ] ∈ F 0 U dr n \ U dr n . For K v -algebras L, the L-points of this coset space turn out to classify isomorphism classes of admissible U dr n -torsors over L (Proposition 1 in loc.cit.).
Following Lemma 1 in loc.cit., P dr n (x) will be an admissible torsor. Therefore, we can associate to it a unique element [P dr n (x)] := [u P dr n (x) ] ∈ F 0 U dr n \ U dr n . We then define the map j dr n by j dr n : X v (K v ) → F 0 U dr n \ U dr n ; x → [P dr n (x)] To determine [P dr n (x)], we need to find a φ-invariant p cr n (x) ∈ P dr n (x), a trivialisation p H (x) ∈ F 0 P dr n (x) and u n (x) ∈ U dr n such that p cr n (x) = p H n (x)u n (x). Then [P dr n (x)] = [u n (x)]. Recall from loc.cit. that U is the dual of the co-ordinate ring of the canonical right-U dr torsor P dr n with fibre P dr (x) over x. Similarly, the dual of U n is the co-ordinate ring for the canonical right-U dr n torsor P dr n . In particular, the Hodge filtration on U n induces the Hodge filtration on P dr n , and thus on U dr n and P dr n . Using the identification above between P dr n (x) and group-like elements of x * U n , we may compute the Hodge filtration on P dr n (x) and a trivialisation for this Hodge structure, p H n (x). To identify p cr n (x), note that this is the analytic continuation along Frobenius of 1 ∈ b * U n to x * U n described in [9,Section 3], . Then p cr n is the unique horizontal section of U n convergent on each residue disk of Y such that p cr n (b) = 1. Thus Here, the iterated integrals are iterated Coleman integrals defined by Balakrishnan has developed algorithms for computing double iterated Coleman integrals on elliptic and hyperelliptic curves (see [2], [6]) which have seen applications to Kim's non-abelian Chabauty. In [4], it is shown that when C is an elliptic curve, the map j dr 2 is given by This is actually the logarithm of j dr 2 . However, as there is a bijection between the primitive elements of LieU dr 2 and elements of U dr 2 , it is more convenient to use this description. In what follows we will make use of two properties of iterated integrals ( [5, Proposition 5. where in (5.3) we sum over all permutations of shuffle type (r, s). Note that (5.4) is a simple consequence of (5.3).

The level 3 map on elliptic curves
Here we determine the map j dr 3 on elliptic curves, using the results of Section 4. First note that as per 5.1, we have Recall that the Hodge filtration of F 0 U 3 is generated by We want to find a primitive p H 3 (x) in the algebra generated by these, and a primitive u 3 (x) ∈ x * R n such that p cr 3 (x) = exp(p H 3 (x)) exp(u 3 (x)). Then we define j dr 3 (x) = u 3 (x). Proposition 5.1. The level 3 unipotent Albanese map j dr 3 : Proof. First, using properties (5.3),(5.4) of iterated integrals, we rewrite p cr 3 (x) as As computed in Proposition 4.13 A 1 + λ[[A 0 , A 1 ], A 1 ] ∈ F 0 U 3 and is primitive, and hence we deduce that the map j dr 3 is as in the statement of the proposition.
where B n−1 = A 1 + n−1 t=2 2 t −1 s=1 a t,s 1 w t s is the generator of F 0 U n−1 over X with lowest degree term A 1 . Suppose also that all the B n are primitive. Then we compute However, as we are now in R n and not R n−1 , exp( x b α 1 B n−1 ) exp(j dr n−1 (x)) = p cr n−1 (x)+terms of degree n. It is easy to see using (5.6) that the remaining terms of (5.7) have degree n. Therefore exp(p cr n (x)−exp( x b α 1 B n ) exp(j dr n−1 (x))) commutes with exp( x b α 1 B n ) exp(j dr n−1 (x)), and we find that If the expression (5.7) is primitive, then we can define j dr n (x) = j dr n−1 (x) + p cr n (x) − exp Otherwise, we need to find a replacement for exp( x b α 1 B n ) in x * F 0 U n and calculate u n (x) using this. The immediate consequence of this is that if we could show that the expressions are primitive in R n for all n, then we would have a closed form for j dr n (x).

The level 2 map on hyperelliptic curves
We can use the computation of the Hodge filtration on U 2 for a generic genus g odd hyperelliptic curve C that we determined in Section 4 to determine the level 2 unipotent Albanese map on X := C − {P ∞ }. Fix a basepoint b ∈ X(K). Recall that by Proposition 4.21 F 0 U 2 will be generated by with r, s ≥ g, Z g = {(i, j) ∈ Z 2 : i, j < g or i < g, j ≥ g or j < g, i ≥ g}, b ijk constants with b ijk = −b jik and a ij ∈ H 0 (X, O X ) with a ij = −a ji . As above, we will find that Proof. A careful computation shows that where Therefore, we conclude that Remark 5.4. So in the case that C is the genus 2 hyperelliptic curve considered previously, and α i = x i ω where ω = dx y , we can conclude that the map j dr 2 on X will be given by

Tangential Basepoints
In previous sections we've been using basepoints b ∈ X(K). If we cannot easily find any such points, we need to resort to using an alternative: tangential basepoints. Not only will the use of tangential basepoints circumvent the obstacle of finding a rational basepoint, but from a philosophical point of view we ought to get a range of canonical maps at our disposal. To define the maps j dr n using tangential basepoints, we need to be able to analytically continue the maps above. We briefly outline the method by which we can compute this, following the description given by Besser and Furusho in [10].
We assume once more that C is a smooth projective curve over K, punctured at a single point P , and let X := C − {P }. The following generalises in the obvious way if we remove more than one point. Let U be the universal unipotent connection on X v , and U it's logarithmic extension to C v . Let t be a parameter at P , inducing a parameter t on T P C, which is a normal vector at P taking the value 1 at the tangent vector d dt . Let T 0 P C := T P C − {0}. Then we can associate to each U n a connection on T 0 P C: Definition 6.1. Define Res P U n := (R n ⊗ O T 0 P C , d + Res P C ′ n d log t) where C ′ n = (ω ij ) is the connection matrix of U n and Res P C ′ n = (res P ω ij ) is the residue matrix of C ′ n at P . This construction does not depend on the choice of parameter t at P . Thus we have a functor Res P : Un(X) → Un(T 0 P C), and this is in fact an exact tensor functor. Thus we get more fibre functors at P , and we use these to identify de Rham paths between rational points and tangent vectors. In [10][Theorem 4.1] Besser and Furusho show that for any two points x, y which are either in X v (K v ) or T 0 P C(K v ) there exists a canonical path (the analytic continuation along Frobenius) p x,y from x to y. These paths are compatible with composition and are functorial. In Proposition 4.5 they identify the path p x,y from x to y for given points x ∈ X v (K v ) and y ∈ T 0 P C(K v ). There is a basis of global solutions to Res P U n = 0 with coefficients in k[log t], with all solutions having the form exp(Res P C ′ n log t) · g with g ∈ R n . The exponential will be finite since U n and hence Res P U n are unipotent. We can also find formal local horizontal solutions s of U n near P with coefficients in the ring K v [[t]][log t]. Here log t is treated as a formal variable with the property that d log t = dt/t.
To analytically continue the horizontal section s near P to the tangent vector d dt we specialise to the fibre R n ⊗ O T 0 P C by taking the constant term of this formal solution: that is, we let t = log t = 0. Call this constant term s 0 . In Proposition 4.5, Besser and Furusho show that the path a x,y is the path U n → (s → exp(Res P C ′ n log t) · s 0 )). The specialisation of exp(Res P C ′ n log t) · s 0 at t = 1 then is the analytic continuation along Frobenius of the horizontal section s to the punctured tangent space, and this is precisely the constant term s 0 .
Taking a tangential basepoint b, the Hodge structure on π 1,dr (X; b, x) is a limit Hodge structure of that on π 1,dr (X; y, x) as y varies over X(K v ) (see [21]). Therefore, our previous computations of the Hodge filtration at rational basepoints are sufficient for the purposes of computing the maps j dr n . We need to be careful here, however. If we simply take the maps j dr n described previously for n = 3, 4 on an elliptic curve and replace b with a tangential basepoint, we run into a problem: notice that we need to compute x b α 1 for example. However, by choice, α 1 has a pole of order 2 at infinity, and we cannot compute this integral directly.
Instead, we need to consider the logarithmic connection U n near P ∞ : the restriction to the open Y is U ′ n which is a logarithmic connection with connection matrix which can be computed using the algorithms of Section 3. Near P ∞ we can compute the parallel transport of 1 ∈ b * Res P U n to x * U n for an x ∈ Y using Proposition 4.5 in loc.cit. as described above.
Let us consider how we do this in practice. Assume for now that C is an elliptic curve and that P is the point at infinity P ∞ , and fix a branch of the p-adic logarithm. The connection U ′ n is given by ∇ ′ = d + C ′ n on Y as computed earlier. This connection matrix then defines an iterated integral which gives the parallel transport of 1 ∈ b * Res P U n to x * U n as before. However, now all the integrals appearing will be integrals of differentials on Y with at most simple poles at P ∞ and no poles anywhere else, and these we can calculate.
To calculate iterated integrals of the form x b ω 1 ...ω n appearing in the above we proceed as follows: first suppose that x ∈]P ∞ [∩Y and let t, b = d dt be as above. Let σ be a dummy variable which we take to be non-zero for now, and let ǫ = (x(σ), y(σ)) and let z = t(x). Suppose that ω i (t) = f i (t)dt is the Laurent series expansion of ω i at P ∞ . Then as x lies in the residue disc of P ∞ , we can compute the iterated integral   f n (t n )dt n dt n−1 ...dt 2 dt 1 (6.1) This will give us a logarithmic Coleman function in the parameter σ; that is, it will be a function of the form a x 0 (σ) + a x 1 (σ) log(σ) + a x 2 (σ) log(σ) 2 + .... (6.2) where the a x i (σ) analytic functions in the variable σ i.e. they have no poles on ]P ∞ [. The iterated integral x b ω 1 ...ω n is then defined to be a x 0 (0) i.e. we set σ = log(σ) = 0 in the above. Note that if the differentials ω i are regular also at P ∞ then in fact the above computation simply gives us x P∞ ω 1 ..ω n If we had chosen a different tangential basepoint b ′ then we need to make a different choice of normalising parameter t ′ with ∂ b ′ t ′ = 1, where ∂ b ′ is the derivation associated to b ′ . In [3,Lemma 3.2] it is shown that the above definition of the integral is independent of the choice of parameter t ′ satisfying this normalisation condition. For points x lying outside ]P ∞ [, we can use the following co-product formula for iterated integrals [5,Lemma 5.2.3]: if x, y, z are points on C such that that a path is to be taken from x to z through y and ω 1 , ..., ω n are holomorphic 1-forms at these points then where the first integral is computed using the algorithms in loc.cit. and the second integral is computed using the above method. All of the above generalises to our hyperelliptic curves as well. We now look at some examples.

Example: level 2 map on elliptic curves
Recall that in [4] it is shown that the level 2 map on elliptic curves is given by In Example 3.20 we deduced that we can take the connection matrix of U 2 over Y to be where α ′ 1 = α 1 − df for f ∈ K(C) with a simple pole at P ∞ such that α ′ 1 has at most a simple pole there also. The parallel transport map from 1 at b = d dt to x ∈ Y v (K v ) then is given by