2.1 Rational points and global heights

Consider a smooth projective curve $X_{\mathbf {Q}}$ of genus $g \geq 2$ whose Jacobian $J$ has rank $r=g$. We also assume that the abelian logarithm induces an isomorphism

(2.1)\begin{equation} \log\colon J(\mathbf{Q})\otimes\mathbf{Q}_p \to \mathrm{H}^0(X_{\mathbf{Q}_p},\Omega^1)^{\vee} \end{equation}

and that $X(\mathbf {Q})$ is non-empty, so we may choose a base point $b$ in $X(\mathbf {Q})$. Suppose that the Néron–Severi rank $\mathrm {rk}_{\mathbf {Z}} \mathrm {NS}(J)$ is at least $2$, so that there exists a non-trivial class

\[ Z \in \mathrm{Ker}\big(\mathrm{NS}(J) \longrightarrow \mathrm{NS}(X) \simeq \mathbf{Z} \big). \]

As explained in Balakrishnan and Dogra [Reference Balakrishnan and DograBD18, Lemma 3.2], we can attach to any such choice of $Z$ a suitable quotient $U_Z$ of the $\mathbf {Q}_p$-pro-unipotent fundamental group of $X_{\bar {\mathbf {Q}}}$, which, via a twisting construction by path torsors, gives rise to a certain family of Galois representations,

\begin{align*} X(K) & \longrightarrow \left\{ G_K \to \mathrm{GL}_{2g+2}(\mathbf{Q}_p) \right\}/\sim \\ x & \longmapsto \mathrm{A}(x) := \mathrm{A}_Z(b,x), \end{align*}

where $K \in \{ \mathbf {Q}, \mathbf {Q}_p \}$ and $G_K$ is the absolute Galois group of $K$. We refer the reader to [Reference Balakrishnan and DograBD18, § 5.1] for the details of this construction (in particular, for the equivalence relation), and merely recall here that with respect to a suitable choice of basis, the representation $\mathrm {A}(x)$ is lower triangular, of the form

(2.2)\begin{equation} g \in G_K \longmapsto \left( \begin{matrix} 1 & & \\ \alpha(g) & \rho_V(g) & \\ \gamma(g) & \beta(g) & \chi_p(g) \end{matrix} \right), \end{equation}

where

\[ \rho_V \colon G_K \longrightarrow \mathrm{GL}_{2g}(\mathbf{Q}_p) \]

is a frame for the Galois action on the $p$-adic étale homology $V = \mathrm {H}^1_{\scriptstyle \mathrm {\acute {e}t}}(X_{\overline {K}}, \mathbf {Q}_p)^{\vee }$, and $\chi _p\colon G_K \to \mathbf {Q}_p^{\times }$ is the $p$-adic cyclotomic character. Representations of this form, which admit a $G_K$-stable filtration with graded pieces $\mathbf {Q}_p(1), V, \mathbf {Q}_p$, are referred to as mixed extensions (see [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, § 3.1]).

The theory of $p$-adic heights due to Nekovář [Reference NekovářNek93, § 2] attaches to any mixed extension $M$ a $p$-adic height $h(M)$. When applied to the family of mixed extensions $\mathrm {A}(x)$, this results in a map

\[ h \colon X(\mathbf{Q}) \longrightarrow \mathbf{Q}_p. \]

The algebraic properties of this map lie at the heart of the quadratic Chabauty method. Most notably, the method relies on the following two facts.

1. − The $p$-adic height is a bilinear function of the pair of cohomology classes $([\alpha ], [\beta ])$ associated to the vectors appearing in (2.2).

2. − It decomposes as a sum of local height functions $h_v$ defined locally at every finite place $v$.

2.2 Local decomposition

We now discuss in more detail the decomposition of the global $p$-adic height $h$ described above, as a sum of local height functions

\[ h_v \colon X(\mathbf{Q}_v) \longrightarrow \mathbf{Q}_p. \]

The nature of these local height functions is as follows.

1. (i) The case $v \neq p$. It follows from Kim and Tamagawa [Reference Kim and TamagawaKT08, Corollary 0.2] that the function $h_v$ has finite image, in the sense that there exists a finite set $\Upsilon _v$ such that

   \[ h_v \colon X(\mathbf{Q}_v) \longrightarrow \Upsilon_v \subset \mathbf{Q}_p. \]

2. (ii) The case $v=p$. The map $h_p$ is locally analytic and has a simple description in terms of linear algebra data of the filtered $\phi$-module

   \[ M(x) := \big( \mathrm{A}(x) \otimes_{\mathbf{Q}_p} \mathrm{B}_{\scriptstyle \mathrm{ crys}}\big)^{G_{\mathbf{Q}_p}}, \]
   where $\mathrm {B}_{\scriptstyle \mathrm { crys}}$ is Fontaine's crystalline period ring. A crucial point in the method of quadratic Chabauty is that the definition of the family of Galois representations $\mathrm {A}(x)$ comes from a motivic quotient of the fundamental group of $X$, and non-abelian $p$-adic Hodge theory yields an analogous de Rham realisation in the form of a filtered connection $(\mathscr {M},\nabla )$ on $X$ with a Frobenius structure, together with an isomorphism of filtered $\phi$-modules
   \[ x^* \mathscr{M} \simeq M(x) \]
   (see [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, § 5]). We have a pair of elements $\pi _1 (M(x))$ and $\pi _2 (M(x))^\vee (1)$ of $\mathrm {H}^0 (X_{\mathbf {Q} _p },\Omega )^\vee$ associated to the filtered $\phi$-module $M(x)$, via the isomorphism
   \[ \mathrm{Ext} ^1 _{\scriptstyle \mathrm{ Fil} ,\phi }(\mathbf{Q} _p ,\mathrm{H} ^1 _{\scriptstyle \mathrm{dR}}(X_{\mathbf{Q} _p })^\vee )\simeq \mathrm{H} ^0 (X_{\mathbf{Q} _p },\Omega )^\vee . \]

2.3 Finiteness

The decomposition $h = \sum _v h_v$ can be used to leverage the bilinear nature of $h$ against the properties of the functions $h_v$. By (1) in § 2.2, we know that there exists a finite set $\Upsilon = \Upsilon _Z \subset \mathbf {Q}_p$ such that

(2.3)\begin{equation} h(x) - h_p(x) \in \Upsilon \end{equation}

for any $x$ in $X(\mathbf {Q})$. In § 3, we describe how the terms in this equation may be computed explicitly.

1. − The set $\Upsilon$ is given by $\{\sum _v \epsilon _v :\epsilon _v \in \Upsilon _v \}$, where the sum is over primes of bad reduction, and $\Upsilon _v$ is the set of values of $h_v (x)$ for $x\in X(\mathbf {Q} _v ).$ For $v\ne p$, the map $h_v$ is made more explicit in § 3.1 using the results of Betts and Dogra [Reference Betts and DograBD19] to compute $\Upsilon _v$ when a regular semi-stable model $\mathcal {X}$ is known. The map $h_v$ factors through the reduction map to the irreducible components of the special fibre of $\mathcal {X}$.

2. − The map $h_p$ may be computed using [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, §§ 4,5], where it is explained how the universal properties of the bundle $\mathscr {M}$ rigidify the (known) structures on the graded pieces, enough to allow us to compute them explicitly (see § 3.2).

3. − Using the isomorphism (2.1), we may view the global height as a pairing

   \[ h \colon \mathrm{H}^0(X_{\mathbf{Q}_p},\Omega^1)^{\vee} \otimes \mathrm{H}^0(X_{\mathbf{Q}_p},\Omega^1)^{\vee} \longrightarrow \mathbf{Q}_p. \]
   Using global information, such as an abundance of global points $x \in X(\mathbf {Q})$ if available, we can solve for the height pairing. This is discussed in § 3.3, where we also explain what to do when too few rational points are available.

Via the above, the map $h$ may be extended to a bilinear map

(2.4)\begin{equation} h\colon X(\mathbf{Q}_p) \to \mathbf{Q}_p,\quad x\mapsto h(\pi _1 (\mathrm{A}(x)),\pi _2 (\mathrm{A}(x))^\vee (1)). \end{equation}

The resulting map

(2.5)\begin{equation} \rho = h - h_p \colon X(\mathbf{Q}_p) \longrightarrow \mathbf{Q}_p \end{equation}

is known to be Zariski dense on every residue disk. We call $\rho$ a quadratic Chabauty function, and we write $\rho _Z$ if we want to emphasise the dependence on $Z$. Hence (2.3) implies that $X(\mathbf {Q})$ is finite. Moreover, the computable nature of the quantities involved in (2.3), discussed at length in the next section, allows us to explicitly determine a $p$-adic approximation of the finite set

\[ \{ x\in X(\mathbf{Q} _p )\colon h(x)-h_p (x)\in \Upsilon \} \supset X(\mathbf{Q} ). \]

As explained in [Reference Balakrishnan and DograBD18, Proposition 5.5], this finite set contains the Chabauty–Kim set $X(\mathbf {Q} _p )_2$. In particular, a proof that this set equals $X(\mathbf {Q} )$ gives a verification of Kim's conjecture [Reference Balakrishnan, Dan-Cohen, Kim and WewersBDCKW18, Conjecture 3.1] for the curve $X$ (we refer the reader to [Reference Balakrishnan, Dan-Cohen, Kim and WewersBDCKW18, Definition 2.7] for the definitions of the set $X(\mathbf {Q} _p )_2$).

3.1 Local heights away from $p$

Let $\ell \neq p$ and let $F$ be an endomorphism of $J$ whose class $Z$ lies in $\mathrm {Ker}\big (\mathrm {NS}(J) \to \mathrm {NS}(X) \big )$. In  [Reference Betts and DograBD19], a description of the map

\[ h_\ell \colon X(\mathbf{Q} _{\ell }) \longrightarrow \mathrm{H}^1(G_\ell, U_Z) \longrightarrow \mathrm{H}^1(G_\ell, \mathbf{Q}_p(1)) \longrightarrow \mathbf{Q} _{p} \]

associated to $F$ and $\chi$ is given, in terms of harmonic analysis on the reduction graph in the sense of Zhang [Reference ZhangZha93].

To explain the result, we introduce some notation. Over some finite extension $K/\mathbf {Q} _{\ell }$, the curve $X$ admits a regular semi-stable model $\mathcal {X}_{\mathrm {reg}}/\mathcal {O}_K$, and a stable model $\mathcal {X}_{\mathrm {st}}/\mathcal {O}_K$. Let $\Gamma _{\mathrm {reg}}$ and $\Gamma _{\mathrm {st}}$ denote the dual graphs of the special fibres of these models. Recall that the dual graph of the special fibre is by definition the graphFootnote ¹ whose vertices are the irreducible components of the special fibre, and whose edges are the singular points of the special fibre. The endpoints of an edge $e$ are defined to be the irreducible components containing the point (by semi-stability, a singular point $e$ lies on at most two irreducible components). By regularity, we have a reduction map

\[ \mathrm{red}\colon X(\mathbf{Q} _{\ell }) \longrightarrow V(\Gamma _{\mathrm{reg}}) \]

from $X(\mathbf {Q} _{\ell })$ to the vertices of the dual graph $\Gamma _{\mathrm {reg}}$.

The definition is the natural one: given $x\in X(\mathbf {Q} _{\ell })$, there is a unique extension to an $\mathcal {O}_K$-section $x\in \mathcal {X}_{\mathrm {reg}}(\mathcal {O}_K )$. Let $k$ be the residue field of $\mathcal {O}_K$. By regularity, the specialisation of $x$ to $k$ lies on a unique irreducible component of $\mathcal {X}_{\mathrm {reg},k}$.

We may give $\Gamma _{\mathrm {reg}}$ and $\Gamma _{\mathrm {st}}$ the structure of rationally metrised graphs (i.e. graphs whose edges $e$ have associated lengths $\ell (e)\in \mathbb {\mathbf {Q} }_{>0}$) by defining the length of an edge $e$ to be $i(e)/r$, where $i$ is the intersection multiplicity of the corresponding singular point and $r$ is the ramification degree of $K/\mathbf {Q} _{\ell }$.

Choose an orientation of the edges of $\Gamma := \Gamma _{\mathrm {st}}$, so that each $e\in E(\Gamma )$ has a source $s(e)$ and target $t(e)$ in $V(\Gamma )$. We define the (rational) homology of $\Gamma$, $H_1 (\Gamma )\subset \mathbf {Q} E(\Gamma )$, to be the kernel of the map

\[ s-t\colon \mathbf{Q} E(\Gamma )\to \mathbf{Q} V(\Gamma ), \]

where $\mathbf {Q} E(\Gamma )$ and $\mathbf {Q} V(\Gamma )$ are the free $\mathbf {Q}$-vector spaces generated by $E(\Gamma )$ and $V(\Gamma )$, respectively.

Define $\Gamma _{\mathbf {Q} }$ to be the set of points on $\Gamma$ whose distance from a vertex is rational. Formally,

\[ \Gamma _{\mathbf{Q}} =\bigsqcup _{e\in E(\Gamma _{\scriptstyle \mathrm{st}})}\{e\}\times ([0,\ell (e)] \cap \mathbf{Q} ) /\sim , \]

where the equivalence relation is that $(e_1 ,1 )\sim (e_2 ,0)$ whenever $t(e_1 )=s(e_2 )$. Since $\mathcal {X}_{\mathrm {reg}}$ is obtained from $\mathcal {X}_{\scriptstyle \mathrm {st}}$ by taking each singular point (corresponding to an edge $e$) and blowing up $i(e)$ times, we have an inclusion $V(\Gamma _{\mathrm {reg}})\subset \Gamma _{\mathbf {Q} }$ (in the terminology of [Reference Betts and DograBD19, 3.7.1], we may view $\Gamma _{\mathrm {reg}}$ as a rational subdivision of $\Gamma _{\scriptstyle \mathrm {st} }$). In this way we can think of the reduction map, $\mathrm {red}$, as a map from $X(K)$ to $\Gamma _{\mathbf {Q} }$ (see [Reference Betts and DograBD19, Definition 1.3.1]). The rationally metrised graph we obtain is independent of the choice of extension over which $X$ acquires stable reduction [Reference Chinburg and RumelyCR91, Proposition 2.6], and in fact there is an equivalent definition of $\Gamma _{\mathbf {Q} }$ as the limit of the dual graphs of special fibres of regular semi-stable models of $X_L$ over all finite extensions $L$ of $K$ (see [Reference Chinburg and RumelyCR93, § 2]).

In [Reference Betts and DograBD19, Lemma 12.1.1], a map

\[ j_{\Gamma }\colon\Gamma _{\mathbf{Q} }\to \mathbf{Q} _p \]

is defined such that $h_\ell =c\cdot j_{\Gamma }\circ \mathrm {red}$, where $c$ is a constant. The map $j_{\Gamma }$ is defined in terms of the Laplacian operator associated to $\Gamma _{\mathrm {st}}$, which we now define. We say a function

\[ \Gamma _{\mathbf{Q}} \to \mathbf{Q} _p \]

is piecewise polynomial if on each edge it is the restriction of a polynomial function $\mathbf {Q} \to \mathbf {Q} _p$. As in [Reference Betts and DograBD19, Definition 7.2.2], we define the Laplacian $\nabla ^2 (g)$ of a piecewise polynomial function $g\colon \Gamma _{\mathbf {Q} }\to \mathbf {Q} _p$ to be the formal sum

\[ -\sum _{e\in E(\Gamma )} g '' (x_e )\cdot e +\sum _{v\in V(\Gamma )}\biggl(\sum _{s(e)=v}g'(0)-\sum _{t(e)=v}g'(1)\biggr)\cdot v. \]

Here we write the function $g$ restricted to the edge $e$ as a polynomial in $\mathbf {Q} _p [x_e ]$ for notational simplicity, where $x_e$ is the inclusion from the edge $e$, thought of as a line segment $[0,\ell (e)]\cap \mathbf {Q}$, into $\mathbf {Q}$. Hence, we have

\[ \nabla ^2 (g) \in \bigoplus _{e\in E(\Gamma )}\mathbf{Q} _p [x_e ]\cdot e \oplus \bigoplus _{v\in V(\Gamma )}\mathbf{Q} _p \cdot v . \]

The Laplacian is linear on piecewise polynomial functions, and its kernel consists of constant functions. Thus $g$ is uniquely determined by $\nabla ^2(g)$ and its value at one point.

In [Reference Betts and DograBD19], an explicit construction is given of a piecewise polynomial function that corresponds, via $\mathrm {red}$, to the local height function we wish to compute. Recall that $F$ is an element of $\mathrm {End} (J)\otimes \mathbf {Q} _p$ whose image in $\mathrm {NS} (J)$ lies in the kernel of $\mathrm {NS} (J)\to \mathrm {NS} (X)$, and $b\in X(\mathbf {Q} )$ is a rational point.

Theorem 3.2 [Reference Betts and DograBD19, Theorem 1.1.2, Lemma 12.1.1 and Corollary 12.1.3]

Let $\Gamma$ be the dual graph of $X$ corresponding to a regular semi-stable model of $X$ over $\mathcal {O}_K$, where $K/\mathbf {Q} _{\ell }$ is a finite extension. Let $\mathrm {red} \colon X(\mathbf {Q} _{\ell } )\to V(\Gamma )$ be the reduction map. For an irreducible component $X_w$ of the special fibre of the regular semi-stable model, let $V_p (X_w )$ denote the $\mathbf {Q} _p$-Tate module of its Jacobian. The morphism $j_{\Gamma }$ is the unique piecewise polynomial function

\[ j_\Gamma \colon \Gamma _{\mathbf{Q} }\to \mathbf{Q} _p \]

satisfying $j_{\Gamma }(\mathrm {red} (b))=0$ and $\nabla ^2 (j_{\Gamma })=\mu _F$, where

\[ \mu _F := \sum _{e\in E(\Gamma )}\frac{1}{\ell (e)} e^* F(\pi (e)) \cdot e+\frac{1}{2}\sum _{w\in V(\Gamma )}\mathrm{Tr} (F|V_p (X_w ))\cdot w . \]

Here, the morphism $\pi$ is by definition the orthogonal projection

\[ \mathbf{Q} E(\Gamma )\to \mathrm{H}_1 (\Gamma ,\mathbf{Q} ) \]

with respect to the pairing $e\cdot e'=\delta _{ee'}$ on $\mathbf {Q} E(\Gamma )$, and $e^*$ is the functional $\mathbf {Q} E(\Gamma )\to \mathbf {Q}$ projecting onto the $e$ component. Recall (e.g. [Reference GrothendieckSGA7, 12.3.7]) that $V_p (X)$ admits a $G_K$-stable filtration

\[ V_p (X)=W_0 V_p (X)\supset W_1 V_p (X)\supset W_2 V_p (X)\supset W_3 V_p (X)=0, \]

and we have isomorphisms of $G_K$-representations

\begin{align*} \mathrm{gr}^W _0 V_p (X) & \simeq H_1 (\Gamma )\otimes \mathbf{Q} _p , \\ \mathrm{gr}^W _1 V_p (X) & \simeq \bigoplus _{w\in V(\Gamma )}V_p (X_w ), \\ \mathrm{gr}^W _2 V_p (X) & \simeq H_1 (\Gamma )^* \otimes \mathbf{Q} _p (1). \end{align*}

The action of $F$ on $V_p (X)$ preserves this filtration since it is a morphism of Galois representations, and hence induces an action of $F$ on the weight $-1$ part of $V_p (X)$, which is isomorphic to $\bigoplus _w V_p (X_w )$. Although the action of $F$ need not respect the direct sum decomposition, the decomposition

\[ \mathrm{End}\Bigl(\bigoplus_w V_p (X_w )\Bigr)\simeq \bigoplus _{w_1 ,w_2 }\mathrm{Hom} (V_p (X_{w_1 }),V_p (X_{w_2} )) \]

implies that we can define $\mathrm {Tr} (F|V_p (X_w ))$ as the trace of the $\mathrm {End} (V_p (X_w ))$-component of $F$.

To determine the possible local heights, it suffices to compute the action of $F$ on $\mathrm {H}_1 (\Gamma )$ and on $V_p (X_v )$. In this paper, we do not discuss methods for the algorithmic computation of the action of $F$ on $\mathrm {H}_1 (\Gamma )$, but algorithms for these computations in the case when the curve $X$ is hyperelliptic will be discussed in forthcoming joint work of the first, second and fifth authors with David Corwin, Sachi Hashimoto, Benjamin Matschke, Oana Padurariu, Ciaran Schembri and Tian Wang.

As we explain in § 5.4, one can sometimes use partial information deduced from Theorem 3.2 to determine the possible local heights without computing the action of $F$ on $\mathrm {H}_1 (\Gamma )$ (e.g. if one has enough rational points on $X$ that are suitably independent in $J(\mathbf {Q})$ and $\Gamma _{\mathbf {Q} }$).

3.2 Local heights at $p$

We discuss the local height component

\[ h_p \colon X(\mathbf{Q}_p) \longrightarrow \mathbf{Q}_p, \]

which appeared in [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, § 5]. Recall that $h_p$ is a locally analytic function, described in terms of the filtered $\phi$-module $M(x)$ discussed in § 2.2. Concretely, we may find two unipotent isomorphisms

\[ \lambda^{\star}(x) \colon \ \mathbf{Q}_p \oplus V_{\scriptstyle \mathrm{dR}} \oplus \mathbf{Q}_p(1) \ \stackrel{\sim }{\longrightarrow } \ M(x), \quad \text{for} \ \star \in \{ \phi, \mathrm{Fil} \}, \]

where $\lambda ^{\phi }$ respects the Frobenius action and $\lambda ^{\mathrm {Fil}}$ respects the Hodge filtration, which with respect to a suitable basis for $M(x)$ may be represented in ($1+2g+1$)-block matrix form as

(3.1) \begin{equation} \lambda^\phi(x) =\left( \begin{array}{@{}ccc@{}} 1 & 0 & 0 \\ \boldsymbol{\alpha }_{\phi} & 1 & 0 \\ \gamma_{\phi} & \boldsymbol{\beta }^{\intercal}_{\phi} & 1 \\ \end{array} \right), \quad \lambda^{\mathrm{Fil}}(x) = \left( \begin{array}{@{}ccc@{}} 1 & 0 & 0 \\ \boldsymbol{\alpha }_{\mathrm{Fil}} & 1 & 0 \\ \gamma_{\mathrm{Fil}} & \boldsymbol{\beta }^{\intercal}_{\mathrm{Fil}} & 1 \\ \end{array} \right) \end{equation}

(see [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, § 5.3] and [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, § 4.5], respectively). The isomorphism $\lambda ^\phi$ is uniquely determined, whereas $\lambda ^{\mathrm {Fil}}$ is only well defined up to the stabiliser of the Hodge filtration $\mathrm {Fil}^0$. A suitable choice gives $\boldsymbol {\alpha }_{\mathrm {Fil}} =0$.

The splitting $s$ of the Hodge filtration (see Remark 3.1) defines idempotents $s_1,s_2$ on $V_{\scriptstyle \mathrm {dR}}$ with images $s(V_{\scriptstyle \mathrm {dR}}/\mathrm {Fil}^0 V_{\scriptstyle \mathrm {dR}})$ and $\mathrm {Fil}^0 V_{\scriptstyle \mathrm {dR}}$ respectively, with respect to which the local height at $p$ is

(3.2)\begin{equation} h_p (x)= \gamma_{\phi} - \gamma_{\mathrm{Fil}} -\boldsymbol{\beta}^\intercal_{\phi} \cdot s_1 (\boldsymbol{\alpha}_{\phi}) - \boldsymbol{\beta }^\intercal_{\mathrm{Fil}} \cdot s_2 (\boldsymbol{\alpha}_{\phi}) \end{equation}

by [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, Equation (17)].

In [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19] we outline a method to compute these quantities explicitly as functions of the local point $x$ in $X(\mathbf {Q}_p)$, which exploits the existence of the connection $(\mathscr {M},\nabla )$ discussed in § 2.2. The Hodge filtration and Frobenius structures of this bundle are characterised by suitable universal properties, discussed at length in [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, §§ 4 and 5]. We have made the algorithms for the computation of $h_p$ more general and streamlined and have added a precision analysis in § 4 but have not made further contributions to this part of the method beyond what is already contained in [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19].

3.3 The global height pairing

One key step in the construction of a quadratic Chabauty function is to write the global height pairing $h$ in terms of a basis of the space of bilinear pairings on $\mathrm {H}^0(X_{\mathbf {Q}_p}, \Omega ^1)^{\vee }$. In [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19], we had as a working hypothesis that our curve $X$ had sufficiently many rational points, in the following sense. For $x\in X(\mathbf {Q}_p)$, the Galois representation $\mathrm {A}(x)$ can be projected onto $\mathrm {H}^1_f(G_T, V)$ (respectively, $\mathrm {H}^1_f(G_T, V^*(1))$), where $G_T$ is the maximal quotient of $G_{\mathbf {Q}}$ unramified outside $T =\{p\}\cup \{$bad primes for $X\}$. With respect to the dual basis $\omega ^\ast _0,\ldots,\omega ^\ast _{g-1},$ the image is the vector $\alpha$ (respectively, $\beta )$ in (2.2). Both of these cohomology groups are isomorphic, under our running assumptions, to $\mathrm {H}^0(X_{\mathbf {Q}_p}, \Omega ^1)^{\vee }$, so we obtain

\[ \pi(\mathrm{A}(x))= \big(\pi_1(\mathrm{A}(x)), \pi_2(\mathrm{A}(x))\big) \in \mathrm{H}^0(X_{\mathbf{Q}_p}, \Omega^1)^{\vee}\times \mathrm{H}^0(X_{\mathbf{Q}_p}, \Omega^1)^{\vee}. \]

Suppose that we can find a basis of $\mathrm {H}^0(X_{\mathbf {Q}_p}, \Omega ^1)^{\vee } \otimes \mathrm {H}^0(X_{\mathbf {Q}_p}, \Omega ^1)^{\vee }$ consisting of elements of the form $\pi (\mathrm {A}_Z(b,x))$, where the $Z$ are cycles on $J$ pulling back to degree 0 cycles on $X$, and the $x$ are rational points on $X$. Then we can compute the coefficients of $h$ in terms of the dual basis by evaluating $h_p(\mathrm {A}_Z(b,x))$ (and, if necessary, $h_{\ell }(\mathrm {A}_Z(b,x))$ for primes $\ell \ne p$). With this choice of basis, the extension of $h$ to a locally analytic function $h\colon X(\mathbf {Q}_p)\to \mathbf {Q}_p$ is immediate.

The number of rational points required can be reduced by working with symmetric heights that are $\mathrm {End}(J)$-equivariant. By the latter we mean that $h(f(x), y) = h(x, f(y))$ for all $f\in \mathrm {End}(J)$, using (2.1). This holds if the splitting $s$ of the Hodge filtration on $V_{\scriptstyle \mathrm {dR}}$ commutes with $\mathrm {End}(J)$ and has the property that $\ker (s)$ is isotropic with respect to the cup product (see [Reference NekovářNek93, § 4.11] and [Reference Balakrishnan and DograBD21, § 4.1]). For instance, if $p$ is a prime of ordinary reduction for the Jacobian, then the height associated to the unit root splitting (see Remark 3.15) is symmetric and $\mathrm {End}(J)$-equivariant. Henceforth we shall assume that $s$ satisfies these assumptions, and we say that $X$ has sufficiently many rational points if the approach outlined above succeeds.

3.3.1 Heights on the Jacobian

If our curve does not have sufficiently many rational points in the above sense, then, in light of (2.1), it is natural to solve for the height pairing using rational points on the Jacobian. In this case, we do not have an algorithm at our disposal to compute $h$ using Nekovář's construction, but we can use the equivalence between this construction and that of Coleman and Gross [Reference Coleman and GrossCG89], proved by Besser [Reference BesserBes04]. In the case when the curve is hyperelliptic and given by an odd-degree model over $\mathbf {Q}_p$ (but see Remark 3.7), we can further use the algorithm of Balakrishnan and Besser [Reference Balakrishnan and BesserBB12, Reference Balakrishnan and BesserBB21]. In the discussion that follows, we will assume that we are in this situation. We will also assume that we know $g$ independent points on the Jacobian.

Recall from Remark 3.1 that we have fixed a continuous idèle class character $\chi \colon \mathbf {A}_{\mathbf {Q}}^\times / \mathbf {Q}^{\times } \longrightarrow \mathbf {Q}_p$ ramified at $p$ and a splitting $s\colon V_{\scriptstyle \mathrm {dR}}/\mathrm {Fil}^0 V_{\scriptstyle \mathrm {dR}} \longrightarrow V_{\scriptstyle \mathrm {dR}}$ of the Hodge filtration on $V_{\scriptstyle \mathrm {dR}} = \mathrm {H}^1_{\scriptstyle \mathrm {dR}}(X_{\mathbf {Q}_p})^{\vee }\,.$ The latter corresponds to a subspace $W\subset H^1_{\scriptstyle \mathrm {dR}}(X_{\mathbf {Q}_p})$, complementary to the image of $\mathrm {H}^0(X_{\mathbf {Q}_p},\Omega ^1)$. With respect to these choices, Coleman and Gross define the local $p$-adic height pairing $h_v(D_1,D_2) \in \mathbf {Q}_p$ at a finite prime $v$ for divisors $D_1,D_2\in \mathrm {Div}^0(X_{\mathbf {Q}_v})$ with disjoint support. The local pairing is bi-additive, and we have $h_v(D_1,D_2) = \chi _v(f(D_2))$ if $D_2 = \operatorname {div}(f)$ is principal. For $v\ne p$, the pairing $h_v$ is also symmetric; $h_p$ is symmetric if and only if $W$ is isotropic with respect to the cup product pairing, which we will assume from now on. Moreover, for $D_1,D_2\in \mathrm {Div}^0(X)$ with disjoint support, only finitely many $h_v(D_1,D_2):= h_v(D_1\otimes \mathbf {Q}_v ,D_2\otimes \mathbf {Q}_v)$ are non-zero. Therefore $h:= \sum _v h_v$ defines a symmetric bilinear pairing $h\colon J(\mathbf {Q})\times J(\mathbf {Q})\to \mathbf {Q}_p$ (see [Reference Coleman and GrossCG89, § 6]).

If we have algorithms to compute the local height pairings, we can solve for the global height pairing in terms of the basis of symmetric bilinear pairings on $J(\mathbf {Q})\otimes \mathbf {Q}_p$ defined by

(3.3)\begin{equation} g_{ij}(D,E):= \tfrac{1}{2}(\log(D) (\omega_i)\log(E)(\omega_j) + \log(D)(\omega_j)\log(E)(\omega_i)) ,\quad 0\le i\le j\le g-1. \end{equation}

Since we can express $\pi _1(\mathrm {A}(x))$ and $\pi _2(\mathrm {A}(x))$ in terms of the dual basis $\{\omega _i^*\}$, we can compute $g_{ij}(\pi (\mathrm {A}(x)))$ for $x\in X(\mathbf {Q}_p)$ (with the obvious abuse of notation) and extend $h$ to a locally analytic function $h\colon X(\mathbf {Q}_p)\to \mathbf {Q}_p$.

It remains to discuss the computation of the local heights. For $D_1,D_2\in \mathrm {Div}^0(X_{\mathbf {Q}_p})$ with disjoint support, the local height is the Coleman integral certain differential with residue divisor $\mathrm {Res}(\omega _{D_1})=D_1$, and $c_p$ is a constant so that $c_p^{-1}\chi _p$ extends to a branch $\mathbf {Q}_p^\times \to \mathbf {Q}_p$ of the $p$-adic logarithm; the Coleman integral is taken with respect to this branch. The differential $\omega _{D_1}$ is normalised with respect to the splitting $s$ using a homomorphism

\[ \Psi\colon T(\mathbf{Q}_p)/T_l(\mathbf{Q}_p)\rightarrow \mathrm{H}_{\scriptstyle \mathrm{dR}}^1(X) \]

from $T(\mathbf {Q}_p)$, the group of differentials of the third kind with integer residues on $X$, quotiented by $T_l(\mathbf {Q}_p)$, the group of logarithmic differentials $\frac {df}{f}$ with $f\in \mathbf {Q}_p(X)^*$, as in the algorithm below. We restrict to degree zero divisors of the form $P-Q$ where $P,Q$ are non-Weierstrass points in $X(\mathbf {Q}_p)$ that do not reduce to a Weierstrass point in $X(\mathbf {F}_p)$ since we will need to compute Coleman integrals between $P,Q$, and our implementation assumes that these points are in non-Weierstrass disks and defined over $\mathbf {Q}_p$.

Algorithm 3.4 The local height $h_p(D_1,D_2)$ at $p$ of the global $p$-adic height [Reference Balakrishnan and BesserBB12]

Input:

1. − Hyperelliptic curve $X/\mathbf {Q}_p$, given by an affine model $y^2=f(x)$, where $f\in \mathbf {Z}_p[x]$ is square-free of degree $2g+1>2$.

2. − Prime $p>2g-1$ of good reduction.

3. − Choice of isotropic subspace $W$ of $H^1_{\scriptstyle \mathrm {dR}}(X_{\mathbf {Q}_p})$, complementary to the subspace of regular 1-forms $\mathrm {H}^0(X_{\mathbf {Q}_p},\Omega ^1)$.

4. − Divisors $D_1 = P-Q, D_2 = R-S$, where $P,Q,R,S$ are non-Weierstrass points in $X(\mathbf {Q}_p)$ that do not reduce to a Weierstrass point in $X(\mathbf {F}_p)$, and $R,S$ do not lie in the residue disks of $P,Q$.

Output: The local height $h_p(D_1,D_2)$ at $p$ of the Coleman–Gross global $p$-adic height

1. (i) Choose $\omega$ a differential in $T(\mathbf {Q}_p)$ with $\text {Res}(\omega )=D_1$.

2. (ii) Solve for the coefficients $b_i$ of $\Psi (\omega )=\sum _{i=0}^{2g-1}b_i\omega _i\in \mathrm {H}_{\scriptstyle \mathrm {dR}}^1(X)$ by computing residues, as in [Reference Balakrishnan and BesserBB12, § 5.2]. Then $\Psi (\omega )-\sum _{i=0}^{g-1}b_i\omega _i\in W$. Let

   \[ \omega_{D_1}:= \omega-\sum_{i=0}^{g-1}b_i\omega_i. \]

3. (iii) Set $\alpha := \phi ^*(\omega )-p(\omega )$. Use Frobenius equivariance of the map $\Psi$ (and the matrix of Frobenius computed with respect to the basis $\{\omega _i\}$ of $\mathrm {H}_{\scriptstyle \mathrm {dR}}^1(X)$) to compute

   \[ \Psi(\alpha)=\phi^*\Psi(\omega)-p\Psi(\omega). \]

4. (iv) Let $\beta$ be a $1$-form with $\mathrm {Res}(\beta )=(R)-(S)$. Compute $\Psi (\beta )$.

5. (v) Compute

   \[ h_p(D_1,D_2):= \int_{D_2}\omega_{D_1} = \int_{S}^R \biggl(\omega - \sum_{i=0}^{g-1}b_i\omega_i\bigr), \]
   where
   \[ \int_{S}^{R}\omega=\frac{1}{1-p}\biggl(\Psi(\alpha)\cup\Psi(\beta)+\sum_{A \in X(\mathbf{C}_p)}{\rm Res}_{A}\biggl(\alpha\int\beta\biggr)- \int_{\phi(S)}^{S}\omega-\int_{R}^{\phi(R)}\omega\biggr) \]
   (see [Reference Balakrishnan and BesserBB12, Remark 4.9]).

Remark 3.5 Note that in the last step above, $\int _{\phi (S)}^S\omega$ and $\int _{R}^{\phi (R)}\omega$ are tiny integrals, that is, Coleman integrals between points in the same residue disk. Such integrals may be computed merely using a uniformising parameter at any point in the residue disk. The computation $\sum _{A \in X(\mathbf {C}_p)}\mathrm {Res}_{A}\big (\alpha \int \beta \big )$ will, in most cases, require working over various extension of $\mathbf {Q}_p$ to pick up all contributions at all poles (see [Reference Balakrishnan and BesserBB12, Remark 4.10]).

Remark 3.6 If our hyperelliptic curve $X$ does not admit an odd-degree model over $\mathbf {Q}$, we may choose our prime $p$ such that $X$ has an odd-degree model over $\mathbf {Q}_p$ and compute local heights at $p$ on this model. This follows from the fact that $\Psi (\varphi ^*\omega ) = \varphi ^*(\Psi (\omega ))$ for $\varphi$ an isomorphism of curves and $\omega$ a differential of the third kind.

Remark 3.7 In his thesis [Reference GajovićGaj22], Gajović has improved Algorithm 3.4 and extended it to even-degree models of hyperelliptic curves.

The local height at a prime $\ell \ne p$ is defined in terms of intersection theory. We can extend $D_1$ and $D_2$ to divisors $\mathcal {D}_1$ and $\mathcal {D}_2$ on a regular model of $X_{\mathbf {Q}_\ell }$ so that both $\mathcal {D}_i$ have trivial intersection multiplicity with all vertical divisors; then by [Reference Coleman and GrossCG89, Proposition 1.2], we have

\[ h_{\ell}(D_1,D_2) = -(\mathcal{D}_1\cdot \mathcal{D}_2)\chi_p(\ell). \]

3.4 Mordell–Weil sieving

The idea of the Mordell–Weil sieve, originally due to Scharaschkin [Reference ScharaschkinSch99], is to deduce information on rational points on $X$ via the intersection of the images of $X(\mathbf {F}_v)$ and $J(\mathbf {Q})$ in $J(\mathbf {F}_v)$ (or suitable quotients) for several primes $v$ of good reduction. It is often applied to verify that $X(\mathbf {Q})=\varnothing$, but it can also be combined with $p$-adic techniques to compute $X(\mathbf {Q})$ when there are rational points.

We review the basic idea, which is straightforward. Making the sieve perform well in practice is a different matter (see [Reference Bruin and StollBS10] for an elaborate discussion of the issues one encounters and detailed strategies). For ease of exposition, we assume that $J(\mathbf {Q})$ is torsion-free and that we have generators $P_1,\ldots,P_r$ of $J(\mathbf {Q})$. Let $M>1$ be an integer and let $S$ be a finite set of primes of good reduction for $X$. Then the diagram

is commutative. In the situation of interest to us, the horizontal maps are induced by our choice of base point $b \in X(\mathbf {Q})$.

In our work, we use the Mordell–Weil sieve in two ways. On the one hand, we apply it to show that for a fixed prime $p$, a given residue disk in $X(\mathbf {Q}_p)$ does not contain a rational point. To this end, we set $M = M'\cdot p$ for some suitable auxiliary integer $M'$, and we choose $S$ to consist of primes $\ell$ so that $\gcd (\#J(\mathbf {F}_{\ell }), \#J(\mathbf {F}_q))$ is large for some prime divisors $q\mid pM'$. We can then hope that the image of the reduction of the disk under $\prod \beta _{S,M}$ does not meet the image of the map $\prod \alpha _{S,M}$.

On the other hand, we use the sieve to show for fixed $M>1$ that a given coset of $MJ(\mathbf {Q})$ does not contain the image of a point in $X(\mathbf {Q})$ under the Abel–Jacobi map $P\mapsto [P-b]$. Suppose a point $P\in X(\mathbf {Q}_p)$ is given to finite precision $p^N$. If $P$ is rational, then there are integers $a_1,\ldots,a_g$ such that

\[ [P-b]= a_1P_1+\cdots+a_gP_g. \]

Via the abelian logarithm, we compute a tuple $(\tilde {a}_1,\ldots,\tilde {a}_g) \in \mathbf {Z}/p^N\mathbf {Z}$ satisfying $a_i \equiv \tilde {a}_i \pmod {p^N}$ for all $i\in \{1,\ldots,g\}$. To show that $P$ is not rational, it suffices to show that the corresponding coset of $p^NJ(\mathbf {Q})$ does not contain the image of such a point.

In our implementation, we have not tried to optimise the interplay between quadratic Chabauty and the Mordell–Weil sieve. Such an optimisation is discussed in [Reference Balakrishnan, Besser and MüllerBBM17, § 7]. Let us only note here that we may combine quadratic Chabauty information coming from several primes, and that we can enhance that information using an auxiliary integer $M'$ similar to the above. Another account of combining quadratic Chabauty with the Mordell–Weil sieve can be found in [Reference Balakrishnan, Best, Bianchi, Lawrence, Müller, Triantafillou and VonkBBB⁺21, § 6.7].

3.5 Implementation and scope

We have implemented the algorithms described in this section in the computer algebra system Magma [Reference Bosma, Cannon and PlayoustBCP97]. Our code is freely available at [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺]. It extends the code used for $X_{\mathrm {s}}^+(13)$ in [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19] and can be used to recover that example. It is applied to new examples, as discussed in § 5.

We begin by summarising our discussion so far and describe the general procedure to determine the finite set $X(\mathbf {Q}_p)_2$ as it would apply to the modular curve $X$ attached to a general congruence subgroup, and Atkin–Lehner quotients thereof. In this generality, several steps cannot be easily automated, so we discuss the extent to which our implementation has automated the procedure, and point out which steps require additional action from the user. See Example 5.3 for a fairly detailed worked example.

Our techniques are built on prior work of Tuitman on computing the action of Frobenius on rigid cohomology [Reference TuitmanTui17]. We recall some of the underlying structures present in Tuitman's work and a set of assumptions on these auxiliary structures.

Suppose our modular curve $X/\mathbf {Q}$ is given by a (possibly singular) plane model $Q=0$ with $Q(x,y) \in \mathbf {Z}[x,y]$ a polynomial that is irreducible and monic in $y$. Let $d_x$ and $d_y$ denote the degrees of the morphisms $x$ and $y$, respectively, from $X$ to the projective line. Let $\Delta (x) \in \mathbf {Z}[x]$ denote the discriminant of $Q$ with respect to the variable $y$. Moreover, define $r(x) \in \mathbf {Z}[x]$ to be the square-free polynomial with the same zeros as $\Delta (x)$, in other words, $r=\Delta /(\gcd (\Delta,{d\Delta }/{{d} x}))$.

Definition 3.9 Let $W^0 \in \mathrm {GL}_{d_x}(\mathbf {Q}[x,1/r])$ and $W^{\infty } \in \mathrm {GL}_{d_x}(\mathbf {Q}[x,1/x,1/r])$ denote matrices such that, if we denote

\[ b^0_j = \sum_{i=0}^{d_x-1} W^0_{i+1,j+1} y^i \quad \text{and} \quad b^{\infty}_j = \sum_{i=0}^{d_x-1} W^{\infty}_{i+1,j+1} y^i \]

for all $0 \leq j \leq d_x-1$, then

1. (i) $[b^{0 \;}_0,\ldots,b^{0 \;}_{d_x-1}]$ is an integral basis for $\mathbf {Q}(X)$ over $\mathbf {Q}[x]$,

2. (ii) $[b^{\infty }_0,\ldots,b^{\infty }_{d_x-1}]$ is an integral basis for $\mathbf {Q}(X)$ over $\mathbf {Q}[1/x]$,

where $\mathbf {Q}(X)$ denotes the function field of $X$. Moreover, let $W \in \mathrm {GL}_{d_x}(\mathbf {Q}[x,1/x])$ denote the change-of-basis matrix $W=(W^0)^{-1} W^{\infty }$.

To further determine the subset of rational points $X(\mathbf {Q})$ from the finite set of points produced by our algorithm, we carry out the Mordell–Weil sieve. In practice it may happen (see below) that $X(\mathbf {Q})$ is returned by the algorithm, but this is typically not the case when $X$ has genus $2$.

3.5.2 The Néron–Severi classes $Z_i$

Under the assumption that $T_p$ generates the endomorphism ring of the Jacobian, which we made for convenience above, one may proceed precisely as in [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, § 6.4] to determine a non-trivial class

\[ Z \in \mathrm{Ker}\big(\mathrm{NS}(J) \longrightarrow \mathrm{NS}(X) \big). \]

Indeed, the matrix $A_p$ of the Hecke operator $T_p$ acting on $\mathrm {H}^1_{\scriptstyle \mathrm {dR}}(X_{\mathbf {Q}_p})$ is easily determined from the matrix of Frobenius $F_p$ (which is already a byproduct of the algorithms for the local height at $p$), by the Eichler–Shimura relation:

\[ T_p = F_p +pF_p^{-1}. \]

Under our assumption, the matrices of the classes $Z_i$ acting on $\mathrm {H}^1_{\scriptstyle \mathrm {dR}}(X_{\mathbf {Q}_p})$ may then be computed as linear combinations of powers of $A_p$.

Remark 3.17 This is the only part of our algorithm specific to modular curves, since it relies on the Eichler–Shimura relation. It should, however, be noted that this is mainly a matter of convenience adopted for the purpose of automation. More generally, for a smooth projective curve $X/\mathbf {Q}$ satisfying the assumptions of § 2.1, one could find $p$-adic approximations of the action of the non-trivial classes $Z_i$ on $\mathrm {H}^1_{\scriptstyle \mathrm {dR}}(X_{\mathbf {Q}_p})$ using just $p$-adic linear algebra. Indeed, the space of correspondences which are symmetric under the Rosati involution and induce endomorphisms of trace zero on the Tate module maps under the cycle class into the intersection of the $\mathrm {Fil} ^{1}$ and $\phi =p$ subspaces of

(3.4)\begin{equation} \ker \Big( \wedge ^2 \mathrm{H} ^1 _{\scriptstyle \mathrm{dR}}(X_{\mathbf{Q} _p } )\stackrel{\cup }{\longrightarrow } \mathrm{H} ^2 _{\scriptstyle \mathrm{dR}}(X_{\mathbf{Q} _p } ) \Big) . \end{equation}

In fact, by the Tate conjecture, the rank of the space of (crystalline) cohomology classes of such correspondences over $\mathbf {F} _p$ is equal to the dimension of the $\phi =p$ subspace of (3.4), and by the $p$-adic Lefschetz-$(1,1)$ theorem of Berthelot and Ogus [Reference Berthelot and OgusBO83, § 3.8] such a correspondence over $\mathbf {F} _p$ lifts to $\mathbf {Q} _p$ if and only if its cycle class lies in $\mathrm {Fil} ^{1}$. Note that the dimension of the space of correspondences symmetric under the Rosati involution need not equal the dimension of $\wedge ^2 \mathrm {H} ^1 _{\scriptstyle \mathrm {dR}}(X_{\mathbf {Q} _p })^{\phi =p}\cap \mathrm {Fil} ^1$, as was erroneously claimed in [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, Lemma 4.5], since the rank of the intersection of a $\mathbf {Z}$-lattice with a $\mathbf {Q} _p$-subspace may be less than the dimension of the intersection with the $\mathbf {Q} _p$-subspace it spans. However, if one knows a set of generators of a finite index subgroup of $\mathrm {End} (J)$ in advance (e.g. using algorithms for rigorous computation of the endomorphism algebra of the Jacobian [Reference Costa, Mascot, Sijsling and VoightCMSV19])) then one can use this to compute the classes of generators in cohomology.

Therefore the assumption that $T_p$ generates the endomorphism algebra could be circumvented in this step with a little work, although it is used in the computation of the local heights away from $p$ (see below). When the assumption is not satisfied, our implementation throws an error, urging the user to try a different choice of prime $p$.

4.1 Hodge filtration

We first bound the loss of precision in steps (ii)–(v) of Algorithm 3.12. For simplicity, we restrict to one class $Z$; the extension to $\mathrm {rk}\mathrm {NS}(J) - 1$ such classes is immediate. Let $Y/\mathbf {Q}$ be an affine open subset of $X$, birational to a curve given by an equation that satisfies Assumption 3.10. We may compute an integral, symplectic basis $\boldsymbol {\omega }=(\omega _0 ,\ldots,\omega _{2g-1})$ of de Rham cohomology over $\mathbf {Q}$ exactly, and extend this to an integral basis of $H^1 _{\scriptstyle \mathrm {dR}}(Y)$ via differentials $(\omega _{2g},\ldots,\omega _{2g+d-2})$ of the third kind. Using such a basis, we may compute the action of the Frobenius operator $F$ on $\mathrm {H} ^1 _{\scriptstyle \mathrm {dR}}(X/\mathbf {Q} _p )$ to any desired $p$-adic precision using Tuitman's algorithm [Reference TuitmanTui16, Reference TuitmanTui17], from which we obtain the action of the Hecke operator $T_p = F + pF^{-1}$ on $\mathrm {H}^1 _{\scriptstyle \mathrm {dR}}(X/\mathbf {Q} _p)$ by the Eichler–Shimura relation. The inversion of $F$ results in a finite and computable loss of precision, which the code takes into account. This results in an algorithm that returns the action of the correspondence $Z$ correctly modulo $p^n$ for some $n\geq 1$ that is returned by the algorithm.

Using this, we may compute a matrix $\Lambda$ with entries in $H^0 (Y,\Omega _{Y_{\mathbf {Q}}})$, of the form

\[ \Lambda := -\left( \begin{array}{@{}ccc@{}} 0 & 0 & 0 \\ \boldsymbol{\omega} & 0 & 0 \\ \eta & \boldsymbol{\omega}^{\intercal}Z & 0 \\ \end{array} \right) \]

such that $d+\Lambda$ extends to a flat connection on $X$. From this, we may compute $\gamma _{\mathrm {Fil} }$ and $\boldsymbol {\beta }_{\mathrm {Fil} }$ from (3.1). We recall from [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, § 4] that the defining properties of $\eta$, the $\boldsymbol {\beta }_{\mathrm {Fil} }$ and $\gamma _{\mathrm {Fil}}$ are as enumerated below. For $x\in (X-Y)(\overline {\mathbf {Q} })$, we let $t_x$ denote a parameter, and $\boldsymbol {\Omega }_x$ denote the vector of formal integrals of the basis differentials $\omega _i$:

\begin{align*} d\boldsymbol{\Omega }_{x,i}=\omega _i \in\overline{\mathbf{Q} }[\! [t_x]\!]. \end{align*}

1. (i) The first $g$ entries of $\boldsymbol {\beta }_{\mathrm {Fil} }$ are zero, and the last $g$ are given by a vector $\mathbf {b}_{\mathrm {Fil} }$ of constants specified below.

2. (ii) $\eta$ is a linear combination of $\omega _{2g},\ldots,\omega _{2g+d-2}$, unique by [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, Lemma 4.10], such that

   (4.1)\begin{equation} d\boldsymbol{\Omega }_x^{\intercal }Z \boldsymbol{\Omega }_x -\eta \end{equation}
   has vanishing residues at all $x\in (X-Y)(\overline {\mathbf {Q} })$.

3. (iii) $\mathbf {b}_{\mathrm {Fil} }$ and $\gamma _{\mathrm {Fil} } \in \mathcal {O}(Y)$ are the unique solutions to the equation $\gamma _{\mathrm {Fil} }(b)=0$ and

   (4.2)\begin{equation} g_x +\gamma _{\mathrm{Fil}}-\mathbf{b}_{\mathrm{Fil} }^\intercal N^\intercal \boldsymbol{\Omega }_x -\boldsymbol{\Omega }_x ^\intercal Z N N^\intercal \boldsymbol{\Omega }_x \in L[\! [t_x ]\! ] \end{equation}
   for all $x\in (X-Y)(\overline {\mathbf {Q} })$, where $g_x \in \overline {\mathbf {Q} }[\! [t_x ]\! ]$ is defined to be the formal integral of $d\boldsymbol {\Omega }_x ^\intercal Zd\boldsymbol {\Omega }_x -\eta$ and $N$ is the block $2g \times g$ matrix with top block zero and lower block a $g \times g$ identity matrix.

Given our basis $\boldsymbol {\omega }$, we may calculate $\boldsymbol {\Omega }_x$ to any given $t_x$-adic precision. Note that to solve (4.1), we only need to know $\boldsymbol {\Omega }_x$ modulo $t_x ^{m_x}$, where $m_x$ is the maximum of the order of the poles of the entries of $\boldsymbol {\Omega }_x$. Similarly, to solve for $\gamma _{\mathrm {Fil}}$ and $\mathbf {b}_{\mathrm {Fil}}$ in (4.2), we need only compute the principal parts of $\boldsymbol {\Omega }_x$ and $\boldsymbol {\Omega }_x ^\intercal Z N N^\intercal \boldsymbol {\Omega }_x$. Hence, given the above, we may calculate $\eta,\gamma _{\mathrm {Fil} }$ and $\mathbf {b}_{\mathrm {Fil} }$ to precision $p^{n-2\nu }$, where $\nu$ is minus the minimum of the valuations of the $t_x^i$ coefficients of the entries of $\boldsymbol {\Omega }_x$, for $i\leq m_x$.

4.2 Frobenius-equivariant splitting

We now bound the loss of precision in the computation of the Frobenius-equivariant splitting

\[ \lambda^\phi(x) =\left( \begin{array}{@{}ccc@{}} 1 & 0 & 0 \\ \boldsymbol{\alpha }_{\phi}(b,x) & 1 & 0 \\ \gamma_{\phi}(b,x) & \boldsymbol{\beta }^{\intercal}_{\phi}(b,x) & 1 \\ \end{array} \right) \]

from (3.1) for $x \in X(\mathbf {Q}_p)\cap ]\mathcal {U}[$, where $\mathcal {U}$ is an open of $Y_{\mathbf {F} _p }$ on which we have an overconvergent lift of Frobenius. This computation is the content of [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, § 5].

The first step is to find the Frobenius structure on the filtered $\phi$-module $M(b)$. By [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, § 5.3.2], the inverse of the Frobenius structure is given by a matrix

\[ G \in (\mathrm{H}^0(\,]Y[\,,j^{\dagger} \mathcal{O}_Y))^{(2g+2)\times(2g+2)} \]

such that

(4.3)\begin{equation} \Lambda_{\phi}G + dG = G\Lambda, \end{equation}

where $j^{\dagger} \mathcal {O}_Y$ is the overconvergent structure sheaf on the tube $]Y[$.

Compared to [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, § 5.3.2], we give a slightly more detailed account of the algorithm to find $G$. We first apply the algorithms in [Reference TuitmanTui16, Reference TuitmanTui17] (see [Reference Balakrishnan and TuitmanBT20, Algorithm 2.18]) to compute the action of Frobenius on $\mathrm {H}^1_{\scriptstyle \mathrm {{rig}}}(X\otimes \mathbf {Q}_p)$ as

(4.4)\begin{equation} \phi^*\boldsymbol{\omega} = F\boldsymbol{\omega} + d\mathbf{f} \end{equation}

for a matrix $F \in M_{2g}(\mathbf {Q}_p)$ and a column vector $\mathbf {f}$ with entries in $\mathrm {H}^0(\,]Y[\,,j^{\dagger} \mathcal {O}_Y)$, uniquely determined by the condition that $\mathbf {f}(b_0 ) = \mathbf {0}$, where $b_0$ is the Teichmüller point in the disk of $b$.

Next, we define a vector of functions $\mathbf {g}_0 := -F^{\intercal}Z \mathbf {f}$. Then the differential

(4.5)\begin{equation} \xi :=(\phi^*\boldsymbol{\omega}^{\intercal})Z\mathbf{f} + (\phi^*\eta - p \eta) -\mathbf{g}_0 ^{\intercal} \boldsymbol{\omega} \end{equation}

is of the second kind, and therefore the reduction algorithms in $\mathrm {H}^1_{\mathrm {rig}}(Y)$ from [Reference TuitmanTui16, Reference TuitmanTui17] can be applied to compute a vector of constants $\mathbf {c} \in \mathbf {Q}_p^{2g}$ and a function $H$Footnote ⁴ $\in \mathrm {H}^0(\,]Y[\,,j^{\dagger} \mathcal {O}_Y)$ such that

(4.6)\begin{equation} \mathbf{c}^{\intercal} \boldsymbol{\omega} + dH = \xi. \end{equation}

Hence, the function $\mathbf {g}:= \mathbf {g}_0 +\mathbf {c}$ satisfies

\[ d\mathbf{g}^\intercal = d\mathbf{f}^\intercal ZF\quad\text{and }\quad dH = \boldsymbol{\omega}^\intercal F^\intercal Z\mathbf{f} + d\mathbf{f}^\intercal Z\mathbf{f}-\mathbf{g}^\intercal \boldsymbol{\omega} +\phi^*\eta - p\eta, \]

and we normalise $H$ by requiring that $H(b_0)=0$. The matrix

(4.7)\begin{equation} G = \left( \begin{array}{@{}ccc@{}} 1 & 0 & 0 \\ \mathbf{f} & F & 0\\ H & \mathbf{g}^\intercal & p \\ \end{array} \right) \end{equation}

then satisfies (4.3).

4.2.1 Frobenius-equivariant splitting for Teichmüller points

Suppose that $x_0\in X(\mathbf {Q}_p)\cap ]\mathcal {U}[$ is a Teichmüller point. As described in [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, § 5.3.2], the Frobenius-equivariant splitting of $M(x_0)$ is given by

(4.8)\begin{equation} \lambda^\phi(x_0) = \left( \begin{array}{@{}ccc@{}} 1 & 0 & 0 \\ (I-F)^{-1}\mathbf{f} & 1 & 0\\ \dfrac{1}{1-p}\big(\mathbf{g}^{\intercal}(I-F)^{-1}\mathbf{f} +H\big) & \mathbf{g}^{\intercal}(F-p)^{-1} & 1 \\ \end{array} \right)(x_0). \end{equation}

The loss of precision in the computation of $\mathbf {f}$ and $F$ is estimated in [Reference TuitmanTui17]. Hence, it is easy to bound the precision loss in the computation of $\lambda ^\phi (x_0)$ using the following result.

Proposition 4.1 Suppose that the entries of the matrix $G$ and a point $P\in X(\mathbf {Q}_p)\cap ]\mathcal {U}[$ are accurate to $n$ digits of precision. Then $G(P)$ is also accurate to $n$ digits of precision.

Our proof of Proposition 4.1 is somewhat similar to but more involved than the proofs in [Reference Balakrishnan and TuitmanBT20, § 4], where the loss of precision in the evaluation of $\mathbf {f}$ and of single Coleman integrals is estimated. We may expand

(4.9)\begin{equation} \xi = \sum_{j \in \mathbf{Z}}\biggl(\sum_{k=0}^{d_x - 1} \frac{w_{j,k}(x)}{r(x)^j} b_k^0\biggr)\frac{{d} x}{r}. \end{equation}

The hardest part of the proof of Proposition 4.1 is to find lower bounds on the valuation of the coefficients $w_{j,k}$, which we now describe. Let $e_0$ (respectively, $e_\infty$) be the maximum of the ramification indices of the map $x\colon X\to \mathbf {P}^1$ with respect to our chosen model at points lying in affine (resp., infinite) disks.

Lemma 4.2 There is a constant $\kappa$ such that for all $j,k$ we have

(4.10)\begin{equation} \mathrm{ord}_p(w_{jk}) \geq \begin{cases} \left\lfloor\dfrac{j}{p}\right\rfloor + 1 - \log_p(je_0) +\kappa , & j \neq 0,\\ \kappa, & j = 0. \end{cases} \end{equation}

Proof. Looking at the constituent parts of (4.5), we start with $(\phi ^*\boldsymbol {\omega }^{\intercal})Z\mathbf {f}$. We write

\[ (\phi^*\boldsymbol{\omega}^{\intercal})_i = \sum_{j_1 \in \mathbf{Z}} \biggl(\sum_{k_1 = 0}^{d_x - 1} \frac{d_{j_1, k_1}^{(i)}(x)}{r^{j_1}} b_{k_1}^0\biggr) \frac{{d} x}{r}. \]

Then $\mathrm {ord}_p(d_{j_1, k_1}^{(i)}) \geq \lfloor {j_1}/{p} \rfloor + 1$ by [Reference TuitmanTui17, Proof of Proposition 4.9]. We have

\[ f_i = f_{i,0} + f_{i, \infty} + f_{i, \mathrm{end}}, \]

where $f_{i,0}, f_{i,\infty }$ and $f_{i,\mathrm {end}}$ correspond to the three reduction steps (2), (3) and (4) in the reduction algorithm from [Reference TuitmanTui17], summarised in [Reference Balakrishnan and TuitmanBT20, Algorithm 2.18]. By (1), (3) and (4) of [Reference Balakrishnan and TuitmanBT20], there are $\mu _1, \lambda _1\ge 0$ such that

\begin{gather*} f_{i,0} = \sum_{j_2 = 1}^{\infty} \biggl(\sum_{k_2 = 0}^{d_x -1} \frac{c_{j_2, k_2}^{(i)}(x)}{r^{j_2}}b_{k_2}^0\biggr),\\ f_{i, \infty} = \sum_{k_3 = 0}^{d_x-1} \sum_{l = 0}^{\mu_1} e_{k_3, l}^{(i)} x^l b_{k_3}^0, \quad f_{i,\mathrm{end}} = \sum_{k_4 = 0}^{d_x-1}\sum_{m=0}^{\lambda_1} u_{k_4, m}^{(i)}x^m b_{k_4}^0. \end{gather*}

Equation (2) of [Reference Balakrishnan and TuitmanBT20] implies the lower bound $\mathrm {ord}_p(c_{j_2, k_2}^{(i)}) \geq \left \lfloor {j_2}/{p} \right \rfloor + 1 - \log _p \lfloor j_2 e_0\rfloor.$ Let

(4.11)\begin{equation} \kappa^{(i)} := \min(\{0,\mathrm{ord}_p(e_{k_3,l}^{(i)})\} \cup \{\mathrm{ord}_p(u_{k_4,m}^{(i)})\})\quad \text{and}\quad \kappa_1 := \min_i\{\kappa^{(i)}\}. \end{equation}

Without loss of generality, the matrix $Z$ has $p$-integral entries. Hence, every $(Z\mathbf {f})_i$ is of the form

(4.12)\begin{equation} (Z\mathbf{f})_i = \sum_{j_2 = 0}^{\infty} \sum_{k_2 = 0}^{d_x -1} \frac{g^{(i)}_{j_2,k_2}(x)}{r^{j_2}}b_{k_2}^0 \end{equation}

where, for all $k_2$, we have

(4.13)\begin{equation} \mathrm{ord}_p(g^{(i)}_{j_2,k_2}) \geq \begin{cases} \left\lfloor\dfrac{j_2}{p}\right\rfloor + 1 - \log_p\lfloor j_2 e_0\rfloor, & \text{if}\ j_2>0,\\ \kappa_1, & \text{if}\ j_2=0.\\ \end{cases} \end{equation}

Let us now consider, for each $i$,

\begin{align*} \big(\phi^*\boldsymbol{\omega}^{\intercal}\big)_i \big(Z\mathbf{f}\big)_i &= \sum_{j_1 \in \mathbf{Z}} \biggl(\sum_{k_1 = 0}^{d_x-1} \frac{d_{j_1,k_1}^{(i)}}{r^{j_1}}b_{k_1}^0\biggr) \biggl(\sum_{j_2 = 0}^{\infty}\sum_{k_2 = 0}^{d_x-1} \frac{g^{(i)}_{j_2,k_2}}{r^{j_2}}b_{k_2}^0\biggr)\frac{{d} x}{r}\\ &=\sum_{j = j_1 + j_2 \in \mathbf{Z}, j_1 \in \mathbf{Z}, j_2 \geq 0} \frac{1}{r^j}\biggl(\sum_{k = k_1 + k_2, k_i \in \{0,\ldots,d_x-1\}} \big(d^{(i)}_{j_1,k_1} g^{(i)}_{j_2,k_2}\big)b_k^0\biggr)\frac{{d} x}{r}\\ &=:\sum_{j\in \mathbf{Z}}\biggl(\frac{1}{r^j}\sum^{d_x-1}_{k=1} \tau_{jk}b_k^0\biggr) \frac{{d} x}{r}. \end{align*}

We distinguish two cases. If $j_2 > 0$ then

(4.14)\begin{equation} \mathrm{ord}_p(d^{(i)}_{j_1,k_1}g^{(i)}_{j_2,k_2}) \geq \left\lfloor\frac{j_1}{p}\right\rfloor + 1 + \left\lfloor\frac{j_2}{p}\right\rfloor + 1 - \log_p(j_2e_0) \geq \left\lfloor \frac{j}{p} \right\rfloor + 1 - \log_p((j-1)e_0). \end{equation}

If $j_2 = 0$, then $\mathrm {ord}_p(d^{(i)}_{j_1,k_1}g^{(i)}_{j_2,k_2})\geq \left \lfloor {j_1}/{p}\right \rfloor + 1 + \kappa _1$. Together, these yield

(4.15)\begin{equation} \mathrm{ord}_p(\tau_{jk}) \geq \left\lfloor \frac{j}{p}\right\rfloor + 1 - \log_p((j-1)e_0) + \kappa_1. \end{equation}

The next term to consider in (4.5) is $\phi ^* \eta - p \eta$, where $\eta$ is constructed in [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, § 4]. Let $\kappa _2$ denote the $p$-adic valuation of the vector of coefficients of $\eta$ in terms of the basis differentials $\omega _{2g},\ldots,\omega _{2g+2-d}$ (see [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, § 4.1]). Write

\[ \phi^*\eta - p\eta = \sum_{j \in \mathbf{Z}}\biggl(\sum_{k = 0}^{d_x-1} \frac{s_{jk}(x)}{r^j} b_k^0\biggr) \frac{{d} x}{r}. \]

Then the $s_{jk}$ satisfy $\mathrm {ord}_p(s_{jk}) \geq \kappa _2 + \left \lfloor {j}/{p} \right \rfloor + 1$ if $j\neq 0$ and $\mathrm {ord}_p(s_{0k}) \geq \kappa _2 +1$, so

(4.16)\begin{equation} \mathrm{ord}_p(s_{jk}) \geq \kappa_2 + \left\lfloor \frac{j}{p}\right\rfloor + 1\quad \text{for all}\ j. \end{equation}

For the final summand $\mathbf {g}_0 ^{\intercal}\boldsymbol {\omega }$ in (4.5) note that since $F$ has $p$-integral entries, every $(F^{\intercal}Z \mathbf {f})_i$ has an expansion as in (4.12). Because $\omega _i$ is integral for all $i$, the lower bounds in (4.13) remain valid for $\mathbf {g}_0 ^{\intercal}\boldsymbol {\omega }$. The proof of Lemma 4.2 follows from this and from (4.14) and (4.15) upon setting $\kappa = \min \{\kappa _1,\kappa _2\}$.

We now estimate the precision loss that can occur during the application of the reduction algorithm from [Reference TuitmanTui17] to the differential $\xi$. Our proof is similar to the proof of [Reference TuitmanTui17, Prop 4.9], which estimates the precision loss in the reduction of $F^*(\omega _i)$. Suppose that $\xi$ is correct to $n$ digits of $p$-adic precision. First consider terms in (4.9) with $j>0$. It follows from (4.10) that $j - p\log _p(je_0) \leq pm - p\kappa$ (note that $\kappa \le 0$). By [Reference TuitmanTui17, Proposition 3.7], the precision loss at pole order $j$ during the reduction at finite points is at most $\lfloor \log _p(j_{\max }e_0)\rfloor,$ where $j_{\max }$ is the largest integer $j$ such that $j - p\log _p(je_0) \leq pn - p\kappa.$ As in the proof of [Reference TuitmanTui17, Proposition 4.9], this might introduce small poles above $\infty$, but by the same reasoning as in [Reference TuitmanTui17], the reduction of these poles leads to a loss of precision bounded by $\lfloor \log _p(-(\mathrm {ord}_\infty W^{-1})+1)e_{\infty }\rfloor$. We set

\[ g_1(n) := \lfloor \log_p(j_{\max}e_0)\rfloor + \lfloor\log_p(-(\mathrm{ord}_\infty W^{-1})+1)e_{\infty}\rfloor. \]

If we write

\[ \xi = \biggl(\sum_{i=0}^{d_x-1} \alpha_i(x,x^{-1})b_i^{\infty}\biggr)\frac{{d} x}{r}\quad\text{and}\quad m_\infty= -\min_i \{\mathrm{ord}_\infty\alpha_i-\deg(r) + 1\}, \]

then the loss of precision during the reductions above infinity (where $j\leq 0$) is bounded by $g_2:= \lfloor \log _p(m_\infty e_\infty )\rfloor.$

Hence, we have established the following lemma.

Lemma 4.3 Suppose that $\xi$ is correct to $n$ digits of precision. Then $\mathbf {c}$ and $H$ are correct to $n-\max \{g_1(n), g_2\}$ digits of precision.

Proof of Proposition 4.1 Similar to the $f_i$, we may decompose $H$ as $H = H_0 + H_{\infty } + H_{\mathrm {end}}$, corresponding to steps (2), (3) and (4), respectively, in [Reference Balakrishnan and TuitmanBT20, Algorithm 2.18]. By the above, the reduction above finite points introduces a denominator of valuation at most $\log _p(je_0)$ for pole order $j$, therefore we have

(4.17)\begin{equation} H_0 = \sum_{j \geq 1}\sum_{k = 0}^{d_x - 1} \frac{c_{jk}(x)}{r^j} b_k^0,\quad\text{where}\ \mathrm{ord}_p(c_{jk}) \geq \left\lfloor \frac{j}{p} \right\rfloor - 2\log_p(je_0) + \kappa. \end{equation}

Recall that the matrix $G$ is defined in (4.7). There is no loss of precision when evaluating $\mathbf {f}(P)$ by [Reference Balakrishnan and TuitmanBT20, Proposition 4.5]. By our assumption that $F$ and $Z$ are $p$-integral, there is no precision loss when evaluating $\mathbf {g_0}(P)$. Using the bounds (4.17), the proof of [Reference Balakrishnan and TuitmanBT20, Proposition 4.5] shows that $H(P)$ is accurate to $n$ digits of precision as well. Since $\mathbf {g} = \mathbf {g_0} + \mathbf {c}$, the proposition follows.

4.2.2 Frobenius-equivariant splitting for general points

For $x\in X(\mathbf {Q}_p)\cap ]\mathcal {U}[$, not necessarily Teichmüller, the Frobenius-equivariant splitting $\lambda ^\phi (x)$ of $M(x)$ is given by

(4.18) \begin{equation} \begin{pmatrix} 1 & 0 & 0 \\ \int_{x}^{x_0} \boldsymbol{\omega} & 1 & 0 \\ \int_{x}^{x_0} \eta+\int ^x _{x_0 }\boldsymbol{\omega }^{\intercal }Z \boldsymbol{\omega } & \int_{x}^{x_0} \boldsymbol{\omega}^{\intercal}Z & 1 \\ \end{pmatrix}\cdot \begin{pmatrix} 1 & 0 & 0 \\ \int_{b_0}^b \boldsymbol{\omega} & 1 & 0 \\ \int_{b_0}^b \eta +\int ^b _{b_0 }\boldsymbol{\omega }^{\intercal }Z \boldsymbol{\omega } & -\int_{b_0}^b \boldsymbol{\omega}^{\intercal}Z & 1 \\ \end{pmatrix}\cdot \lambda^\phi(x_0), \end{equation}

where $x_0$ is the Teichmüller point in the disk of $x$. The first two matrices in (4.18) correspond to parallel transport of $\Lambda$ from $x$ to $x_0$ and from $b_0$ to $b$, respectively.

For the local height $h_p(A(x))$, we need the Frobenius-equivariant splitting $\lambda ^\phi (x)$ both for fixed $x$ and for $x$ varying inside a residue disk. We start by bounding the valuations of the coefficients of power series expansions of the differentials in the parallel transport matrices of $\Lambda$ in terms of a local coordinate $t$ at a fixed affine point $y_0 \in X(\mathbf {Q}_p)\cap ]\mathcal {U}[$. By assumption, the entries of the expansions of $\boldsymbol {\omega }$ and $\boldsymbol {\omega }^{\intercal }Z$ all have integral coefficients, so their integrals have entries whose $i$th coefficient has valuation $-\geq \mathrm {ord}_p(i)$. Therefore, we have

(4.19)\begin{equation} \boldsymbol{\omega}(t)^{\intercal}Z\int \boldsymbol{\omega}(t) = \sum_{i\ge 1} a_it^i,\quad\text{where}\ \mathrm{ord}_p(a_i)\ge -\lfloor \log_p(i)\rfloor. \end{equation}

It follows that

(4.20)\begin{equation} \int\biggl(\boldsymbol{\omega}(t)^{\intercal}Z\int \boldsymbol{\omega}(t)\biggr) = \sum_{i\ge 1} b_it^i,\quad\text{where}\ \mathrm{ord}_p(b_i)\ge -2\lfloor \log_p(i)\rfloor. \end{equation}

By construction, the coefficients of $\eta$ in terms of $\omega _{2g},\ldots,\omega _{2g+d-2}$ are polynomials in $x$. Define $d_i(\eta )$ to be the valuation of the $i$th coefficient if $i$ is smaller than the maximum of the degrees of these coefficients and 0 otherwise. Then the $i$th coefficient of the integral of $\eta$ has valuation at least $-\mathrm {ord}_p(i)-d_i(\eta )$. Hence, the $i$th coefficient of every expansion of the parallel transport matrix in $t$ has valuation at least

(4.21)\begin{equation} \varphi(i) := -\lfloor \log_p(i)\rfloor+ \min\{d_i(\eta), -\lfloor \log_p(i)\rfloor\}. \end{equation}

For definite parallel transport from $y_0$ to another $\mathbf {Q}_p$-point $y_1$ in the same residue disk, we need to evaluate the integrals above. Suppose that $y_0,y_1$, and the coefficients of the expansions of $\boldsymbol {\omega }$ and $\eta$ are correct to $n$ digits of $p$-adic precision, and suppose that the expansions are truncated modulo $t^l$. Let

\[ \nu_1:= 1+\min_{i\ge l}\{i-\lfloor\log_p(i+1)\rfloor\}\quad\text{and}\quad \nu_2:= n+\min_{0\le i\le l-1}\{i-\lfloor\log_p(i+1)\rfloor\}. \]

Then $\int _{y_0}^{y_1}\omega _j$ and $\int _{y_0}^{y_1}(Z\boldsymbol {\omega })_j$ are correct to $\min \{\nu _1,\nu _2\}$ digits by [Reference Balakrishnan and TuitmanBT20, Proposition 4.1]. The proof of [Reference Balakrishnan and TuitmanBT20, Proposition 4.1] requires that the differential we integrate has integral coefficients. A modification of this proof yields that the integral $\int _{y_0}^{y_1}\eta$ is correct to $\min \{\nu _1', \nu _2\}$ digits, where $\nu '_1=1+\min _{i\ge l}\{i-\lfloor \log _p(i+1)\rfloor -d_i(\eta )\}.$ A similar modification shows that the double integral $\int ^{y_1} _{{y_0}}\boldsymbol {\omega }^{\intercal }Z \boldsymbol {\omega }$ is correct to $\min \{\nu _1'', \nu _2'\}$ digits, where $\nu _1''= 1+\min _{i\ge l}\{i-2\lfloor \log _p(i+1)\rfloor \}$ and

\[ \nu_2'= n-\lfloor\log_p(n)\rfloor+\min_{0\le i\le l-1}\{1-\lfloor\log_p(i+1)\rfloor\}. \]

Hence, we obtain the following lemma.

Lemma 4.4 The parallel transport matrix from ${y_0}$ to ${y_1}$ is correct to $\min \{\nu _1',\nu ''_2, \nu _2'\}$ digits of precision.

Using (4.18), we can finally bound the loss of precision in the computation of $\lambda ^\phi (x)$ for fixed points $x\in X(\mathbf {Q}_p)\cap ]\mathcal {U}[$ by combining Lemma 4.4 and Proposition 4.1.

4.3 Global heights

We now discuss the possible precision loss in the computation of the global height $h$. In step (vi(b)) of Algorithm 3.12 we solve for $d_1,\ldots,d_g$ such that

(4.22)\begin{equation} h =\sum_{i}d_i\Psi_i \end{equation}

in terms of a basis $\{\Psi _i\}$ of bilinear pairings on $\mathrm {H}^0(X_{\mathbf {Q}_p}, \Omega ^1)^{\vee }$ by evaluating $h$ and the $\Psi _i$. Recall that our method for determining the coefficients depends on whether there are sufficiently many rational points on $X$ in the sense of § 3.3. If this is the case, meaning that we can use a basis consisting of $\pi (\mathrm {A}_Z(b,x))$ for rational points $x\in X(\mathbf {Q})\cap ]\mathcal {U}[$, then we need to compute $h_p(\mathrm {A}_Z(b,z)$ and $\pi (\mathrm {A}_Z(b,x))$, and then apply simple linear algebra. The precision loss in the computation $h_p(\mathrm {A}_Z(b,x))$ has already been bounded and $\pi (\mathrm {A}_Z(b,x))$ can be obtained directly from the same data (see [Reference Balakrishnan, Best, Bianchi, Lawrence, Müller, Triantafillou and VonkBBB⁺21, Equation (41)]). The loss of precision in the linear algebra computations is easy to detect in practice, so we do not bound it explicitly here.

In the other case, the basis $\Psi _i$ is given in terms of products of abelian integrals. As mentioned above, the loss of precision in their computation is estimated in [Reference Balakrishnan and TuitmanBT20]. It remains to discuss precision loss in the computation of Coleman–Gross local heights $h_p(D_1,D_2)$, where $D_1, D_2$ are divisors in $\mathrm {Div}^0(X)(\mathbf {Q}_p)$ for $X$ a hyperelliptic curve subject to the hypotheses of Algorithm 3.4 (see [Reference Balakrishnan and BesserBB12, § 6.2] for further details). Choosing $\omega$ in step (1) can be done up to the precision of the points in the support of the divisor $D_1.$ To compute $\Psi (\omega )$ and $\omega _{D_1}$ to $O(p^n)$ in step (2) (see § 5.2 and § 6.2.3 of [Reference Balakrishnan and BesserBB12]): one needs to compute the local coordinates $(x(t),y(t))$ at infinity, with $x(t)$ to precision $t^{2(2g-1)}$ and $y(t)$ to precision $t^{2g-1}$, where these $t$-adic estimates are made based on the maximal pole order in the basis of $H^1_{\scriptstyle \mathrm {dR}}(X)$. Step (4) proceeds similarly to this step as well.

In step (5), the tiny integrals are computed as in [Reference Balakrishnan and BesserBB12, § 6]. In previous steps, we wrote $\Psi (\alpha )$ and $\Psi (\beta )$ as $\mathbf {Q}_p$-linear combinations of the basis elements of $H_{\scriptstyle \mathrm {dR}}^1(X)$, up to precision $O(p^n)$. Note that the hypothesis that $p> 2g-1$ is to ensure that the cup product matrix has entries that are $p$-integral, so no precision loss comes from the cup product matrix.

Finally, for $\sum _{A \in X(\mathbf {C}_p)}\mathrm {Res}_{A}\big (\alpha \int \beta \big )$, we consider the cases of $A$ a non-Weierstrass point (where we describe the computation in the annulus of $A$) versus $A$ Weierstrass (where we have just one contribution, at the Weierstrass point). If $A \neq (0,0)$ is a Weierstrass point, we compute the local coordinate $(x(t),y(t))$ at $A$ to precision $t^{2pn-p-1}$ (see the corrected Proposition 6.5 in [Reference Balakrishnan and BesserBB21]) so that $\mathrm {Res}_A(\alpha \int \beta )$ is computed to $n$ digits of $p$-adic precision.

Now we consider the non-Weierstrass poles of $\alpha$. For the annulus of a non-Weierstrass pole $A$, the generic situation is handled by [Reference Balakrishnan and BesserBB12, Corollary 6.4]. By [Reference Balakrishnan and BesserBB12, Remark 4.10], we consider all $A \in \{P_i, Q_j\}_{i,j}$ where $x(P_i)$ corresponds to a root of an irreducible factor of $x^p - x(P)$ (and similarly where $x(Q_j)$ corresponds to a root of an irreducible factor of $x^p - x(Q)$). For these $i,j$, we compute $\int _P^{P_i}\beta$ and $\int _Q^{Q_j} \beta$ and trace down to $\mathbf {Q}_p$. We suppose $P \in X(\mathbf {Q}_p)$ has precision $O(p^n)$. Fix $m$ and suppose $\beta$ is computed to $t^{d_i m}$ at $P_i$, where $d_i = [\mathbf {Q}_p(P_i): \mathbf {Q}_p]$. Let $\pi _i$ be a uniformiser of $\mathbf {Q}_p(P_i)$. Note that $P$ is known to $d_i n$ $\pi _i$-adic digits, and suppose that $P_i$ is known to $n_i$ $\pi _i$-adic digits. Then the $\pi _i$-adic precision of $\int _P^{P_i} \beta$ is at least $\min \{n_i, d_i n, \lfloor d_i m + 1\rfloor - \log _p(d_i m + 1)\}.$ We similarly repeat this for $Q$ and the corresponding $Q_j$. Hence, $\sum _A\mathrm {Res}_A(\alpha \int \beta )$, where the sum is over all non-Weierstrass poles $A$ of $\alpha$, is correct to $p$-adic precision

\[ \min_{i,j} \{n_i, d_i n, \lfloor d_i m + 1\rfloor - \log_p(d_i m + 1), n_j, d_j n, \lfloor d_j m + 1\rfloor - \log_p(d_j m + 1)\}, \]

where we consider the corresponding $i,j$ for all $P_i$ and all $Q_j$.

4.4 Coefficients of the quadratic Chabauty function and root finding

The previous results of this section bound the loss of precision in the computation of the quadratic Chabauty function $\rho = h-h_p$. Let $D\subset X(\mathbf {Q}_p)\cap ]\mathcal {U}[$ be a residue disk and let $x_0$ be the Teichmüller point in $D$. We now bound the valuations of the coefficients of the expansion of $\rho$ in $D$ and show how to provably compute its roots to desired precision.

In our algorithm, we fix a point $x_1\in D$, and we compute the Frobenius-equivariant splitting $\lambda ^\phi (x)$ on $D$ as a power series in a local coordinate $t$ in $x_1$ by first computing $\lambda ^\phi (x_1)$ from $\lambda ^\phi (x_0)$ and then multiplying this by the parallel transport matrix from $x_1$ to $x$. To bound the valuations of the coefficients of the entries of $\lambda ^\phi (x)$, we first compute

\[ c_1:= \mathrm{ord}_p(\lambda^\phi(x_1)) \]

using Lemma 4.4. By the above, we find that the $i$th coefficient of every entry of the expansion of $\lambda ^\phi (x)$ has valuation at least $\varphi (i) + c_1$. We use this to bound the valuations of the coefficients of the local height $h_p$. Recall from § 3.3 that we use a height with respect to an $\mathrm {End}(J)$-equivariant splitting of the Hodge filtration; let $v_{\mathrm {spl}}$ be the smallest valuation of the coefficients of this splitting in terms of our basis $\boldsymbol {\omega }$. We denote by $\mathrm {ord}_p(\gamma _{\mathrm {Fil}})$ the smallest valuation in the coefficients of the rational function $\gamma _{\mathrm {Fil}}$, and we set

\[ c_2 := \min\{0,v_{\mathrm{spl}}, \mathrm{ord}_p(\beta_{\mathrm{Fil}}),v_{\mathrm{spl}} +\mathrm{ord}_p(\beta_{\scriptstyle \mathrm{ Fil}}) \}. \]

Lemma 4.5 Let

\[ h_p(x(t)) = \sum_{i\ge 0} h_it^i \]

be the expansion of $h_p$ on the residue disk $D$ in the local parameter $t$. Then we have

(4.23)\begin{equation} \mathrm{ord}_p(h_i) \ge \min\{\mathrm{ord}_p(\gamma_{\mathrm{Fil}}), \varphi(i) +c_2\}. \end{equation}

We set $c_3:= \min _i\{\mathrm {ord}_p(d_i)\}$, where the $d_i$ are the coefficients in (4.22). Let $i_0\ge 0$ be such that

\[ -\lfloor\log_p(i)\rfloor \le \min\left\{d_{i}(\eta), \left\lfloor \frac{\mathrm{ord}_p(\beta_{\mathrm{Fil}})}{2}\right\rfloor, \left\lfloor \frac{\mathrm{ord}_p(\gamma_{\mathrm{Fil}})-c_2}{2}\right\rfloor \right\} \]

for all $i\ge i_0$. Then we have $\varphi (i) = -2\lfloor \log _p(i)\rfloor $ for all $i\ge i_0$. This proves the following proposition.

Proposition 4.6 Let

\[ \rho(t) =\sum_{i\ge 0} \rho_it^i \]

be the expansion of the quadratic Chabauty function $\rho =h-h_p$ on $D$. If $i\ge i_0$, then we have

\[ \mathrm{ord}_p(\rho_i) \ge -2\lfloor\log_p(i)\rfloor + c_1+ \min\{c_2, c_3\}. \]

Together with Proposition 4.6, the following result allows us to provably determine the roots of $\rho$ to any desired precision.

Lemma 4.7 Suppose $F(x)=\sum _{i\geq 0}F_i x^i \in \mathbf {Q} _p [\! [x]\! ]$ is such that there are integers $k,m,n$ satisfying

\[ \min \{\mathrm{ord}_p (F_i )+i:i\geq 0\} =k \]

and

\[ \max \{i\geq 0:\mathrm{ord}_p (F_i )+i=n \} < m, \]

and furthermore that $F$ has at most $d$ roots in the closed disk $\{ \mathrm {ord}_p (x)\geq 1\}$. Then the roots of $F$ in the ball $\{\mathrm {ord}_p (x)\geq 1\}$ can be determined, with multiplicity, to precision $(n-k)/d$, by computing $F_0 ,\ldots,F_{m-1}$ modulo $p^n$.

5.1 The exceptional curve $X_{S_4}(13)$

Recall that for a prime $\ell \geq 5$, any proper subgroup of $\mathrm {GL}_2(\mathbf {F}_{\ell })$ is conjugate to a subgroup of a Borel subgroup, the normaliser of a Cartan subgroup, or an ‘exceptional’ subgroup with projective image isomorphic to $S_4, A_4$, or $A_5$. The field of definition of the modular curves attached to the exceptional subgroups is the unique quadratic subfield $\mathbf {Q}(\sqrt {\pm \ell })$ of the cyclotomic field $\mathbf {Q}(\zeta _{\ell })$, with the exception of the curves $X_{S_4}(\ell )$ for $\ell \equiv \pm 3 \pmod {8}$, which are defined over $\mathbf {Q}$. For such values of $\ell$, we would therefore like to determine $X_{S_4}(\ell )(\mathbf {Q})$.

Serre [Reference SerreSer72] shows by a monodromy argument that such tetrahedral modular curves have no points defined over $\mathbf {Q}_{\ell }$ when $\ell$ is large enough, and in particular he obtains

\[ X_{S_4}(\ell)(\mathbf{Q}) = \varnothing, \quad \text{if} \ \ell > 13. \]

The curves $X_{S_4}(3)$ and $X_{S_4}(5)$ are both of genus zero, and contain a unique rational cusp. Ligozat [Reference LigozatLig77] showed that $X_{S_4}(11)$ is an elliptic curve of conductor $11^2$ whose Mordell–Weil group is trivial, where the unique rational point is CM, corresponding to discriminant $D=-3$. This leaves only the curve $X_{S_4}(13)$, which has genus $3$. In fact, this curve is the last remaining modular curve of level $13^n$ whose rational points have not been determined.

Using modular symbols algorithms, Banwait and Cremona [Reference Banwait and CremonaBC14] show that the curve $X_{S_4}(13)$ is a smooth plane quartic whose canonical model is given by

\begin{align*} &4x^3y - 3x^2y^2 + 3xy^3 - x^3z + 16x^2yz - 11xy^2z \\ &\quad +5y^3z + 3x^2z^2 + 9xyz^2 + y^2z^2 + xz^3 + 2yz^3 = 0. \end{align*}

Furthermore, they exhibit the four rational points

\[ \{ \big(1 : 3 : -2\big), \big(0 : 0 : 1\big), \big(0 : 1 : 0\big),\big(1 : 0 : 0\big)\} \subseteq X_{S_4}(13)(\mathbf{Q}), \]

where the rational point $(0 : 0 : 1)$ corresponds to an elliptic curve with CM by the order of discriminant $D = -3$, and the three other rational points correspond to non-CM elliptic curves over $\mathbf {Q}$ with projective mod $13$ image equal to $S_4$, whose $j$-invariants are given by

\begin{gather*} j = \frac{2^4\cdot 5 \cdot 13^4 \cdot 17^3}{3^{13}}, \quad j = - \frac{2^{12}\cdot 5^3 \cdot 11 \cdot 13^4}{3^{13}}, \\ j = \frac{2^{18}\cdot 3^3 \cdot 13^4\cdot 127^3 \cdot 139^3 \cdot 157^3 \cdot 283^3 \cdot 929}{5^{13} \cdot 61^{13}}. \end{gather*}

The Jacobian of $X_{S_4}(13)$ is isogenous to that of $X_{\rm s}^+(13)$, so it is absolutely simple and has Mordell–Weil rank 3 over $\mathbf {Q}$ by the results of [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19, § 6]. The curve has potential good reduction at $p = 13$, as can be seen, for instance, using the Sage toolbox MCLF.

We determine the set of rational points on the curve $X_{S_4}(13)$ using quadratic Chabauty with $p=11$ for the affine patches

\begin{gather*} y^4 + (18x + 9)y^3 + (160x^2 + 176x + 52)y^2 + (560x^3 + 832x^2 + 384x + 48)y\\ + 192x^4 + 512x^3 + 384x^2 + 64x = 0 \end{gather*}

and

\begin{gather*} y^4 + (9x + 9)y^3 + (52x^2 + 72x + 36)y^2 + (48x^3 + 240x^2 + 208x + 64)y \\ + 64x^3 + 192x^2 - 64x = 0. \end{gather*}

The computation is analogous to the computation of $X^+_{\rm s}(13)(\mathbf {Q})$ in [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM⁺19]. The Hecke operator $T_{11}$ generates the Hecke algebra, as can be verified, for instance, by checking the analogous statement for $X^+_{\rm s}(13)$. Hence, we may construct suitable cycles $Z_1, Z_2$ from $T_{11}$ and its square, respectively. The set of common zeros of the resulting quadratic Chabauty functions consists precisely of the known rational points, so we obtain Theorem 1.1.

In order to solve for the height pairing, we use the four known rational points and the cycle $Z_1$, so the resulting function automatically vanishes there. However, since the cycles $Z_1$ and $Z_2$ are independent, and $Z_2$ is not used to solve for the height, the vanishing of the resulting function in the rational points provides a check for the correctness of our code.

5.2 The Atkin–Lehner quotients $X_0^+(N)$

For a positive integer $N$, consider the Atkin–Lehner involution $w_N$ acting on the modular curve $X_0(N)$. Then the quotient

\[ X_0^+(N ) := X_0(N)/\langle w_N \rangle \]

is a smooth projective curve defined over $\mathbf {Q}$ whose non-cuspidal points classify unordered pairs $\{E_1,E_2\}$ of elliptic curves admitting an $N$-isogeny between them. The study of rational points on these curves is also important in an ongoing research programme aiming to compute quadratic points on the modular curves $X_0(N)$ (see, for instance, recent work of Box [Reference BoxBox21]). Among the rational points, we distinguish between cusps, CM points and exceptional points, those which are neither cusps nor CM points. The exceptional points correspond to quadratic $\mathbf {Q}$-curves without CM.

In this subsection we restrict to prime values $N$ such that $X_0^+(N)$ has genus 2 or 3. Galbraith [Reference GalbraithGal96] has computed models for all these curves (and many more) by finding relations in the vector space spanned by the newforms of level $N$ and weight $2$ that are invariant under $w_N$. Up to conjugation, there is a unique such newform.

By work of Ogg, for prime level $N$, the curve $X_0^+(N)$ has genus 2 if and only if

(5.1)\begin{equation} N \in \{67, 73, 103, 107, 167, 191 \}. \end{equation}

It has genus 3 if and only if

(5.2)\begin{equation} N \in \{97, 109, 113, 127, 139, 149, 151, 179, 239\}. \end{equation}

Models for all these curves were communicated to us by Elkies; one can also find such models in Galbraith's thesis [Reference GalbraithGal96] or by using the Magma command X0NQuotient.

Via a search for small rational points, Galbraith [Reference GalbraithGal99] found exceptional rational points on $X_0^+(N)$ for $N=73,\ 91,\ 103,\ 191$ (genus 2) and $N=137,\ 311$ (genus 4). The latter examples disproved an earlier conjecture of Elkies that there are no exceptional rational points on non-hyperelliptic $X_0^+(N)$ for prime level $N$. In [Reference GalbraithGal02], Galbraith also finds an exceptional point on $X_0^+(125)$ and conjectures that there are no further exceptional points on modular curves $X^+_0(N)$ of genus $2\le g\le 5$.

Together with [Reference Balakrishnan, Best, Bianchi, Lawrence, Müller, Triantafillou and VonkBBB⁺21] and [Reference GalbraithGal96], our computations described below prove Theorem 1.3. We first check that for level $N$ as in (5.1) and (5.2) the curves $X_0^+(N)$ satisfy the requirements to apply our algorithm. The Jacobian $J_0^+(N)$ of $X_0^+(N)$ has real multiplication over $\mathbf {Q}$, so the Picard number is at least $g$. Using Magma, we computed the $L$-function of the corresponding newforms to show that the analytic rank is $g$, so the work of Gross and Zagier and of Kolyvagin and Logachev proves that the rank of $J(\mathbf {Q})$ is exactly $g$. For the genus 2 examples, we also applied $2$-descent on $J_0^+(N)$, as implemented in Magma, as an independent check.