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# A semi-numerical algorithm for
the homology lattice and periods of complex elliptic surfaces over
$\mathbb{P}^{1}$

Eric Pichon-Pharabod

###### Abstract

We provide an algorithm for computing an effective basis of homology of
elliptic surfaces over $\mathbb{P}_{\mathbb{C}}^{1}$ on which integration of
periods can be carried out. This allows the heuristic recovery of several
algebraic invariants of the surface, notably the Néron-Severi lattice, the
transcendental lattice, the Mordell-Weil group and the Mordell-Weil lattice.
This algorithm comes with a SageMath implementation.

###### Contents

  1. 1 Introduction
  2. 2 Elliptic surfaces
  3. 3 Homology and periods of elliptic surfaces
  4. 4 Application: Mordell-Weil group and lattice

##  1 Introduction

An elliptic surface is a surface $S$ together with a map $f:S\to C$ to a
curve, such that the generic fibre is an elliptic curve. Such surfaces benefit
from a rich structure, bridging the elegance of elliptic curves with
properties of higher-dimensional varieties. As such, they exhibit a weave of
algebraic and topological phenomena.

The set of rational points of an elliptic curve benefits from a group
structure, called the Mordell-Weil group. This group’s structure and
properties have far-reaching implications in various fields, such as algebraic
number theory with the modular theorem Wiles1995; cryptography with Elliptic
Curve Cryptography Miller1986, Koblitz1987; or the study of Diophantine
equation Lang1978. The Mordell-Weil theorem, a central result in this context,
establishes that this group is finitely generated Mordell1922, Weil1928 in the
case of elliptic curves over a number field. Determining the Mordell-Weil
group, and in particular determining its rank, however, remains a notoriously
difficult problem hindry2007.

Viewing elliptic surfaces as elliptic curves over the function field of the
base space reveals a map between rational points of the curve and sections of
the fibration. In this context the Mordell-Weil group gains additional
structure, induced from the lattice structure of the second homology group of
the surface. Furthermore in this setting, Shioda provided an isomorphism
between the Mordell-Weil group of $S$ and a group derived from the Néron-
Severi lattice of the surface Shioda1990. This isomorphism not only simplifies
the computation of the group’s rank, yielding the Shioda-Tate formula, but
also imparts a lattice structure to the Mordell-Weil group.

In light of these observations, we provide in this paper a semi-numerical
algorithm for efficiently computing the homology of elliptic surfaces, as well
as high precision numerical approximations of its holomorphic periods. In
turn, we obtain a heuristic method for computing their Mordell-Weil group and
lattice. All in all, this yields practical computational methods to fields
where elliptic surfaces naturally arise, such as the study of Feynman
integrals BlochKerrVanhove2018, Bonishetal2021, doran2023motivic and mirror
symmetry CoxKatz1999, Horietal2003.

### Contributions

We provide an algorithm for computing the full homology lattice, with its
intersection product, of an elliptic surface $S$ given its defining equation
as a polynomial $P_{t}\in\mathbb{Q}[t,X,Y,Z]$. When $S$ is not isotrivial, the
algorithm also provides:

  * 1.

the holomorphic period map $H^{2,0}(S)\times H_{2}(S)\to\mathbb{C}$ given
numerically with certified precision bounds in quasilinear time with respect
to precision;

  * 2.

an embedding of the Néron-Severi lattice in $H_{2}(S)$, obtained heuristically
as the kernel of the holomorphic period map;

  * 3.

the Mordell-Weil group, with the lattice structure of its torsion free part,
also depending on the heuristic of the Néron-Severi lattice.

By “heuristically”, we mean that it is possible that the algorithm misses
elements of the kernel or finds fake ones, in the case of numerical
coincidence – we provide more detail at the end of Section 4.1. The algorithm
presented here is implemented in Sagemath, as part of the package _lefschetz-
family_ 111https://gitlab.inria.fr/epichonp/lefschetz-family.

### Previous works

Periods of algebraic curves are well studied as they coincide with Riemann
surfaces
DeconinckVanHoeij2001,Swierczewski2017,BruinSijslingZotine2019,MolinNeurohr2019,Neurohr2018.
Computation of the periods of some varieties realised as double cover ramified
along hyperplane arrangements have been carried out in two distinct cases: in
CynkvanStraten2019 in the case of Calabi-Yau manifolds given as double covers
of $\mathbb{P}^{3}$ ramified along $8$ planes; and in ElsenhansJahnel2022 in
the case of K3 surfaces given as a double cover of $\mathbb{P}^{2}$ ramified
along $6$ lines.

A method for arbitrary dimensions was first given by Sertoz2018 in the case of
hypersurfaces. A computationally cheaper and more general approach was
developed in LairezPichonVanhove23. The work presented in this paper is an
improvement on the methods of LairezPichonVanhove23, adapted to the case of
elliptic surfaces.

### Outline

We consider elliptic surfaces with section. In the case where the elliptic
surface only has simple singular fibres – i.e. singular fibres of type $I_{1}$
– semi-numerical methods for computing the homology and periods were developed
in LairezPichonVanhove23. One may always reduce to this case by deforming the
elliptic surface in a way that separates its singular fibres into $I_{1}$
fibres, thus obtaining a Lefschetz fibration. By continuity, the homology of
this deformation is the same as that of the initial surface. As the monodromy
representation of the deformation only depends on the type of the singular
fibres, this can be done formally, without having to consider an explicit
realisation of such a morsification.

This provides a way to compute an effective basis of the full homology lattice
of an elliptic surface. Certain cycles can be expressed as _extensions_ , i.e.
lifts of paths in $\mathbb{P}^{1}$ avoiding the critical values to
$H_{1}(S_{t})$. The lattice generated by these cycles and the fibre components
is a full rank sub-lattice of “primary” cycles $\operatorname{Prim}(S)\subset
H_{2}(S)$. Numerical approximations of the holomorphic periods of primary
cycles can be computed through numerical integration. We may thus recover a
numerical approximation of the full holomorphic period mapping
$H^{2,0}(S)\times H_{2}(S)\to\mathbb{C}$. In turn, we may use the LLL
algorithm to heuristically recover the Néron-Severi group in our basis of
homology. This allows the computation of several algebraic invariants of the
fibration, notably the Mordell-Weil group, and the lattice associated to its
torsion-free part.

##  2 Elliptic surfaces

In this section, we recall the definition of an elliptic surface, as well as
related notions that are useful to our discussion, notably the action of
monodromy and Kodaira’s classification of singular fibres. For further reading
on elliptic surfaces, we recommend SchuttShioda10, Miranda1989.

######  Definition 1.

Let $V$ be a Riemann surface. An elliptic surface over $V$ is an algebraic
surface $S$ along with a proper surjective map $f:S\to V$ such that

  * 1.

for all but finitely many $t\in V$, the fibre $F_{t}=f^{-1}(t)$ is a smooth
genus $1$ algebraic curve (i.e. an elliptic curve);

  * 2.

no fibre contains a smooth rational curve of self-intersection $-1$.

The second condition is there to ensure that the surface is minimal, as such
rational curves can always be blown down. We denote by $\Sigma$ the finite set
of values $t\in V$ over which the fibre $F_{t}$ is not a smooth elliptic
curve.

In the following, we consider an elliptic surface $f:S\to\mathbb{P}^{1}$ over
$V=\mathbb{P}^{1}$.

######  Definition 2.

A section of $S$ is a map $\pi:\mathbb{P}^{1}\to S$ such that
$f\circ\pi=\operatorname{id}_{V}$.

Sections of $S$ are in bijection with $\mathbb{C}(t)$-points on the generic
fibre $E$, which is an elliptic curve over $\mathbb{C}(t)$. Indeed, given a
section $\pi$, the intersection of its image with the generic fibre
$\operatorname{im}\pi\cap E$ yields a point in $E$. Conversely, given a point
$P\in E(\mathbb{C}(t))$, its specialisation to any smooth fibre yields a point
of the fiber. The closure $\Gamma$ of the union of all these points yields a
birational isomorphism $f|_{\Gamma}:\Gamma\to\mathbb{P}^{1}$. The inverse of
this map gives the section associated to $P$, which we denote $\bar{P}$.

Throughout this paper, we will only consider elliptic surfaces with section.
We will notably fix a section $O$ of $S$, which we call the _zero section_. It
will serve as the zero of the group of rational points $E(\mathbb{C}(t))$ of
the generic elliptic curve. Furthermore, to avoid the trivial case of a
product $E\times\mathbb{P}^{1}$, we require in the rest of this text that the
elliptic surface has at least one singular fibre.

###  2.1 Monodromy and extensions

We briefly recall the notions of monodromy and extensions, following lamotke
and Section 2.1.2 of LairezPichonVanhove23. The restriction of $f$ to
$f^{-1}(\mathbb{P}^{1}\setminus\Sigma)$ is a locally trivial fibration: if
$U\subset\mathbb{P}^{1}\setminus\Sigma$ is open and simply connected, there is
a trivialisation $f^{-1}(U)\simeq F_{b}\times U$ of the fibration, for all
$b\in U$. In particular a non-self-intersecting path
$\ell:[0,1]\to\mathbb{P}^{1}\setminus\Sigma$ induces a diffeomorphism
$F_{\ell(0)}\simeq F_{\ell(1)}$ which is unique up to some automorphism of
$F_{\ell(1)}$ that is isotopic to the identity. Thus $\ell$ induces an
isomorphism $\ell_{*}:H_{1}(F_{\ell(0)})\to H_{1}(F_{\ell(1)})$. For $\ell$,
$\ell^{\prime}$ two non-intersecting such paths compatible for concatenation
that do not intersect, one may show that

$(\ell^{\prime}\ell)_{*}=\ell^{\prime}_{*}\circ\ell_{*}\,,$ (1)

(where $\ell^{\prime}\ell$ is the path that goes through $\ell$ first, then
through $\ell^{\prime}$). Using this formula, we can extend the notion of
monodromy to self intersecting paths, and to loops. Furthermore, one may show
that the map $\ell_{*}$ depends only on the homotopy class of $\ell$.

Let $b\in\mathbb{P}^{1}\setminus\Sigma$. The above construction yields a map

$\begin{cases}\pi_{1}(\mathbb{P}^{1}\setminus\Sigma,b)\to\operatorname{Aut}(H_{1}(F_{b}))\\\
[\ell]\mapsto\ell_{*}\end{cases}\,,$ (2)

where $[\ell]$ denotes the homotopy class of $\ell$. The map $\ell_{*}$ is
called the action of monodromy along $\ell$ on $H_{1}(F_{b})$.

As $F_{b}$ is an elliptic curve, its first homology group $H_{1}(F_{b})$ is a
lattice of rank $2$, equipped with a skew-symmetric intersection product. As
readily seen from the trivialisation of the fibration, monodromy preserves the
intersection product. Therefore, a simple computation shows that the matrix of
the action of monodromy in a symplectic basis of $H_{1}(F_{b})$ belongs to
$\operatorname{SL}_{2}(\mathbb{Z})$. It is possible to compute this matrix
with semi-numerical computations involving the Picard-Fuchs equation of
$F_{t}$, see [§3.5.2]LairezPichonVanhove23.

A related notion to monodromy is that of extensions. Given a non-intersecting
path $\ell$, a simply connected neighbourhood $V$ of $\operatorname{im}\ell$
and a $1$-chain $\Delta$ of $F_{\ell(0)}$, the identification of
$\Delta\times\operatorname{im}\ell$ in $f^{-1}(V)\subset S$ produces a
$2$-chain with boundary in $F_{\ell(0)}\cup F_{\ell(1)}$. Once again, the
relative homology class of this $2$-chain is unique in the relative homology
group $H_{2}(S,F_{\ell(0)}\cup F_{\ell(1)})$, and only depends on the homotopy
class of $\ell$ and the homology class of $\Delta$. Therefore we define the
extension map $\tau_{\ell}:H_{1}(F_{\ell(0)})\to H_{2}(S,F_{\ell(0)}\cup
F_{\ell(1)})$. Similarly to monodromy, extensions satisfy a composition rule
which allows to extend their definition to self-intersecting paths:

$\tau_{\ell^{\prime}\ell}(\gamma)=\tau_{\ell}(\gamma)+\tau_{\ell^{\prime}}(\ell_{*}(\gamma))\,,$
(3)

in $H_{2}(S,F_{\ell(0)}\cup F_{\ell(1)}\cup F_{\ell^{\prime}(1)})$. In
particular, when $\ell$ is a loop pointed at $b$, we obtain a map

$\tau:\begin{cases}\pi_{1}(\mathbb{P}^{1}\setminus\Sigma,b)\times
H_{1}(F_{b})\to H_{2}(S,F_{b})\\\
[\ell],\Delta\mapsto\tau_{\ell}(\Delta)\end{cases}\,.$ (4)

Extensions and monodromy are closely related by the formula

$\delta(\tau_{\ell}(\gamma))=\ell_{*}\gamma-\gamma\,,$ (5)

where $\delta:H_{2}(S,F_{b})\to H_{1}(F_{b})$ is the boundary map. This is
represented in Fig. 1(b)

###  2.2 The Kodaira classification

Kodaira provides a classification of the singular fibres of an elliptic
fibration Kodaira63: Let $\sigma\in\Sigma$ be a critical value and $\ell$ be
the counter-clockwise simple loop around $\sigma$ pointed at $b$. As stated in
the previous section, the monodromy action $\ell_{*}$ is represented by a
matrix $M\in\operatorname{SL}_{2}(\mathbb{Z})$ in a symplectic basis of
$H_{1}(F_{b})$. The $\operatorname{SL}_{2}(\mathbb{Z})$-conjugation class of
$M$ determines the type of the singular fibre.

These conjugacy classes are classified in two infinite families $I_{\nu}$ and
$I_{\nu}^{*}$, $\nu\in\mathbb{N}$ and six classes $II$, $III$, $IV$, $II^{*}$,
$III^{*}$ and $IV^{*}$. Representatives $M_{T}$ of these conjugacy classes are
given in Table 1 (which is a reproduction of [Table 1]CadavidVelez08), along
with a factorisation as a product of $I_{1}$-type monodromy matrices which
will prove useful in Section 3.2.1, and the Euler characteristic of the fibre.
These singular fibres have been extensively studied – for further reading on
this topic, we recommend [Chap. 7]Esole2017.

Type | $M_{T}$ |  Minimal normal factorisation |  Euler characteristic of the fibre
---|---|---|---
$\mathit{I}_{\nu},\nu\geq 1$ | $\left(\begin{array}[]{cc}1&\nu\\\ 0&1\end{array}\right)$ | $U^{\nu}$ | $\nu$
$\mathit{II}$ | $\left(\begin{array}[]{cc}1&1\\\ -1&0\end{array}\right)$ | $VU$ | $2$
$\mathit{III}$ | $\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right)$ | $VUV$ | $3$
$\mathit{IV}$ | $\left(\begin{array}[]{cc}0&1\\\ -1&-1\end{array}\right)$ | $(VU)^{2}$ | $4$
$\mathit{I}_{\nu}^{*},\nu\geq 0$ | $\left(\begin{array}[]{cc}-1&-\nu\\\ 0&-1\end{array}\right)$ | $U^{\nu}(VU)^{3}$ | $\nu+6$
$\mathit{II}^{*}$ | $\left(\begin{array}[]{cc}0&-1\\\ 1&1\end{array}\right)$ | $(VU)^{5}$ | $10$
$\mathit{III}^{*}$ | $\left(\begin{array}[]{cc}0&-1\\\ 1&0\end{array}\right)$ | $VUV(VU)^{3}$ | $9$
$\mathit{IV}^{*}$ | $\left(\begin{array}[]{cc}-1&-1\\\ 1&0\end{array}\right)$ | $(VU)^{4}$ | $8$
Table 1: The singular fibre types of the Kodaira classification,
representatives of their $\operatorname{SL}_{2}(\mathbb{Z})$ conjugacy class
of the monodromy matrix, and the minimal normal factorisation of this
representative in terms of the $I_{1}$-type matrices
$U=\left(\begin{smallmatrix}1&1\\\ 0&1\end{smallmatrix}\right)$ and
$V=\left(\begin{smallmatrix}1&0\\\ -1&1\end{smallmatrix}\right)$.

##  3 Homology and periods of elliptic surfaces

In this section, we discuss means to recover the full homology lattice from
the knowledge of the monodromy matrices of an elliptic fibration. Let us first
fix some notations. Let $f:S\to\mathbb{P}^{1}$ be an elliptic fibration.

  * 1.

$F_{v}$ denotes the fibre above $v\in\mathbb{P}^{1}$. We will also use this
notation for the homology class of $F_{v}$ in $H_{2}(S)$ when there is no
ambiguity;

  * 2.

$O$ denotes the zero section of $S$;

  * 3.

$c_{1},\dots,c_{r}\in\mathbb{P}^{1}$ denote the critical values of $f$, and
$\Sigma=\\{c_{1},\dots,c_{r}\\}$;

  * 4.

$m_{v}$ denotes the number of irreducible components of the fibre $F_{v}$;

  * 5.

$\Theta_{0}^{v}$ denotes the zero component of $F_{v}$, i.e. the irreducible
component intersecting the zero section.

  * 6.

$\Theta_{1}^{v},\dots,\Theta_{m_{v}-1}^{v}$ denote the irreducible singular
components of $F_{v}$ that are orthogonal to $O$;

  * 7.

$\mathcal{T}\subset H_{2}(S,F_{b})$ denotes the image of extension maps. For a
basis $\ell_{1},\dots,\ell_{r-1}$ of
${\pi_{1}(\mathbb{P}^{1}\setminus\Sigma,b)}$, we have
$\mathcal{T}=\bigoplus_{i=1}^{r-1}\operatorname{im}\tau_{\ell_{i}}$ as a
direct consequence of (3);

We begin by a simple lemma connecting the homology groups $H_{2}(S)$ and
$H_{2}(S,F_{b})$.

######  Lemma 1.

Let $\delta:H_{2}(S,F_{b})\to H_{1}(F_{b})$ be the boundary map. We have

$H_{2}(S)/\langle F_{b}\rangle\simeq\ker\delta\,.$ (6)

###### Proof.

This is a direct consequence of the long exact sequence of relative homology
of the pair $(S,F_{b})$:

$\langle F_{b}\rangle=H_{2}(F_{b})\to H_{2}(S)\to
H_{2}(S,F_{b})\to^{\delta}H_{1}(F_{b})$ (7)

∎

Several cycles in $H_{2}(S)$ are distinguished in that their associated
holomorphic periods (that is, periods of holomorphic forms) are directly
computable, see Section 4.1 below. In the next paragraph, we define a lattice
$\operatorname{Prim}(S)\subset H_{2}(S)$ of such cycles. When the fibration is
Lefschetz, $\operatorname{Prim}(S)$ coincides with the full homology lattice
$H_{2}(S)$. In general, however, $\operatorname{Prim}(S)$ may be a proper
sublattice of $H_{2}(S)$. Nevertheless, we will show in Section 4.1 that
$\operatorname{Prim}(S)$ always has full rank. In particular, all the periods
of $S$ can be recovered from the periods of $\operatorname{Prim}(S)$.

More precisely, the periods associated to singular components and the section
are $0$. Furthermore, LairezPichonVanhove23 provides a way to compute the
periods of extensions. We call such cycles primary and define
$\operatorname{Prim}(S)$ as follows:

######  Definition 3.

The primary lattice $\operatorname{Prim}(S)$ is the sublattice of $H_{2}(S)$
generated by extensions, fibre components and the zero section:

$\operatorname{Prim}(S)=\phi(\mathcal{T}\cap\ker\delta)\oplus\langle
O,F_{b}\rangle\oplus\bigoplus_{v\in\Sigma}\bigoplus_{i=1}^{m_{v}-1}\Theta^{v}_{i}\,,$
(8)

where $\phi:\ker\delta\to H_{2}(S)$ is a lift of the quotient of Lemma 1. Note
that the choice of this lift does not matter.

######  Remark 2.

Although not apparent in the notation, $\operatorname{Prim}(S)$ depends on the
fibration of the surface, and not solely on $S$.

###  3.1 The Lefschetz case

When all the singular fibres are of type $I_{1}$, $f$ is said to be a
Lefschetz fibration of $S$. In this setting, LairezPichonVanhove23 provides an
algorithm for computing an effective basis of $H_{2}(S)$ from the list of the
monodromy matrices of a certain basis of the homotopy group
$\pi_{1}(\mathbb{C}\setminus\Sigma)$ (where
$\mathbb{C}\simeq\mathbb{P}^{1}\setminus\\{\infty\\}$ with $\infty$ a regular
value). In particular, we have the following theorem.

######  Theorem 3.

When $f:S\to\mathbb{P}^{1}$ is a Lefschetz fibration,
$\operatorname{Prim}(S)=H_{2}(S)$.

In order to prove this theorem, we first recall the elements of the
construction in LairezPichonVanhove23 that are relevant to our discussion. Let
$V\subset\mathbb{P}^{1}$ be diffeomorphic to a disk, let $S^{*}=f^{-1}(V)$,
let $\Sigma_{V}=\Sigma\cap V$, $s=|\Sigma_{V}|$ and pick $b\in
V\setminus\Sigma_{V}$.

figs/homotopy_basis/homotopy_basis

(a) A distinguished basis. The composition $\ell_{1},\dots,\ell_{r}$ is
represented by the loop encircling all the critical values once
counterclockwise.

figs/thimbles/thimbles

(b) The thimble $\Delta_{i}\in H_{2}(S,F_{b})$ above a critical values $c_{i}$
is the nontrivial extension of a $1$-cycle $\gamma\in H_{1}(F_{b})$ along a
loop $\ell_{i}$ around a unique critical value $c_{i}$. For a given such loop
$\ell_{i}$, there is a unique thimble (up to sign). Its boundary is
$\delta\Delta_{i}={\ell_{i}}_{*}\gamma-\gamma$.

Figure 1: Distinguished bases and thimbles.

######  Definition 4.

A basis $\ell_{1},\dots,\ell_{s}$ of $\pi_{1}(V\setminus\Sigma_{V},b)$ is
distinguished if for every $i$, $\ell_{i}$ is isotopic to a simple (i.e. with
winding number $1$) counterclockwise loop around a critical value of
$\Sigma_{V}$, and the composition $\ell_{r}\cdots\ell_{1}$ is isotopic to a
simple counterclockwise loop around all the points of $\Sigma_{V}$. This is
represented in Fig. 1(a).

If $\ell_{1},\dots,\ell_{s}$ is a distinguished basis of
$\pi_{1}(V\setminus\Sigma,b)$, then for each $i$,
$\operatorname{im}\tau_{\ell_{i}}\subset H_{2}(S^{*},F_{b})$ has rank $1$ –
its generator (up to sign) is called the thimble above $c_{i}$, denoted
$\Delta_{i}$. The border of $\Delta_{i}$ is called the vanishing cycle at
$c_{i}$ and is a generator of the image of ${\ell_{i}}_{*}-\operatorname{id}$,
as can be readily seen from (5). An illustration of a thimble is given in Fig.
1(b).

Thimbles serve as building blocks for $H_{2}(S^{*})$, as the following lemma
demonstrates.

######  Lemma 4 ([Main lemma]lamotke).

$H_{2}(S^{*},F_{b})=\bigoplus_{i=1}^{s}\mathbb{Z}\Delta_{i}\hskip
10.00002pt\text{ and }\hskip 10.00002ptH_{1}(S^{*},F_{b})=0\,.$ (9)

In particular when $V=\mathbb{P}^{1}\setminus\\{\infty\\}$ for a regular point
$\infty\notin\Sigma$, the following lemmas shows how to recover $H_{2}(S)$
from $H_{2}(S^{*},F_{b})$.

######  Lemma 5.

$\ker\left(\iota:H_{2}(S^{*},F_{b})\to
H_{2}(S,F_{b})\right)=\operatorname{im}\tau_{\infty}$, where
$\tau_{\infty}=\tau_{\ell_{r}\cdots\ell_{1}}$. In other words, an element of
$H_{2}(S^{*},F_{b})$ is trivial in $H_{2}(S,F_{b})$ if and only if it is a sum
of extensions along isotopy classes of $\pi_{1}(V\setminus\Sigma,b)$ that are
trivial in $\pi_{1}(\mathbb{P}^{1}\setminus\Sigma,b)$.

###### Proof.

This is the second line of diagram (16) in LairezPichonVanhove23. ∎

######  Lemma 6.

There is a split short exact sequence

$0\to\mathcal{T}\to H_{2}(S,F_{b})\to H_{0}(F_{b})\to 0\,,$ (10)

where the first map is the inclusion and the second map is the intersection
with the generic fibre $F_{b}$. In other words,
$H_{2}(S,F_{b})=\mathcal{T}\oplus\langle O\rangle$.

###### Proof.

We have the long exact sequence of the triplet $(S,S^{*},F_{b})$:

$H_{2}(S^{*},F_{b})\to H_{2}(S,F_{b})\to H_{2}(S,S^{*})\to
H_{1}(S^{*},F_{b})\,.$ (11)

It follows from Lemma 4 that the image of the first map is $\mathcal{T}$. As
$(S,S^{*})\simeq F_{b}\times(D,S^{1})$, the Künneth formula yields

$H_{2}(S,S^{*})\simeq H_{0}(F_{b})\,,$ (12)

where the identification is the intersection product with the generic fibre
$F_{b}$. Finally, from Lemma 4, $H_{1}(S^{*},F_{b})=0$. The sequence splits
because $H_{0}(F_{b})$ is free. ∎

###### Proof of Proposition 3.

The proposition follows from Lemma 6 and Lemma 1. ∎

###  3.2 Morsification

figs/morsification_2/morsification_2

figs/morsification/morsification

Figure 2: The morsification of a neighbourhood of a single critical value.
Left: A neighbourhood $V$ of a single critical value of the elliptic fibration
$f$, along with a chosen basepoint $b$. The homotopy group
$\pi_{1}(V\setminus\\{c\\},b)$ is generated by the counterclockwise loop
$\ell_{\infty}$. Right: The neighbourhood after morsification. The critical
fibre has split into five Lefschetz fibres. The homotopy group
$\pi_{1}(V\setminus\Sigma)$ is generated by the $5$ counterclockwise loops
$\ell_{1},\dots,\ell_{5}$. Thus $H_{2}(f^{-1}(V),F_{b})$ has rank $5$ – it
follows from Table 1 that the original singular fibre was of type $I_{5}$.
Furthermore we see that $\tau_{\ell_{\infty}}=\tau_{\ell_{5}\dots\ell_{1}}$.

We have seen in the previous section that the data of the monodromy matrices
of the elliptic surface is sufficient to recover the homology in the Lefschetz
case. In this section we extend this result to general elliptic surfaces.

In a nutshell, it is possible to deform the elliptic surface in a way that
splits the non-Lefschetz singular fibres into several Lefschetz fibres.
Therefore it is possible to compute the homology of the deformed elliptic
surface from its monodromy matrices, which is by construction diffeomorphic to
the original one. Interestingly, the monodromy matrices of the Lefschetz
fibres of the deformation are determined by the monodromy matrices of the
initial surface. In particular, this implies that we do not have to do any
computations with (or even find) an explicit realisation of such a
deformation. Instead it only serves as a formal computational tool.

More precisely, let $V\subset\mathbb{P}^{1}$ be open and
$D=\\{z\in\mathbb{C}\mid|z|\leq 1\\}$ denote the unit disk.

######  Definition 5.

Let $S\to V$ be an elliptic surface. Consider a commutative diagram

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{22.57442pt}{-1.47363pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,,$ (13)

where $p$ is the projection onto the second coordinate. For $u\in D$, denote
$S_{u}=\eta^{-1}(u)$ and $V_{u}=p^{-1}(u)$. Such a diagram is a morsification
of $S\to V$ if

  * 1.

$\eta:\tilde{S}\to D$ is a locally trivial smooth fibration;

  * 2.

$\tilde{f}|_{S_{0}}:S_{0}\to V_{0}$ coincides with $S\to V$;

  * 3.

for $u\in D\setminus\\{0\\}$, $\tilde{f}|_{S_{u}}:S_{u}\to V_{u}$ is a
Lefschetz elliptic surface;

  * 4.

$\eta$ has no critical values.

######  Remark 7.

Such a deformation is sometimes also called a splitting deformation of the
elliptic surface.

Morsifications of elliptic surfaces are useful as they allow use of the
results for Lefschetz fibrations to obtain information about general elliptic
surfaces. Indeed as $\eta$ is a trivial fibration, it induces an isometry
$H_{2}(S_{0})\simeq H_{2}(S_{u})$ for all $u\in D\setminus\\{0\\}$, and we may
apply the results of Section 3.1 to the describe the latter. The existence of
a morsification of any elliptic surface is guaranteed by the following theorem
of Moishezon77.

######  Theorem 8 ([Thm. 8]Moishezon77).

Let $S\to V$ be an elliptic surface. There exists a morsification of $S$.
Moreover, the number of singular fibres of $S_{u}$ for $u\in
D\setminus\\{0\\}$ does not depend on $u$.

######  Remark 9.

In our setting, the existence of a section prevents the possibility of
multiple fibres mentioned in Moishezon77.

We will apply this result to neighbourhoods of each critical value to obtain
local morsifications. For $c\in\Sigma$, define $D_{c}$ a disk around $c$ in
$\mathbb{P}^{1}$ such that $b\notin D_{c}$ Let $\ell_{c}$ be a path connecting
$b$ to a point $b_{c}\in\partial D_{c}$. Assume that for $c\neq c^{\prime}$,
$D_{c}\cap D_{c^{\prime}}=\emptyset$ and the interior of $\ell_{c}$ and
$\ell_{c^{\prime}}$ do not intersect. Let
$\infty\notin\bigcup_{c\in\Sigma}D_{c}\cup\\{b\\}$, identify
$\mathbb{C}=\mathbb{P}^{1}\setminus\\{\infty\\}$, and let
$S^{*}=f^{-1}(\mathbb{C})$. Define $T_{c}=f^{-1}(D_{c})$ and
$F_{b_{c}}=f^{-1}(b_{c})$. Then we have the following lemma.

######  Lemma 10.

The inclusion yields an isomorphism

$\bigoplus_{c\in\Sigma}H_{*}(T_{c},F_{b_{c}})\to H_{*}(S^{*},F_{b})\,,$ (14)

where the identification $F_{b_{c}}\simeq F_{b}$ is given by $\ell_{c}$.

###### Proof.

The main line of the argument is that the retraction of $\mathbb{C}$ to
$\bigcup_{c\in\Sigma}D_{c}\cup\ell_{c}$ lifts through the fibration. For more
details, see [§5.3]lamotke. ∎

Notice that $T_{c}\to D_{c}$ is an elliptic surface with a single singular
fibre.

####  3.2.1 Local morsification

Let $V\subset\mathbb{P}^{1}$ be a disk in $\mathbb{P}^{1}$ and consider an
elliptic surface $S\to V$ with a single singular fibre. Let $c$ denote the
critical value and $b$ a regular value on the boundary of $V$. Fix a
symplectic basis of $H_{1}(F_{b})$ and let $M_{\infty}$ be the monodromy
matrix around $c$ in this basis. From Theorem 8, there exists a morsification
of $S\to V$

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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (15)

Let $t\neq c$ and $S^{\prime}=\eta^{-1}(t)$. Then $S^{\prime}\to V_{t}$ is a
Lefschetz fibration. Denote by $r$ its number of singular fibres and by
$\Sigma$ its set of critical values.

######  Lemma 11.

Following the terminology of CadavidVelez08, the number $r$ of singular fibres
of $S^{\prime}\to D$ is the number of factors in the minimal normal
factorisation of $M_{T}$ (see Table 1). Let $G_{r}\dots G_{1}$ be this
factorisation. Let $A\in\operatorname{SL}_{2}(\mathbb{Z})$ be a matrix such
that $M_{\infty}=AM_{T}A^{-1}$. Then there is a distinguished basis of
$\pi_{1}(D\setminus\Sigma_{t},b)$ such that the corresponding monodromy
matrices $M_{1},\dots,M_{r}$ are given by $M_{i}=AG_{i}A^{-1}$.

###### Proof.

The first part is simply the observation that

$r=\chi(S)=\chi(S^{\prime})=\sum_{v\in\Sigma}\chi(F_{v})\,,$ (16)

and $\chi(F_{v})=1$ for every $v$ as $F_{v}$ is of type $I_{1}$. For the
second part, let $\ell_{1},\dots,\ell_{r}$ be a distinguished basis of
$\pi_{1}(D\setminus\Sigma,b)$. Let $1\leq i\leq r-1$ and notice that the bases

$(\ell_{1},\dots,\ell_{i-1},\ell_{i}\ell_{i+1}\ell_{i}^{-1},\ell_{i},\ell_{i+2},\dots,\ell_{r})$
(17)

and

$(\ell_{1},\dots,\ell_{i-1},\ell_{i+1},\ell_{i+1}^{-1}\ell_{i}\ell_{i+1},\ell_{i+2},\dots,\ell_{r})$
(18)

are also distinguished bases. Then the result is a direct application of
[Thm.19]CadavidVelez08, as Hurwitz moves on the product $M_{r}\dots M_{i}$ can
be achieved by the above changes of basis, see (1). ∎

######  Remark 12.

As shown by [Thm. 21]CadavidVelez08, the choice of the factorisation of
$M_{T}$ as a product of $I_{1}$ monodromy matrices does not matter. We could
equivalently take any such minimal factorisation, such as the ones in [Table
5]Naruki87.

Pick such a distinguished basis $\ell_{1},\dots,\ell_{r}$ of
$\pi_{1}(D\setminus\Sigma,b)$ and let $\Delta_{1},\dots,\Delta_{r}$ be the
corresponding thimbles. Then the trivialisation of $\tilde{S}$ through $\eta$
yields an isomorphism $H_{2}(S,F_{b})\simeq
H_{2}(S^{\prime},F^{\prime}_{b})=\bigoplus_{i=1}^{r}\mathbb{Z}\Delta_{i}$. We
conclude with two lemmas linking extensions and singular components of $S$ to
this basis of thimbles.

######  Lemma 13.

Let $\ell_{\infty}$ be the simple loop around $c$ pointed at $b$. Then

$\tau_{\ell_{\infty}}=\sum_{i=1}^{r}\tau_{\ell_{i}}\circ{\ell_{i-1}}_{*}\circ\cdots\circ{\ell_{1}}_{*}\,.$
(19)

###### Proof.

For $t\in D$, let $\Sigma_{t}$ be the set of critical values of
$\tilde{S}_{t}\to D$. Define $\tilde{\Sigma}=\bigcup_{t\in
D}\Sigma_{t}\times\\{t\\}$. $\tilde{\Sigma}$ is an analytic set of $V\times D$
and the projection onto $D$ is a finite morphism of degree $r$, totally
ramified at $c$. Clearly, $\ell_{\infty}$ and $\ell_{r}\dots\ell_{1}$ have the
same homotopy class in $\pi_{1}((V\times D)\setminus\tilde{\Sigma})$. The
lemma is then a direct application of (3). ∎

######  Lemma 14.

The inclusion of the singular components of $F_{0}$ in $H_{2}(S,F_{b})$
coïncides with the kernel of the boundary map:

$\bigoplus_{i=1}^{m_{0}-1}\left\langle\Theta^{c}_{i}\right\rangle=\ker\left(\delta:H_{2}(S^{\prime},F^{\prime}_{b})\to
H_{1}(F^{\prime}_{b})\right)\,.$ (20)

In particular, $r=m_{0}$ if $F_{0}$ has type $I_{\nu}$ and $m_{0}+1$
otherwise.

###### Proof.

The direct inclusion is clear. The rank of this kernel is $r-1$ in the case of
fibres of type $I_{\nu}$ and $r-2$ in the other cases. Therefore, if the
mentioned equality holds, the last statement follows.

Let us detail the proof of the equality in the case of a fibre of type
$I_{3}$. Its morsification splits it into $3$ fibres of type $I_{1}$, for each
of which the monodromy matrix is (up to a global conjugation) $U$. There are
thus $3$ thimbles, the restriction of the boundary map to these thimbles has
rank $1$, and the kernel thus has rank $3-1=2$. The intersection matrix of
(the lift in $H_{2}(S)$ of) this kernel is

$\left(\begin{array}[]{rr}-2&-1\\\ -1&-2\end{array}\right)\,,$ (21)

and it is thus a sublattice of discriminant $3$. As this coincides with the
discriminant of the sublattice generated by the singular components Kodaira63,
and as one is contained in the other, these sublattices are equal.

A similar direct computation gives the same result for each possible fibre
type.

∎

An illustration of the effects of a morsification is provided in Fig. 2.

###  3.3 Global homology and periods

We are now ready to give an algorithm for computing $H_{2}(S)$ (with its
lattice structure) from the action of monodromy on $F_{b}$ of the fibration
$S\to\mathbb{P}^{1}$.

Algorithm 1 Homology of elliptic surface

the monodromy matrices $M_{1},\dots,M_{r}\in\operatorname{SL}_{2}(\mathbb{Z})$

a description of $H_{2}(S)$ with its lattice structure

$N\leftarrow[]$

for $1\leq i\leq r$ decreasing do

Find $A\in SL_{2}(\mathbb{Z})$ and $T$ such that $M_{i}=AM_{T}A^{-1}$
$\triangleright$ See Table 1 for $M_{T}$

for $W$ in the minimal normal factorisation of $M_{T}$ do

Append $AWA^{-1}$ to $N$

for $M_{i}$ in $N$ do

Compute $d_{i}\in\mathbb{Z}^{2\times 1}$ and $m_{i}\in\mathbb{Z}^{1\times 2}$
such that $M_{i}=I_{2}+d_{i}m_{i}$.

$T_{i}\leftarrow(-1)^{n-1}\left(\begin{array}[]{ccc}&\mathbf{0}&\\\
\hline\cr&m_{i}&\\\ \hline\cr&\mathbf{0}&\end{array}\right)$ , where $m_{i}$
is the $i$-th line.

$B\leftarrow\left(\begin{array}[]{c|c|c}&&\\\ d_{1}&\dotsb&d_{r}\\\
&&\end{array}\right)$

$T_{\infty}\leftarrow
T_{1}+T_{2}M_{1}+T_{3}M_{2}M_{1}+\dotsb+T_{r}M_{r-1}\dotsb M_{1}$

$k\leftarrow\ker B$

$i\leftarrow\operatorname{im}T_{\infty}$

$H\leftarrow k/i$ $H$ is identified to the subspace of
$H_{2}(S^{*})/H_{2}(F_{b})$ generated by extensions. The (representatives of)
vectors of $H$ are the coordinates in the basis of thimbles of
$H_{2}(S^{*},F_{b})$. For more details, see [§§3,5]LairezPichonVanhove23.

Note that per Lemma 14 the sublattice generated by the singular components of
a given singular fibre is explicitly identified in this description of
homology; and per Lemma 13, the extensions of $\mathcal{T}\cap\ker\delta$ are
also explicitly identified.

To conclude this section, we provide a way to compute the periods of certain
$2$-forms on this basis of homology. Let $\omega\in H^{2}(S)$ and assume that
$\omega=\omega_{t}\wedge\mathrm{d}t$ for some $1$-form $\omega_{t}\in
H^{1}(S)$, where $t$ denotes the dependance on a coordinate of
$\mathbb{P}^{1}$. Then the integral of $\omega$ on an extension can be
obtained as a path integral of a period of the elliptic fibre via the
observation that

$\int_{\tau_{\ell}(\eta)}\omega=\int_{\ell}\left(\int_{\eta_{t}}\omega_{t}\right)\mathrm{d}t\,,$
(22)

where $\eta_{t}\in H_{1}(F_{t})$ is the unique deformation of $\eta\in
H_{1}(F_{b})$ along $\ell$. In particular, it is possible to recover the
periods of $\omega$ on $\mathcal{T}$ with high precision from the Picard-Fuchs
equation of $\omega_{t}$ using numerical integration methods with quasilinear
algorithmic complexity with respect to precision alone VanDerHoeven1999,
Mezzarobba2010. In practice, we rely on the implementation provided in
Sagemath by Mezzarobba2016 in the _ore_algebra_ package
KauersJaroschekJohansson2015. For further details on the computation of
periods on thimbles, see [§3.7]LairezPichonVanhove23.

As the fibre components are localised in a single fibre, the periods of
$\omega$ on a fibre component are all zero. The following lemma shows that
this information, i.e. the periods on extension and fibre components, is
sufficient to recover the full period mapping.

######  Lemma 15.

The primary lattice $\operatorname{Prim}(S)$ has full rank.

###### Proof.

For each $c\in\Sigma$,
$H_{2}(T_{c},F_{b_{c}})_{\mathbb{Q}}=\left(\ker\delta_{c}\right)_{\mathbb{Q}}\oplus\left(\operatorname{im}\tau_{c}\right)_{\mathbb{Q}}$.
Furthermore, $\mathcal{T}=\bigoplus_{c\in\Sigma}\operatorname{im}\tau_{c}$.
For a $\mathbb{Z}$-module $A$, denote by $A_{\mathbb{Q}}$ the tensor product
$A\otimes\mathbb{Q}$. From Lemma 14 and Lemma 10, we have that

$\begin{split}\left(\mathcal{T}\cap\ker\delta\right)_{\mathbb{Q}}&\oplus\bigoplus_{\begin{subarray}{c}c\in\Sigma\\\
1\leq i\leq m_{c}-1\end{subarray}}\langle\Theta^{c}_{i}\rangle_{\mathbb{Q}}\\\
&=\left(\ker\delta\right)_{\mathbb{Q}}\cap\bigoplus_{c\in\Sigma}\left(\operatorname{im}\tau_{c}\right)_{\mathbb{Q}}\oplus\left(\ker\delta_{c}\right)_{\mathbb{Q}}\,,\\\
&=\left(\ker\delta\right)_{\mathbb{Q}}\,.\end{split}$ (23)

Lemma 1 allows to conclude. ∎

Let $\mathcal{B}=(\eta_{1},\dots,\eta_{s},F_{b},O)$ be the basis of $H_{2}(S)$
obtained from Algorithm 1, where $\eta_{i}$ lies in $\phi(H_{2}(S,F_{b}))$ for
a lift $\phi:\ker\delta\to H_{2}(S)$. Let

$\mathcal{B}^{\prime}=(\Gamma_{1},\dots,\Gamma_{t},\Theta_{1},\dots,\Theta_{s-t},F_{b},O)$
(24)

be a basis of $\operatorname{Prim}(S)$, such that for each $i$,
$\Gamma_{i}\in\mathcal{T}\cap\ker\delta$ is an extension and
$\Theta_{i}\in\ker\delta_{c}$ for some $c\in\Sigma$ is a fibre component. Over
$\mathbb{Q}$, $\mathcal{B}^{\prime}$ is also a basis of
$H_{2}(S)_{\mathbb{Q}}$, and the matrix of change of basis
$M_{\mathcal{B}^{\prime}\to\mathcal{B}}\in\operatorname{GL}(\mathbb{Q})$ has
integer coefficients and can be computed with Lemma 13 and Lemma 14.

Using the methods of [§3.7]LairezPichonVanhove23, we may numerically compute
the periods $\int_{\Gamma_{i}}\omega$. Furthermore, the periods
$\int_{\Theta_{i}}\omega$, $\int_{F_{b}}\omega$ and $\int_{O}\omega$ are all
zero222The vanishing of the periods on the zero section is a consequence of
the fact that parabolic cohomology is the orthogonal complement of the trivial
lattice, see CoxZucker1979. [§1]Stiller87. Let
$\pi_{\mathcal{B}}=\left(\int_{\gamma}\omega\right)_{\gamma\in\mathcal{B}}$ be
the vector of periods of $\omega$ on a basis $\mathcal{B}$. Then

$\pi_{\mathcal{B}^{\prime}}=\left(\int_{\Gamma_{1}}\omega,\dots,\int_{\Gamma_{t}}\omega,0,\dots,0\right)$
(25)

and

$\pi_{\mathcal{B}}=M_{\mathcal{B}^{\prime}\to\mathcal{B}}^{-1}\pi_{\mathcal{B}^{\prime}}\,.$
(26)

##  4 Application: Mordell-Weil group and lattice

In this section, we expalin how to compute explicit embeddings of the Néron-
Severi lattice in the description of $H_{2}(S)$ given in the previous section.
We then use this to recover the Mordell-Weil group, and the lattice structure
of its torsion-free part, the Mordell-Weil lattice.

We start by recalling generalities about these lattices.

######  Definition 6.

The Néron-Severi lattice $NS(S)$ is the sublattice of $H_{2}(S)$ generated by
classes of divisors. Its rank is called the Picard rank or Picard number or
Néron-Severi rank.

######  Definition 7.

The trivial lattice $\operatorname{Triv}(S)$ is the sublattice of
$\operatorname{NS}(S)$ generated by the zero section and the fibre components.
Its orthogonal complement is the essential lattice
$L(S)=\operatorname{Triv}(S)^{\perp}$.

######  Definition 8.

The Mordell-Weil group $E(\mathbb{C}(t))$ of the elliptic curve
$E/\mathbb{C}(t)$ is the group of its $\mathbb{C}(t)$-rational points.

As mentioned in the beginning of Section 2, sections of $S\to\mathbb{P}^{1}$
are in bijection with $E(\mathbb{C}(t))$. The following lemma shows that the
group structure of $E(\mathbb{C}(t))$ coïncides with the lattice structure of
$H_{2}(S)$ modulo the trivial lattice.

######  Theorem 16 ([Thm 6.5]SchuttShioda10).

The map $P\mapsto\bar{P}\mod\operatorname{Triv}(S)$ is an isomorphism from
$E(\mathbb{C}(t))$ to $\operatorname{NS}(S)/\operatorname{Triv}(S)$.

In particular, this equips the torsion-free part of the Mordell-Weil group
with a lattice structure, inherited from the lattice structure on
$\operatorname{NS}(S)\subset H_{2}(S)$. More precisely, the orthogonal
projection of $NS(S)_{\mathbb{Q}}=NS(S)\otimes\mathbb{Q}$ onto
$L(S)_{\mathbb{Q}}=L(S)\otimes\mathbb{Q}$ defines a map
${\phi:E(\mathbb{C}(t))\to L(S)_{\mathbb{Q}}}$. The kernel of this map is the
torsion subgroup, and thus $\phi$ equips
$E(\mathbb{C}(t))/E(\mathbb{C}(t))_{tor}$ with a rational lattice structure.

######  Definition 9.

The Mordell-Weil lattice $MWL(S)$ of $S$ is the resulting lattice
$E(\mathbb{C}(t))/E(\mathbb{C}(t))_{tor}$.

For further reading on this topic, we recommend SchuttShioda10.

###  4.1 Computing the Néron-Severi lattice

The first step toward computing the Mordell-Weil lattice is to compute the
Néron-Severi lattice $NS(S)$. By Lefschetz’s $(1,1)$ theorem
[§1.2]GriffithsHarris78, it is entirely characterised as the kernel of the
holomorphic period mapping.

######  Theorem 17 (Lefschetz (1,1) theorem).

Let $\omega_{1},\dots,\omega_{s}$ be a basis of the space of holomorphic
$2$-forms of $S$, $H^{2,0}(S)$, and consider the period map
$\pi:H_{2}(S)\to\mathbb{C}^{s},\gamma\mapsto(\int_{\gamma}\omega_{1},\dots,\int_{\gamma}\omega_{s})$.
Then

$NS(S)=\ker\pi\,.$ (27)

We compute this kernel heuristically using the LLL method. In order to do this
we need two things: a basis of $H^{2,0}(S)$ and numerical approximations of
the associated periods.

Let $\omega$ be a holomorphic $2$-form on $S$. As an element of
$H^{0}(S,\Omega_{S}^{2})$, it can be written as

$\omega=f(t)\omega_{t}\wedge\mathrm{d}t\,,$ (28)

where $\omega_{t}\in H^{1}(E)$ is a rational section of the holomorphic
$1$-form bundle of the generic fibre, and $f\in\mathbb{Q}(t)$ is a rational
function. This representation is well adapted to the integration algorithm of
[§3.7]LairezPichonVanhove23. Of course the converse is not true: not every
rational function will yield a holomorphic $2$-form on $S$. In fact the
rational functions for which this is true are very tightly controlled by the
_Picard-Fuchs equation_ $\Lambda$ of $\omega_{t}$ – that is, the minimal
differential equation satisfied by $\omega_{t}$ with respect to the connection
inherited from the derivation on $\mathbb{C}(t)$ through the fibration over
$\mathbb{P}^{1}$.

Before focusing on the rational functions, let us briefly recall how
$\omega_{t}$ and its Picard-Fuchs equation can be computed. Let $P_{t}$ be the
defining equation of the elliptic fibration. In particular, the defining
equation for $F_{a}$ is $P_{a}$ whenever $a$ is regular.

######  Proposition 1.

There is a natural isomorphism
$\operatorname{Res}:H^{2}(\mathbb{P}^{2}\setminus F_{a})\to H^{1}(F_{a})$. It
is called the residue mapping.

Since $\mathbb{P}^{2}\setminus F_{a}$ is affine, its De Rham cohomology can be
computed using algebraic forms directly Grothendieck1966. More precisely, it
is the quotient of the space of homogeneous rational fractions of the form
$\frac{A}{P_{a}^{k}}$ with degree $-3$ by derivatives of the same form.
Furthermore, the holomorphic form of $H^{1}(F_{a})$ coincides with the residue
of the image of $\frac{1}{P_{a}}$ in this description [(8.6)]Griffiths1969a.
All in all, $\omega_{t}$ can be taken to be
$\operatorname{Res}\frac{1}{P_{t}}$. Its Picard-Fuchs equation can then be
computed using Griffiths-Dwork reduction [§5.3]CoxKatz1999. For a more
detailed discussion, see [§3.2]LairezPichonVanhove23.

We now turn back to the computation of valid rational coefficients. The
following result of Stiller87 gives a way to compute rational functions
$f_{1},\dots,f_{r}\in\mathbb{Q}(t)$ such that:

  * 1.

$f_{i}(t)\omega_{t}\wedge\mathrm{d}t$ defines a holomorphic $2$-form on $S$;

  * 2.

and
$f_{1}(t)\omega_{t}\wedge\mathrm{d}t,\dots,f_{r}(t)\omega_{t}\wedge\mathrm{d}t$
is a basis of $H^{2,0}(S)$.

######  Theorem 18 ([§3]Stiller87).

There is a divisor $\mathfrak{A}_{0}$ on $\mathbb{P}^{1}$ depending only on
$\Lambda$ such that if $Z\in L(\mathfrak{A}_{0})$, then
$\frac{Z}{W}\omega_{t}\wedge\mathrm{d}t$ is a holomorphic $2$-form on $S$,
where $W\in\mathbb{Q}(t)$ is the Wronskian of $\Lambda$. Furthermore the map
$L(\mathfrak{A}_{0})\to H^{2,0}(S)$ is an isomorphism.

An algorithm for computing $\mathfrak{A}_{0}$ is given in [§3]Stiller87, and
we recall it here for the sake of completeness. Assume that the Picard-Fuchs
equation $\Lambda$ has order two333this may happen when $S$ is an isotrivial
elliptic surface. In that case, the author knows does not know of a way to
identify the holomorphic form.. In particular at any point
$p\in\mathbb{P}^{1}$, the space of local solutions in a slit neighbourhood of
$p$ is generated by two solutions, that are locally of the form

$(t-p)^{q}(h_{1}(t-p)\log(t-p)+h_{2}(t-p))\,,$ (29)

with $q\in\mathbb{Q}$, $h_{1}$ and $h_{2}$ two holomorphic functions in a
neighbourhood of $0$. Let $r\leq s$ be the respective leading exponents of
these two solutions. Then the order of $\mathfrak{A}_{0}$ at $p$ is

$\operatorname{ord}_{p}\mathfrak{A}_{0}=\begin{cases}-\lfloor s\rfloor-3\text{
if }p=\infty\\\ -\lfloor s\rfloor+1\text{ otherwise}\end{cases}$ (30)

In particular, if $p$ is a finite regular point of $\Lambda$,
$\operatorname{ord}_{p}\mathfrak{A}_{0}=0$, meaning we can consider solely the
singular points of $\Lambda$. For further reading on the topic, we recommend
Stiller87 and [§4.3]DoranKostiuk2019.

The local exponents of $\Lambda$ can be obtained symbolically (see
Frobenius1873 and [§4]Mezzarobba2010 for instance), and we thus have a method
to compute a basis of the holomorphic $2$-forms of $S$, with a presentation
that is well suited for the integration methods of the periods mentioned in
Section 3.3. We can thus compute high precision numerical approximations of
the holomorphic periods of $S$. The Néron-Severi group can be heuristically
computed by recovering integer linear relations between these periods. Indeed,
let $\alpha_{i}$ be integers. Then

$\int_{\sum_{i}\alpha_{i}\gamma_{i}}\omega=\sum_{i}\alpha_{i}\left(\int_{\gamma_{i}}\omega\right)=0\text{
for all }\omega\in H^{2,0}(S)$ (31)

if and only if the cycle
$\sum_{i}\alpha_{i}\gamma_{i}\in\operatorname{NS}(S)$, where the
$\gamma_{i}$’s form a basis of $H_{2}(S)$. Thus integer linear relations
between the holomorphic period vectors
$(\int_{\gamma_{i}}\omega_{1},\dots,\int_{\gamma_{i}}\omega_{s})$ are in
bijection with $NS(S)$.

In order to recover these linear relations, we use the LLL algorithm. This
computation is not certified, and may fail in two ways: the algorithm may miss
integer relations with large coefficients, or may recover “fake” linear
relations that hold up to very high precision. More precisely, the algorithm
provides a sublattice $\Lambda\subset H_{2}(X)$ and positive numbers $B$, $N$
and $\varepsilon$ that depend on precision such that

  1. 1.

$\Lambda=\operatorname{NS}(X)$; or

  2. 2.

$\operatorname{NS}(X)$ is not generated by elements of the form
$\sum_{i}\alpha_{i}\gamma_{i}$ with $\sum_{i}\alpha_{i}^{2}\leq B$; or

  3. 3.

There $\sum_{i}\alpha_{i}\gamma_{i}\notin\operatorname{NS}(X)$ such that

$\sum_{j}\left|\sum_{i}\alpha_{i}\int_{\gamma_{i}}\omega_{j}\right|^{2}\leq\varepsilon^{2}\hskip
20.00003pt\text{and}\hskip 20.00003pt\sum_{i}\alpha_{i}^{2}\leq N^{2}\,.$ (32)

In practice, for $300$ recovered decimal digits of precision for the periods
of an elliptic $K3$ surface (which can be obtained in a few seconds), we find
$B\simeq 10^{132}$, $N=3$, and $\varepsilon\simeq 10^{-271}$. For further
discussion on these issues, see LairezSertozPicardRank.

###  4.2 Example: elliptic curve with high Mordell-Weil rank over $\mathbb{Q}$

In this section we detail the workings of our algorithm on an explicit
example. A Sagemath worksheet reproducing the results mentioned here is
available at _example_paper_elliptic.ipynb_
444https://nbviewer.org/urls/gitlab.inria.fr/epichonp/eplt-
support/-/raw/main/example_paper_elliptic.ipynb. The elliptic surface $S$ we
consider is an elliptic K3 with Picard rank 19 used in [§9]ElkiesKlagsbrun2020
(with $u=5$ in their notations) to find the elliptic curve with highest known
Mordell-Weil rank over $\mathbb{Q}$ for which the Mordell-Weil torsion
subgroup is $\mathbb{Z}/2\mathbb{Z}$. Its defining equation is

$X^{3}+4A(t)X^{2}Z+512B(t)XZ^{2}-Y^{2}Z$ (33)

where

$A(t)=93273t^{4}+58840t^{3}+102618t^{2}+35680t+14485$ (34)

and

$\begin{split}B(t)=-8590032t^{8}-78412620t^{7}+17011856t^{6}+241822775t^{5}-19459741t^{4}-127136490t^{3}\\\
+16161642t^{2}+15406335t-2083725\end{split}$ (35)

#### The homology lattice of $S$

This elliptic fibration has $16$ singular fibres above points
$c_{1},\dots,c_{16}$. We pick a basepoint $b$ as well as a distinguished basis
$\ell_{1},\dots,\ell_{16}$ of
${\pi_{1}(\mathbb{C}^{1}\setminus\\{c_{1},\dots,c_{16}\\},b)}$. The
corresponding monodromy matrices in a chosen symplectic basis
$\gamma_{1},\gamma_{2}$ of the homology of the fibre are given by

$\begin{gathered}M_{1}=\left(\begin{array}[]{rr}7&9\\\
-4&-5\end{array}\right)\,,\hskip
10.00002ptM_{i}=\left(\begin{array}[]{rr}1&1\\\ 0&1\end{array}\right)\text{
for $i=2,3,15$}\,\\\ M_{i}=\left(\begin{array}[]{rr}3&1\\\
-4&-1\end{array}\right)\text{ for $i=4,9,11,12$, and }\\\
M_{i}=\left(\begin{array}[]{rr}3&2\\\ -2&-1\end{array}\right)\text{ for
$i=5,6,7,8,10,13,14,16$.}\end{gathered}$ (36)

Computing the $\operatorname{SL}_{2}(\mathbb{Z})$ conjugation class of these
matrices, one finds that eight fibres (those for which the monodromy matrix is
$M_{5}$) are $I_{2}$ fibres, and the remaining eight are Lefschetz, i.e. of
type $I_{1}$. Indeed, we have

$M_{5}=AM_{I_{2}}A^{-1}\text{ with }A=\left(\begin{array}[]{rr}1&0\\\
1&1\end{array}\right)\in\operatorname{GL}_{2}(\mathbb{Z})\,.$ (37)

Per the minimal normal factorisation of Table 1, $M_{I_{2}}=U^{2}$. Therefore
in a morsification $S^{\prime}$ of $S$, there is a distinguished basis
$\ell^{\prime}_{1},\dots,\ell^{\prime}_{24}$ of
$\pi_{1}(\mathbb{C}\setminus\Sigma^{\prime},b)$ consisting of $8+8\times 2=24$
elements, where $\Sigma^{\prime}$ is the set of critical values of the
morsification, and such that the associated monodromy matrices are given by

$\begin{gathered}M^{\prime}_{1}=\left(\begin{array}[]{rr}7&9\\\
-4&-5\end{array}\right),\hskip
10.00002ptM^{\prime}_{i}=\left(\begin{array}[]{rr}1&1\\\
0&1\end{array}\right)\text{ for $i=2,3,22$},\\\
M^{\prime}_{i}=\left(\begin{array}[]{rr}3&1\\\ -4&-1\end{array}\right)\text{
for $i=4,13,16,17$, and }\\\ M^{\prime}_{i}=\left(\begin{array}[]{rr}2&1\\\
-1&0\end{array}\right)=AUA^{-1}\text{ for all other $i$.}\end{gathered}$ (38)

We may then use the methods of [§§3,5]LairezPichonVanhove23 to compute an
effective basis $\Gamma_{1},\dots,\Gamma_{22}$ of the homology of $S^{\prime}$
in terms of the thimbles $\Delta^{\prime}_{1},\dots,\Delta^{\prime}_{24}$, the
fibre class and the zero section. For instance, we find that a non trivial
homology class is given by $\Delta^{\prime}_{2}-\Delta^{\prime}_{22}$: as
$M^{\prime}_{2}=M^{\prime}_{22}$, this relative homology class has empty
boundary and thus lifts to a class in $H_{2}(S^{\prime})$. It is non-trivial
as $\ell^{\prime}_{22}{\ell^{\prime}}^{-1}_{2}$ is non-trivial in
$\pi_{1}(\mathbb{P}^{1}\setminus\Sigma^{\prime})$.

With this description, the singular components coming from an $I_{2}$ fibre of
$S$ above a critical value $c$ can be obtained as the kernel of the thimbles
of critical points flowing together at $c$. More explicitly, the component of
the fibre above $c_{5}$ is the homology class corresponding to the lift of
$\Delta^{\prime}_{5}-\Delta^{\prime}_{6}$.

Furthermore, extensions of $S$ can also be described in this basis. For
example

$\tau_{\ell_{6}^{-1}\ell_{5}}(\gamma_{2})=\tau_{\ell_{8}^{-1}\ell_{7}^{-1}\ell_{6}\ell_{5}}(\gamma_{2})=\Delta_{5}+\Delta_{6}-\Delta_{7}-\Delta_{8}\,.$
(39)

All in all we obtain the coordinates of a basis of $\operatorname{Prim}(S)$ in
the basis of $H_{2}(S)$ obtained from the morsification $S^{\prime}$. From the
Picard-Fuchs equation of the surface, we recover using Theorem 18 that the
space of holomorphic forms of $S$ is generated by

$\omega=\operatorname{Res}\frac{1}{P_{t}}\wedge\mathrm{d}t\,.$ (40)

From then on, we can compute the periods on the primary lattice, and recover
the full period mapping using the coordinates computed above. For example, we
find that the holomorphic period of the first element of the basis of homology
we computed is

$\int_{\Gamma_{1}}\omega=-0.0007064447191\cdots-i0.0002821239749\cdots\,,$
(41)

with certified precision bounds of around 150 digits.

Using the LLL algorithm, we find that the Néron-Severi lattice has rank $19$
as expected. Finally the Mordell-Weil group is obtained as the quotient of the
Néron-Severi lattice by the trivial lattice. We find

$MW(S)\simeq\mathbb{Z}^{9}\times\mathbb{Z}/2\mathbb{Z}\,.$ (42)

It should be noted that the result of [§9]ElkiesKlagsbrun2020 is stronger than
what we have computed here. First our approach for computing the Néron-Severi
group is heuristic as it relies on the LLL algorithm – in particular it is not
a certified computation, and thus does not provide a proof. Secondly we have
merely shown a result for the Mordell-Weil group over $\mathbb{C}(t)$555or
rather over $\bar{\mathbb{Q}}(t)$., whereas ElkiesKlagsbrun2020 shows that
this computation holds over $\mathbb{Q}$ for one of the fibres.

## Acknowledgements

I would like to thank Charles Doran, Pierre Lairez, Erik Panzer, Duco van
Straten, and Pierre Vanhove for valuable and insightful discussions.

## Funding

This work has been supported by the Agence nationale de la recherche (ANR),
grant agreement ANR-19-CE40-0018 (De Rerum Natura), grant agreement
ANR-20-CE40-0026-01 (Symmetries and moduli spaces in algebraic geometry and
physics); and the European Research Council (ERC) under the European Union’s
Horizon Europe research and innovation programme, grant agreement 101040794
(10000 DIGITS)

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