Elliptic Surfaces

EllipticSurface{BaseField<:Field, BaseCurveFieldType} <: AbsCoveredScheme{BaseField}

A relatively minimal elliptic surface defined as follows.

A genus $1$-fibration is a proper map

\[\pi \colon X \to C\]

from a smooth projective surface $X$ to a smooth projective curve $C$ whose generic fiber is a curve of (arithmetic) genus $1$.

The fibration is relatively minimal if its fibers do not contain any $(-1)$-curves. We call the fibration elliptic if it is relatively minimal and comes equipped with a section $\sigma_0\colon \mathbb{P}^1 \to X$. This turns the generic fiber of $\pi$ into an elliptic curve $E/k(C)$ where $k(C)$ is the function field of the curve $C$.

We further require that $\pi$ has at least one singular fiber and that the base field $k$ is perfect.

For now functionality is restricted to $C = \mathbb{P}^1$.

This datatype stores a subgroup of the Mordell-Weil group $E(k(C))$. It is referred to as the Mordell-Weil subgroup of X.

elliptic_surface(generic_fiber::EllipticCurve,
                 euler_characteristic::Int,
                 mwl_gens::Vector{<:EllipticCurvePoint}=EllipticCurvePoint[];
                 is_basis::Bool=true)
                 -> EllipticSurface

Return the relatively minimal elliptic surface with generic fiber $E/k(t)$.

This is also known as the Kodaira-Néron model of $E$.

Input:

• generic_fiber – an elliptic curve over a function field
• euler_characteristic – the Euler characteristic of the Kodaira-Néron model of $E$.
• mwl_gens – a vector of rational points of the generic fiber
• is_basis – if set to false compute a reduced basis from mwl_gens

Examples

julia> Qt, t = polynomial_ring(QQ, :t);

julia> Qtf = fraction_field(Qt);

julia> E = elliptic_curve(Qtf, [0,0,0,0,t^5*(t-1)^2]);

julia> X3 = elliptic_surface(E, 2)
Elliptic surface
  over rational field
with generic fiber
  -x^3 + y^2 - t^7 + 2*t^6 - t^5

elliptic_surface(g::MPolyRingElem, P::Vector{<:RingElem})

Transform a bivariate polynomial g of the form y^2 - Q(x) with Q(x) of degree at most $4$ to Weierstrass form, apply Tate's algorithm and return the corresponding relatively minimal elliptic surface as well as the coordinate transformation.

kodaira_neron_model(E::EllipticCurve) -> EllipticSurface

Return the Kodaira-Neron model of the elliptic curve E.

generic_fiber(X::EllipticSurface) -> EllipticCurve

Return the generic fiber as an elliptic curve.

zero_section(X::EllipticSurface) -> AbsWeilDivisor

Return the zero section of the relatively minimal elliptic fibration $\pi\colon X \to C$.

euler_characteristic(X::EllipticSurface) -> Int

Return the Euler characteristic $\chi(\mathcal{O}_X)$.

fibration(X::EllipticSurface)

Return the elliptic fibration $X \to \mathbb{P}^1$.

weierstrass_chart(X::EllipticSurface)

Return the Weierstrass chart of $X$ on its weierstrass_model.

weierstrass_chart_on_minimal_model(X::EllipticSurface)

Return an affine chart $U$ of $X$ which is isomorphic to the weierstrass_chart of $X$ on its weierstrass_model, but with all singular fibers removed.

More precisely, the affine coordinates of $U$ are $(x,y,t)$ and the chart is constructed as the vanishing locus of

\[y^2 + a_1(t) xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6\]

minus the reducible singular fibers.

weierstrass_model(X::EllipticSurface) -> CoveredScheme, CoveredClosedEmbedding

Return the Weierstrass model $W$ of $X$ and the inclusion in its ambient projective bundle

\[S\subseteq \mathbb{P}( \mathcal{O}_{\mathbb{P}^1}(-2s) \oplus \mathcal{O}_{\mathbb{P}^1}(-3s) \oplus \mathcal{O}_{\mathbb{P}^1}).\]

weierstrass_contraction(X::EllipticSurface) -> SchemeMor

Return the contraction morphism of $X$ to its Weierstrass model.

This triggers the computation of the underlying_scheme of $X$ as a blowup from its Weierstrass model. It may take a few minutes.

fibration_on_weierstrass_model(X::EllipticSurface)

Return the elliptic fibration $W \to \mathbb{P}^1$ where $W$ is the Weierstrass model of $X$.

trivial_lattice(X::EllipticSurface) -> Vector{AbsWeilDivisor}, ZZMatrix

Return a basis for the trivial lattice as well as its Gram matrix.

The trivial lattice is the sublattice of the numerical lattice spanned by fiber components and the zero section of $X$.

reducible_fibers(X::EllipticSurface)

Return the reducible fibers of $X$.

The output format is the following: A list [F1, ..., Fn] where each entry Fi represents a reducible fiber.

The list $F$ has the following entries:

• A point $P \in \mathbb{P}^{1}$ such that $F = \pi^{-1}(P)$;
• The ADE-type of the fiber;
• The fiber $F$ as a Weil divisor, including its multiplicities;
• The irreducible components of the fiber. The first component intersects the zero section;
• Their intersection matrix.

fiber_cartier(X::EllipticSurface, P::Vector = ZZ.([0,1])) -> EffectiveCartierDivisor

Return the fiber of $\pi\colon X \to C$ over $P\in C$ as a Cartier divisor.

fiber_components(X::EllipticSurface, P) -> Vector{<:AbsWeilDivisor}

Return the fiber components of the fiber over the point $P \in C$.

algebraic_lattice(X) -> Vector{AbsWeilDivisor}, ZZLat

Return the sublattice $L$ of the numerical lattice spanned by fiber components, torsion sections and the sections provided at the construction of $X$.

The first return value is a list $B$ of vectors corresponding to the standard basis of the ambient space of L. The second consists of generators $T$ for the torsion part of the Mordell-Weil group. Together $B$ and $L$ generate the algebraic lattice. However, they are linearly dependent. The third return value is the lattice $L$.

mordell_weil_sublattice(X::EllipticSurface) -> Vector{EllipticCurvePoint}, ZZLat

Return the (sublattice) of the Mordell-Weil lattice of $X$ spanned by the sections of $X$ supplied at its construction.

The Mordell-Weil lattice is represented in the same vector space as the algebraic lattice (with quadratic form rescaled by $-1$).

mordell_weil_torsion(X::EllipticSurface) -> Vector{EllipticCurvePoint}

Return the torsion part of the Mordell-Weil group of the generic fiber of $X$.

section(X::EllipticSurface, P::EllipticCurvePoint) -> EllipticSurfaceSection

Given a rational point $P\in E(C)$ of the generic fiber $E/C$ of $\pi\colon X \to C$, return its closure in $X$ as a AbsWeilDivisor.

basis_representation(X::EllipticSurface, D::AbsWeilDivisor)

Return the vector representing the numerical class of $D$ with respect to the basis of the ambient space of algebraic_lattice(X).

set_mordell_weil_basis!(X::EllipticSurface, mwl_basis::Vector{EllipticCurvePoint})

Set a basis for the Mordell-Weil sublattice of $X$ or at least of a sublattice.

This invalidates previous computations depending on the generators of the Mordell Weil lattice such as the algebraic_lattice. Use with care.

The points in mwl_basis must be linearly independent.

update_mwl_basis!(X::EllipticSurface, mwl_gens::Vector{<:EllipticCurvePoint})

Compute a reduced basis of the sublattice of the Mordell-Weil lattice spanned by mwl_gens and set these as the new generators of the Mordell-Weil lattice of $X$.

algebraic_lattice_primitive_closure!(X::EllipticSurface)

Compute the primitive closure of the algebraic lattice of $X$ inside its numerical lattice and update the generators of its Mordell–Weil group accordingly.

The algorithm works by computing suitable division points in its Mordell Weil group.

algebraic_lattice_primitive_closure(X::EllipticSurface, p) -> Vector{<:EllipticCurvePoint}

Return sections $P_1,\dots P_n$ of the generic fiber, such that together with the generators of the algebraic lattice $A$, they generate

\[\frac{1}{p} A \cap N\]

where $N$ is the numerical lattice of $X$.

The algorithm proceeds by computing division points in the Mordell-Weil subgroup of X and using information coming from the discriminant group of the algebraic lattice to do so.

isomorphism_from_generic_fibers(X::EllipticSurface, Y::EllipticSurface, f::Hecke.EllCrvIso) -> MorphismFromRationalFunctions

Given an isomorphism $f$ between the generic fibers of $X$ and $Y$, return the corresponding isomorphism of $X$ and $Y$ over $\mathbb{P}^1$.

isomorphism_from_generic_fibers(X::EllipticSurface, Y::EllipticSurface) -> MorphismFromRationalFunctions

Given given two elliptic surfaces $X$ and $Y$ whose generic fibers are isomorphic, return the corresponding isomorphism of $X$ and $Y$ over $\mathbb{P}^1$.

isomorphisms(X::EllipticSurface, Y::EllipticSurface) -> Vector{MorphismFromRationalFunctions}

Given two elliptic surfaces $X \to \mathbb{P}^1$ and $Y \to \mathbb{P}^1$ return all isomorphisms $X \to Y$ such that there exists Möbius transformation $\mathbb{P}^1 \to \mathbb{P}^1$ fitting in the following commutative diagram.

\[\begin{array}{ccc} X & \to & Y \\ \downarrow & & \downarrow\\ \mathbb{P}^1 & \to & \mathbb{P}^1 \end{array}\]

translation_morphism(X::EllipticSurface, P::EllipticCurvePoint) -> MorphismFromRationalFunctions

Return the automorphism of $X$ defined by fiberwise translation by the section $P$.

linear_system(X::EllipticSurface, P::EllipticCurvePoint, k::Int) -> LinearSystem

Compute the linear system $|O + P + k F|$ on the elliptic surface $X$. Here $F$ is the class of the fiber over $[0:1]$, $O$ the zero section and $P$ any section given as a point on the generic fiber.

The linear system is represented in terms of the Weierstrass coordinates.

two_neighbor_step(X::EllipticSurface, F1::Vector{QQFieldElem})

Let $F$ be the class of a fiber of the elliptic fibration on $X$. Given an isotropic nef divisor $F_1$ with $F_1.F = 2$, compute the linear system $|F_1|$ and return the corresponding generic fiber as a double cover C of the projective line branched over four points.

Input: $F_1$ is represented as a vector in the algebraic_lattice(X) $X$ must be a K3 surface

Output: A tuple (C, (x1, y1, t1)) defined as follows.

• C is given by a polynomial $y_1^2 - q(x_1)$ in $k(t_1)[x_1,y_1]$ with $q$ of degree $3$ or $4$.
• (x1,y1,t1) are expressed as rational functions in terms of the weierstrass coordinates (x,y,t).

elliptic_parameter(X::EllipticSurface, F::Vector{QQFieldElem}) -> LinearSystem

Return the elliptic parameter $u$ of the divisor class F.

The input F must be given with respect to the basis of algebraic_lattice(X) and be an isotropic nef divisor. This method assumes that $X$ is a K3 surface.

pushforward_on_algebraic_lattices(f::MorphismFromRationalFunctions{<:EllipticSurface, <:EllipticSurface}) -> QQMatrix

Return the pushforward $f_*: V_1 \to V_2$ where $V_i$ is the ambient quadratic space of the algebraic_lattice.

This assumes that the image $f_*(V_1)$ is contained in $V_2$. If this is not the case, you will get $f_*$ composed with the orthogonal projection to $V_2$.