Analytic Jacobians of Hyperelliptic Curves#

This section contains descriptions of functions pertaining to the creation and use of analytic Jacobians for hyperelliptic curves.

Suppose \(C\) is a curve of genus \(g\) defined over the complex numbers. The analytic Jacobian of the curve is an abelian torus and is constructed as follows: We view the complex points on the curve, \(C({\mathbb{C}})\), as a compact Riemann surface of genus \(g\). It is known that the dimension of the vector space of holomorphic differentials is equal to the genus of the curve. So let \(\phi_i\), \(i = 1,2,\dots ,g\) be a basis for this vector space and set \(\overline{\phi} = {}^t(\phi_1,\dots,\phi_g)\), a vector of holomorphic 1-forms. For a hyperelliptic curve there is a natural choice of holomorphic differentials, namely \(\phi_i = x^{i-1}dx/y.\) We can now define a mapping \({\rm Int} : C \rightarrow {\mathbb{C}}^g\) by \(P \mapsto \int_P^\infty \overline{\phi}.\) To remove the dependency of this mapping on the path of integration, we define \(\Lambda\) to be the image of the map from the first homology group of \(C\), \(H_1(C,{\mathbb{Z}})\), to \({\mathbb{C}}^g\) which sends a closed path, \(\sigma \in H_1(C,{\mathbb{Z}})\), on the Riemann surface to \(\int_\sigma \overline{\phi}\). Then \(\Lambda\) turns out to be a lattice in \({\mathbb{C}}^g\) and so we get the complex torus \({\mathbb{C}}^g/\Lambda\). We then see that \({\rm Int}\) is a well-defined mapping from \(C\) into \({\mathbb{C}}^g/\Lambda\). We extend this mapping additively to the divisors of degree zero on \(C\). The Abel part of the Abel-Jacobi theorem states that two divisors map to the same image under \({\rm Int}\) if and only if they are linearly equivalent. The Jacobi part of the theorem asserts that the map is onto. So there is a bijection between the points of \({\mathbb{C}}^g/\Lambda\) and the points of the (algebraic) Jacobian.

So far the analytic Jacobian \({\mathbb{C}}^g/\Lambda\) is just a torus, but it can be made into an abelian manifold by choosing a polarization. Let \(A_1,\dots,A_g,B_1,\dots,B_g\) be a symplectic homology basis. That is, the \(A_i\) and \(B_i\) are closed paths on the Riemann surface such that all intersection numbers are \(0\) except that \(A_i\) intersects \(B_i\) with intersection number \(1\) for each \(i\). Let \(\omega_1\) and \(\omega_2\) be the \(g \times g\)-matrices with entries

\[{\omega_1}_{ij} = \int_{B_j} \phi_i,\]

and

\[{\omega_2}_{ij} = \int_{A_j} \phi_i.\]

Then the columns of \(P = (\omega_1, \omega_2)\) form a \({\mathbb{Z}}\)-basis for the lattice \(\Lambda = P {\mathbb{Z}}^{2g}\). The matrix \(P\) is called the (big) period matrix. The space of complex, symmetric \(g \times g\) matrices with positive definite imaginary part is called the Siegel upper-half space of degree \(g\). Because we chose a symplectic basis for the homology it follows that \(\tau = \omega_2^{-1}\omega_1\) is an element of Siegel upper half-space. We will refer to \(\tau\) as the small period matrix.

Set

\[\begin{split}J = \left(\begin{matrix}0 & {\bf 1}_g \\ -{\bf 1}_g & 0\end{matrix}\right).\end{split}\]

If we define \(E(P x, P y) = {}^tx J y,\) for any \(x,y \in {\mathbb{R}}^{2g}\), then \(E\) is a Riemann form for the torus \({\mathbb{C}}^g/\Lambda\) and so the torus acquires a polarization. The existence of a Riemann form on a torus is a necessary and sufficient condition for the torus to be embeddable into projective space. The image is isomorphic to the (algebraic) Jacobian.

In order for the theory describing the map from the analytic Jacobian to the algebraic Jacobian (see Mumford [Mumford, 1984]) to work we actually need a special symplectic basis linked to a particular ordering of the roots of the hyperelliptic polynomial. For an example of such a basis see Mumford [Mumford, 1984, Chap III.5]. The basis used can be obtained using the function HomologyBasis and the roots in the corresponding order are stored as the attribute A`Roots (where A is an analytic Jacobian).

Note: many of the function with analytic Jacobians use complex approximations, and indelicacies involving these can cause the functions to fail. One common remedy is to first transform the hyperelliptic curve before applying the analytic Jacobian machinery, either by translating or by inverting \(x\rightarrow 1/x\). This can often have the effect of changing the internal computations sufficiently that any errors are avoided.

Creation and Access Functions#

AnalyticJacobian(f): RngUPolElt -> AnHcJac#: Given \(f \in {\mathbb{C}}[x]\) where \({\mathbb{C}}\) is a complex field (see Chapter Real and Complex Fields), this function returns the analytic Jacobian of the hyperelliptic curve defined by \(y^2 = f(x)\). The polynomial must have degree at least \(3\) and the complex field precision at least \(20\) and, for the moment, less than \(2000\).

HyperellipticPolynomial(A): AnHcJac -> RngUPolElt#: Returns the polynomial defining the hyperelliptic curve whose Jacobian is \(A\).

HomologyBasis(A): AnHcJac -> SeqEnum, SeqEnum, Mtrx#

The symplectic homology basis used for the computation of the period matrix of the Jacobian \(A\). First assume that HyperellipticPolynomial(A) has odd degree. The function HomologyBasis(A) returns three values which we label basepoints, loops and S. Then basepoints is a list of points in the complex plane. The return valueloops is a list of \(2g\) lists of indices into basepoints. So the list [i1, i2, i3, ...] corresponds to the loop consisting of joining the straight lines from basepoints[in] to basepoints[in+1] for i \(= 1,2,...\). These closed loops form a homology basis for the curve but probably not a symplectic basis. The third return value, S, is a matrix giving the linear combinations of the loops that were used to form a symplectic homology basis and to compute the period matrix.

If HyperellipticPolynomial(A) has even degree then the returned homology basis is not for the Riemann surface for this curve, but for the curve of odd degree obtained by sending \(a =\) A`InfiniteRoot to infinity through the linear fractional transformation given by \(x \mapsto 1/(x-a)\). So one needs to apply the inverse transformation to the returned basis in order to get the basis that was used to compute the period matrix.

Dimension(A): AnHcJac -> RngIntElt#
Genus(A): AnHcJac -> RngIntElt#: The dimension of the Jacobian \(A\) as a complex abelian variety. This is equal to the genus of the curve \(C\) for which \(A\) is the Jacobian.

BaseField(A): JacHyp -> Fld#
BaseRing(A): JacHyp -> Rng#
CoefficientRing(A): JacHyp -> Rng#: The base field of the Jacobian \(A\).

Period Matrices#

SmallPeriodMatrix(A): AnHcJac -> AlgMatElt#: The small period matrix of the analytic Jacobian \(A\). If \(A\) has dimension \(g\) or equivalently the hyperelliptic curve has genus \(g\), then this is a symmetric \(g \times g\) matrix with positive definite imaginary part. If \(P = (\omega_1, \omega_2)\) is the full period matrix (see BigPeriodMatrix) then the small period matrix is defined by \(\omega_2^{-1} \omega_1\).

BigPeriodMatrix(A): AnHcJac -> AlgMatElt#: The full period matrix of the hyperelliptic curve of the Jacobian \(A\). If \(A\) has dimension \(g\) or equivalently the hyperelliptic curve has genus \(g\), then this is a \(g \times 2g\) matrix. The analytic Jacobian as a torus equals \({\mathbb{C}}^g/\Lambda\), where \(\Lambda\) is the \({\mathbb{Z}}\)-lattice spanned by the columns of the period matrix. The period matrix is computed with respect to the holomorphic differentials \(\phi_i = x^{i-1}dx/y\) and a symplectic basis for the homology that can be retrieved using HomologyBasis.

PeriodMapping(A, n): ModSym, RngIntElt -> Map#

Precision: RngIntElt                    Default: 0

The complex period mapping to precision Precision, computed using \(n\) terms of \(q\)-expansion and complex precision 30 if the optional parameter Precision is 0 (default) or using an appropriate number of terms if Precision is non-zero. The period map is a homomorphism \(M \rightarrow C^d\), where \(d\) is the dimension of \(A\).

The period mapping attached to the space of modular symbols \(A\), computed using \(n\) terms of the q-expansions of modular forms associated to \(A\). If the parameter Precision is set to some positive integer, the parameter \(n\) is ignored. Instead, the number of terms of the \(q\)-expansions needed to have a result of complex precision Precision (i.e., \(< 10^{-\rm{Precision}}\)) is computed. The period mapping is a homomorphism from AmbientSpace(A) to a complex vector space of dimension equal to the number of newform conjugates.

Periods(M, n): ModSym, RngIntElt -> SeqEnum#

Precision: RngIntElt                    Default: 0

The complex period lattice associated to \(M\) using \(n\) terms of the \(q\)-expansions and complex precision 30 if the optional parameter Precision is 0 (default) or using an appropriate number of terms is Precision is non-zero.

The complex period lattice associated to the space of modular symbols \(M\), computed using Precision terms of the \(q\)-expansions of modular forms associated to \(M\). If the parameter Precision is set to some positive integer, the parameter \(n\) is ignored. Instead, the number of terms of the \(q\)-expansions needed to have a result of complex precision Precision (i.e., \(< 10^{-\rm{Precision}}\)) is computed.

Maps between Jacobians#

ToAnalyticJacobian(x, y, A): FldComElt, FldComElt, AnHcJac -> Mtrx#: Let \(A\) be the analytic Jacobian of \(y^2 = f(x)\). This function maps the point \((x,y)\) on the curve to the analytic Jacobian. More precisely, let \(a = \infty\) when HyperellipticPolynomial has odd degree and \(a =\)A`InfiniteRoot otherwise. Then it maps the divisor \((x,y) - (a,0)\) to the analytic Jacobian. As any point on the algebraic Jacobian is simply a sum of such divisors, we can get its image by linearity of the map. The function returns a \(g \times 1\) matrix. This should be thought of as an element of \({\mathbb{C}}^g / \Lambda\) where \(\Lambda\) is the \({\mathbb{Z}}\)-lattice generated by the columns of the BigPeriodMatrix.

FromAnalyticJacobian(z, A): Mtrx, AnHcJac -> SeqEnum#: Let \(A\) be an analytic Jacobian of \(y^2 = f(x)\), with small period matrix \(\tau\). Let \(z\) be a \(g \times 1\) complex matrix (thought of as an element of \({\mathbb{C}}^g / \Lambda\) where \(\Lambda\) is the \({\mathbb{Z}}\)-lattice generated by the columns of the BigPeriodMatrix). This function returns a list of \(g\) (the dimension of \(A\)), or fewer, pairs \(P_i = \langle x_i,y_i\rangle\) satisfying \(y^2 = f(x)\). This is an element of the algebraic Jacobian when interpreted as the divisor \(\sum_{i=1}^g P_i - g\infty\).

Example: Analytic Jacobian Addition (ex-56e605)#

We give an example of moving between the algebraic and analytic Jacobians. We take two points on an algebraic Jacobian, map them to the analytic Jacobian, add them there and map the answer back to the algebraic Jacobian. We check that the answer corresponds to adding the points on the algebraic Jacobian.

> P<x> := PolynomialRing(RationalField());
> f := -8*x^5 - 14*x^4 + 28*x^3 - 30*x^2 + 6*x;
> C := HyperellipticCurve(f);
> J := Jacobian(C);
> pts := ReducedBasis(RationalPoints(J:Bound:=500)); pts;
[ (x^2 + 1, -3*x + 5, 2), (x^2 - 4/9*x + 1/9, -55/27*x + 43/27, 2) ]

Analytic Jacobians of Hyperelliptic Curves#

Creation and Access Functions#

Period Matrices#

Maps between Jacobians#

Isomorphisms, Isogenies and Endomorphism Rings of Analytic Jacobians#

From Period Matrix to Curve#

Voronoi Cells#