Hyperexponential solutions of elliptic difference equations

Let us consider an elliptic curve under Weierstrass form $\{(u,v)\in\mathbb{C}^{2},v^{2}=u^{3}+au+b\}$. This curve can be completed by a point at infinity, noted $O$, and the completed curve will be called $\mathcal{C}$. Remark that this completion point is unique because $\sqrt{u^{3}+au+b}$ is ramified above $u=\infty$. We can now define on $\mathcal{C}$ an addition law $\oplus$ with the property that $p\oplus q\oplus r=O$ if $p,q,r$ are aligned, and $\ominus p$ is the symmetric with respect of the abscissa of $p$. Any straight line cuts $\mathcal{C}$ at $3$ points (with possibly multiplicity), and thus given any pair of points $p,q$, the point $\ominus r$ and then $r$ can be computed.

A point is called of torsion if an integer multiple of it is $O$, which we will note $N.p=O$ where $N\in\mathbb{N}^{*}$. Remark in particular that the ramification points of $\mathcal{C}$ are of $2$ torsion. For a given point $\delta\in\mathcal{C}$, the addition of $\delta$ defines an automorphism $\phi_{\delta}$ of $\mathcal{C}$. This automorphism is cyclic if and only if the point $\delta$ is of torsion. In the rest of the article, we will consider a point of $\mathcal{C}$ noted $\delta$ which is not of torsion. In this article we will be interested by difference equations of the form

where $a_{i}$ are polynomial functions over $\mathcal{C}$, i.e. elements of $\mathbb{C}[u,v]/(u^{3}+au+b-v^{2})$. These kind of difference equations where considered by Dreyfus, Arreche and al. [5], [2] with applications in mind [4], and can be also written using the standard shift by parametrizing $\mathcal{C}$ with Weierstrass elliptic functions. We are interested in three types of explicit solutions.

These definitions have a natural equivalence with the classical case of the shift on $\mathbb{C}$, which gives the classical difference equations, see [6]. These type of solutions were analysed by [7] and generalization to systems [1] with algorithms to compute them. The intermediary class we call pseudo-rational solutions is the equivalent of solutions of the form $\lambda^{n}F(n),\;F\in\mathbb{C}(n)$ for the shift in $\mathbb{C}$. They indeed are hyperexponential and only have finitely many roots and poles. In the elliptic case, they appear as the main building block for computing hyperexponential solutions.

Before even beginning, we need to ensure that $\delta$ is not a torsion element. The algorithm TorsionOrder of Combot [3] gives an integer $N$ which is a candidate for the torsion order of $\delta$. If it is zero, then $\delta$ is not of torsion. Else if $\delta$ is of torsion, then its order is $N$. This has then to be checked by performing the computation $N.\delta$. Thus it is possible to decide if the equation is well posed. In the article, for computing on examples, we will always choose the elliptic curve $v^{2}=u^{3}+15$ with the shift $\delta=(1,4)$. This curve has nice symmetries and a Mordell Weil group over $\mathbb{Q}$ of rank $2$, and thus has many rational points which are not multiple of $\delta$.

The first step is to compute the set of possible singularities for a rational solution. There are finitely many orbit under the action of $\phi_{\delta}$ which can encounter a zero of $a_{l}a_{0}$. We however need to know if two roots are on the same orbit. For this we will need to bound a priori the distance between two roots on an orbit of $\phi_{\delta}$.

The definition of $\hat{h}$ is not suitable for its practical computation. This is done through computation of local height functions at many primes. The canonical height of a point is (probably) typically a transcendental number, and can only be computed approximatively. However, this is not a problem for our application, where we need to test if a given point is a multiple of $\delta$.

This implies that the singularities of equation (1) can be studied up to Galois conjugacy: we never need to consider two different Galois conjugated roots, and if two roots are on the same orbit, then their conjugate also are. Remark moreover that if two roots are in a field extension $\mathbb{L}$ over $\mathbb{K}$, the two extension have to be the same as any field extension of $\mathbb{K}$ is stable by $\phi_{\delta}$. This allows to reduce the number of tests to do.

Principal divisor are proper, but all proper divisors are not principal. Recall that a proper divisor is principal if and only if the sum of its points with multiplicities using the elliptic addition law equals $O$. For example, removing a root of the set of roots of a polynomial can define a divisor which is not principal, and thus is not the set of roots (with multiplicities) of another polynomial. Because of this, it is possible that all coefficients of equation (1) have a common root, but that we cannot simplify it nevertheless. Let us present then an algorithm to compute the roots of $a_{0},a_{l}$ which are not common roots of all coefficients of equation (1).

Remark that we cannot assume that $a_{0}$ and $a_{l}$ have distinct roots, so the same $\mathcal{J}_{i}$ can appear both positively and negatively. We now compute what we call the universal divisor.

The output defines a divisor whose support are the roots of the $\mathcal{J}$ and their shifts, and the value of the divisor is the value of the associated list.

This minoration of the divisor of a rational solution allows to bound the degree (which is the valuation at the point $O$), and the order and position of its poles. Remark that even the affine part of the divisor $D$, completed at $O$ such that it becomes proper, is not always principal. Thus this set of poles cannot always be represented by a single polynomial.

Thus adding at most one finite point to $D$ makes it principal. A candidate for the denominator of a rational solution is then a polynomial of minimal degree vanishing on $D$, and it will contain at most one additional unnecessary root. We conclude this section with the algorithm for computing rational solutions.

Remark that building simple examples having rational solution is not immediate: even the simplest rational solutions as above have $3$ or $4$ singularities, which after two shifts give $10$ or more singularities, and some new ones possibly in field extensions.

Even for the classical shift on $\mathbb{C}$, hyperexponential functions with finitely many poles and roots are not always rational as a factor $\lambda^{n}$ can appear. Similarly, a multiplicative factor can appear in front a rational function for elliptic difference equations, by considering a non zero solution of equation $f(p\oplus\delta)-\lambda f(p)$. However, this is not enough to encompass all such pseudo-rational solutions.

In the limit case $(\alpha,\beta)=O$, the $\Theta_{\lambda,O}$ satisfies the much simpler equation

When the point $(\alpha,\beta)\in\mathcal{C}$ is of torsion, the divisor of $\Theta_{\lambda,\alpha,\beta}$ will be $-\mathbb{I}_{(\alpha,\beta)}+\mathbb{I}_{O}$, which is a torsion divisor. Thus a multiple of this divisor can be realized by a rational function $f$ on $\mathcal{C}$. Thus dividing $\Theta_{\lambda,\alpha,\beta}$ be a suitable $n$ th root of $f$ will produce a function without any poles nor roots. As $(f(p\oplus\delta)/f(p))^{1/n}$ will be rational as its divisor is principal by construction, the ratio $\Theta_{\lambda,\alpha,\beta}/f$ will still be hyperexponential, and without any root or poles, and thus equal to a function $\Theta_{\tilde{\lambda},O}$. Now when $\tilde{\lambda}$ is a root of unity, this function will be algebraic (it will have in fact finitely many values), and thus $\Theta_{\lambda,\alpha,\beta}$ will be algebraic.

Remark that we do not try to understand possible analytic properties of these pseudo-rational functions. Only the knowledge along each orbit is enough to determine if they are solution of an elliptic difference equation. Thus a constant function is a function which is constant on all orbits, and is from the point of view of the elliptic difference equation not distinguishable from the constant function everywhere. A similar phenomenon occurs for the classical shift in $\mathbb{C}$ where functions $n\rightarrow e^{2i\pi kn}$ with $k\in\mathbb{Z}$ are for all purpose constant as they are constant on all orbits.

We can nonetheless find possible analytic formulas for $\Theta$ as an exponential of a sum of an elliptic integral of the first kind (for the $\lambda$ parameter) and an elliptic integral of the third kind (for the $\alpha,\beta$ parameter). As already known cite, elliptic integrals of the third kind become elementary for a torsion singular point with logarithmic singularity, and thus taking the exponential we recover the algebraic nature of $\Theta$ function in this case.

We now want to design an algorithm to compute pseudo-rational solutions. The possible parameters $\lambda$ can be obtained by considering a point $p$ on a non singular orbit and computing

where $f$ is a non zero solution of equation (1). The $\lambda$ depends on the choice of $f$, so this limit has to be computed for all solutions. This expression gives the possible $\lambda$’s in a pseudo-rational solution as the rational factor in $f$ will induce in $\ln f(p\oplus n.\delta)$ a logarithmic growth, and the $\Theta$ function will for large $n$ have the behaviour of multiplying by $\lambda$ at each shift of $\delta$ on the orbit. We use $\overline{\lim}$ as it is unclear such quantity would converge for non pseudo-rational solutions.

Now expression (2) is nice but totally unusable for explicit computation. Indeed we will most likely obtain transcendental numbers, and it is even unclear that for all choice of $f$ (a $l$ dimensional vector space) we will obtain a finite set of $\lambda$. In the shift case on $\mathbb{C}$, the possible $\lambda$’s can be obtained through local analysis at infinity, which is a fixed point for the shift. In particular, the $\lambda$ is always algebraic even for non hyperexponential solutions.

In the elliptic case, the solution to this problem is just to not solve it. Indeed, it would be convenient to know $\lambda$ in advance, but it is not necessary. Just replacing $f(p\oplus i.\delta)\rightarrow\lambda^{i}f(p\oplus i.\delta)$ in equation (1) will not change the singularities and thus neither the output of UniversalDivisor. Thus in RationalSolutions, the linear system to solve will simply have $\lambda$ as a parameter appearing polynomially in the matrix of the equation.

The other difficulty is the function $\Theta_{\lambda,\alpha,\beta}$. The parameter $(\alpha,\beta)\in\mathcal{C}$ can be a priori anywhere on $\mathcal{C}$ as it is related to the “default” of principality of a divisor. We already know the locus of the poles, but only a bound on their multiplicities, and for the roots nothing apart a bound on their number. However the reasoning for the construction of the universal divisor stays valid. Indeed, we only use the fact that the solution should have finitely many roots and poles, and thus

Let us look at the algorithm RationalSolutions. After computing a universal denominator $Q$, the resulting divisor could be not principal, and thus for pseudo-rational solutions a function $\Theta$ would be needed. The solution is to relax by $1$ the constraint on the degree of the numerator $P$ and allow the compensation of the additional root with a function $\Theta$. As we cannot know this superfluous root before computing $P$, we need to make the computation of the kernel of the matrix while keeping the parameter $(\alpha,\beta)\in\mathcal{C}$.

The while loop is necessary the ensure that all solutions are linearly independent. However, we can ask if continuum of solutions are possible in step $2d$. If a continuum exists, then by evaluation we obtain arbitrary many pseudo-rational solutions $f_{1},\dots,f_{k}$ of (1). Evaluating these solutions on a non singular orbit, we obtain the asymptotic estimates

Thus if a linear combination of these $f_{i}$ equals zero, so is the right expression. We see that the terms $e^{n\ln\lambda_{i}+o(n)}$ cannot compensate between each other, and thus the only possibility is that all $\lambda_{i}$ are equal. Thus if a continuum exists, $\lambda$ is fixed along it, and thus the parameter $(\alpha,\beta)\in\mathcal{C}$ is free.

Now sending the parameter $(\alpha,\beta)$ to $O$, and up to computing a series expansion for the rational part of the expression of the solution, we obtain a solution of the form $\frac{P}{Q}\Theta_{\lambda,O}$. These solutions are simpler than for a generic $(\alpha,\beta)$ and can be searched independently. Removing them first would ensure that only zero dimensional ideal in the parameter space are encountered, thus possibly reducing the computation time.

The function $H$ plays the role of a product of $\Gamma$ function for the classical shift on $\mathbb{C}$. The main difference is that it is a priori not possible to split each factor to have a product of function for each orbit because all divisors are not principal.

Remark that in the construction of the function $H$, the specific position of the valuation jump on an orbit is not important. We can always shift these jumps by a multiple of $\delta$ (although this would change the pseudo-rational part $F(u,v)\Theta_{\lambda,\alpha,\beta}(u,v)$). Of course the divisor formed by the jumps is not always principal, and this suggests to define a more relaxed version of principality.

A principal divisor is $\delta$ principal, but so are any divisor built by shifting by $\delta$ arbitrary the points of its support, and thus is a property invariant by $\delta$ shifts.

In practice, we can compute bounds for the jump amplitude $m_{p}$ by counting singularities on orbits. Then for any combination of choice of $m_{p}$, we build a divisor, and it should be $\delta$-principal. We can test this property, and if true, we only have to translate by a multiple of $\delta$ one of its points to obtain a principal divisor. This will be a suitable choice for the divisor of $H(p\oplus\delta)/H(p)$ in Proposition 7.

We will test $\delta$-principality for many divisors with the same support. To do it efficiently, we can first compute the regulator matrix of the points of the support $p_{1},\dots,p_{k}$

where $<\;,\;>$ is the scalar product defined by the canonical height. We can now define a norm on divisors by

and then $\|D\|$ gives the value (up to sign) of the number of shifts of $\delta$ to test the $\delta$ principality of $D$. By computing the matrix norm $\|\hbox{Reg}\|_{2}$, we obtain an absolute bound as we can bound $(n_{1},\dots,n_{k})$ as this is simply the number of singularities of equation (1).