Elliptic discrete integrable systems and bi-elliptic addition formulae

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In mathematics, when classifying mathematical objects, one often finds in the class a 'master' object with a number of free parameters from which the other objects in the class can be obtained by taking certain limiting values for those parameters. In 2003 a classification was given of quadrilateral lattice equations, which are partial difference equations for a function depending on discrete variables (e.g. variables that label the sites of a space-time lattice) where the equation only involves the values of the function at the four vertices of an elementary quadrilateral. This is the simplest, (and so far the only) situation of a full classification of integrable partial difference equations of a single function subject to some additional conditions. The equations are nonlinear, but only in an affine-linear way (i.e. each vertex value appears only once in the equation) but the main property is that the equations considered obey the property of 'multidimensional consistency' (abbreviated by MDC), i.e. one can extend the number of dimensions of the lattice from 2 to any dimension, and impose the equation in each pair of directions on that lattice and the equation will still have large classes of nontrivial solutions. This remarkable property is considered to be an indication of the 'integrability' of the equation in question, which means that the MDC guarantees that you can actually construct exact solutions of the equation through a procedure called B"acklund transformation, by which one can obtain a new solution from an already given (possibly trivial) solution. In the classification of quad equations there was a master equation that appeared, discovered by V. Adler in 1998, and which is referred to as Q4. All other quad equations in the class are special parameter cases of this master equation. Because of the pivotal role of the Q4 equation, it is important to have a good insight into the structure of solutions. However, this turned out to be particularly challenging, since the natural parameters of the equation are subject to an algebraic condition which tells us that the parameters are points on an elliptic curve. These elliptic curves have been widely studied since the early 19th century, and are themselves parametrised in terms of a class of functions called 'elliptic functions' (like the circle is parametrised by trigonometric functions). The situation with regard to Q4 is, however, even more complicated, as to obtain even the simplest non-trivial solution of Q4 one needs a combination of two different types of elliptic functions associated with two essentially different elliptic curves. On a single elliptic curve there is a natural group law, that connects three intersection points on a straight line intersecting the curve, which is called an addition formula. It turns out that for solving Q4 there emerges a novel type of addition formulae that mixes the elliptic functions belonging to different elliptic curves. While the theory of elliptic functions and their addition rules is a classic subject, these new rules seem never to have been considered in the vast literature on elliptic functions and curves, so they merit a study in their own right. In the project I will endeavour to attain understanding of these novel 'bi-elliptic' addition formulae which govern the dynamics of the integrable systems that are defined by the Q4 equation. The project has an even more ambitious aim based on the hypothesis that behind the whole parameter-family of Q4 equations lurks a (possibly novel) algebraic object that governs the symmetries of the Q4 equation in terms of both the movable variables on the curve as well as of the parameters that fixes the curve itself, in addition to the independent and dependent variables that describe the complex dynamics encoded in this fascinating but still mysterious equation.