Work in Progress
(August, 2003)
After a period of three years, developing tools in non-Archimedean analysis and algebraic geometry to model flows with scale as the parameter, we are now turning to the computational validation of the following thesis:
1. The coherent structures in a turbulent flow must have their states modelled as L-functions of some adequate elliptic curves (namely, the semi-stable ones, which, beside from being modular, are also isomorphic to rigid analytic tori).
2. The evolution of these later states, across the scales, must obey some principle of preservation of structure, once subjected to the proper constraints, and it must, consequently, follow some uniquely defined path in an adequate rigid analytic variety (that can also be loosely visualized as an unique elliptic surface, with the semi-stable curves as fibres).
The computational validation of these ideas implies that we numerically study the iteration of some polynomials in the maximal ideal of arbitrary p-adic fields. As we are trying to model turbulence in natural systems (specifically, fusion plasmas under disruption and collapse), we are obviously constrained to adopt topological models for these p-adic ideals in the proper Euclidean space. Suggestions on how to develop the most efficient codes to do this would be very much appreciated (we are working with the mathematica system).
(September, 2004)
The main computer codes that we need, in order to validate the above mentioned thesis, were concluded during this summer of 2004. In some cases, they are reasonably efficient but, in others, they show some serious limitations:
1. In what concerns some preliminary characterization of those elliptic curves that we selected, these codes are mostly based upon the algorithms published by J.E. Cremona (Algorithms for Modular Elliptic Curves, 2nd ed., Cambridge Univ. Press, 1997), and we have successfully checked our results against the correspondent, respective ones in his book.
2. In what concerns the dynamics of those polynomials (analytic elements) that we attach to each of these elliptic curves, as well as in what concerns the resolution of their, respective, phase portraits (defined in the topological models of some p-adic integer rings), our codes, although trustable, do show some severe limitations: running times are higher than we expected, as our 1 Ghz microprocessor defines a balance of a few hours of computer time against each set of a few hundred orbits studied.
We are going to proceed with an attempt for optimization, while also considering the possibility to acquire one of the 3.2 Ghz microprocessors that are already available in the market. Nevertheless, our first priority goes to the conception of an adequate strategy, to deal with those huge processing tasks that we are going to execute, working with the information contained in the data base that was also made available by J.E Cremona, in the world wide web ( http://www.maths.nott.ac.uk/personal/jec/ftp/data/INDEX.html ).
Our purpose: the systematic collection of results aiming at the partial validation of our thesis of existence (topological conjugacy between the polynomials that we derive from those semi-stable elliptic curves whose associated modular forms (by the Eichler-Shimura theory) are close to each other).
We are scheduling for summer, 2005 the presentation of a first progress report on the work during the next twelve months.
The few numerical tests that we have already performed were primarily meant to check upon the validity of our codes; nevertheless, they allow us to report:
1. Some occasional observations of flow of coherent structures, through a limited number of scales, in the phase portraits that are defined by iteration of polynomials (inside two topological models for p-adic integer rings). Actually, this is to be acknowledge as an inference made upon qualitative grounds: zooming through the scales in these topological models, we were able to (occasionally) observe the burst, the brief permanence (self-similarity) and the dissipation of some distinct structures, or forms that were defined iterating the polynomials;
2. Some correspondence between closeness, inside the space of modular forms, and similitude, among polynomial phase portraits, both associated to the same semi-stable elliptic curves; this correspondence seems to depend on the proper resolution of that sort of degeneracy that comes from the number of curves in each isogeny class.
We find these observations to be interesting, and even encouraging, but we obviously do not consider them to be conclusive of anything, whatsoever. In fact, they were made in a quite small sample of curves, with conductor up to 100, that were drawn from an available population of curves with conductor up to 25000!
In the near future, any suggestions on how to proceed with a classification of p-adic polynomials will be very much appreciated! For the present, both the Mahler measure and some tools from symbolic dynamics seem to be good alternatives to work with...
(November 2005)
Since 2004, and mostly during the summer of 2005, we proceeded with the qualitative study of those p-adic, polynomial, dynamical systems that we canonically associate with each semi-stable elliptic curve. In particular, as we deduced them from the Weierstrass coefficients of each, correspondent elliptic curve with semi-stable reduction on p = 3, we started to observe the dynamical behavior of these polynomials, under iteration on the 3-adic integer ring,
Although we managed to start working with a new microprocessor (3Ghz), since past September, in a more efficient machine (one, never silent, HP Pavillion zd8000), we are still severely limited by the demands of our computer time consuming codes. For this reason, and also some other practical ones, we are still restricting our numerical simulations to the study of those elliptic curves with conductor less than 100. This is a set with 300 curves, but, as we restrict ourselves to the ones with semi-stable reduction, on p = 3, we are left with just about 100 curves, distributed over 25 isogeny classes.
After the first hundred hours of computation time, we were able to conclude that, among this first group of selected curves, only about 56 of them are such that the 3-adic polynomials that we derive from them show some non-trivial dynamics (in the other cases, the origin is a fixed point whose basin of attraction seems to cover the entire phase space). All the orbits are launched from a randomly chosen initial condition, and it usually takes no more than some 10 iterations for us to see them fall to the origin, whenever that is to be the case.
Subsequently, we defined the escape time as the observable to measure; we defined it as the time needed for an orbit to escape from a fixed size, co-moving cell in phase-space. As we proceeded with its measure, we constructed the symbolic sequence that we aimed at, in order to compute the topological entropy of each polynomial.
The procedure that we have adopted is close to the proposals by M. Abel et al., «Exit-times and e-entropy for dynamical systems, stochastic processes and turbulence», in Physica D 147 (2000) 12-35, by M. Lehrman e A.B. Rechester, «Extracting Symbolic Cycles from Turbulent Fluctuation Data», in Phys. Rev. Lett. 87 (2001) 16, 164501, and also, by S. Benkadda et al., «Exit Times and Chaotic Transport in Hamiltonian Systems», in Phys. Rev. Lett. 72 (1994) 18, 2859-2862.
Meanwhile, as we collected these symbolic sequences, and started to compute their entropies, we also developed an algorithm to quantitatively study the L functions of those elliptic curves that correspond to the 3-adic polynomials generating that symbolic dynamics. These other codes that we developed include some alternative diagnosis methods, like the study of correlation along the orbits, and the numerical study of the Rankin product zeta functions for the modular forms that are connected with the semi-stable elliptic curves that we are working with (see A.W. Knapp, Elliptic Curves, Princeton Univ. Press, Princeton, 1992, and also H. Hida, Elementary theory of L-functions and Eisenstein series, Cambridge Univ. Press, Cambridge, 1993).
Having made these options, we state that our main goal is now actually centered around the study of the following hypothesis: there is some correlation between the values taken, on the one side, by the topological entropies of some p-adic polynomial dynamical systems, and, on the other side, by the L functions that belong to the elliptic curves from which those polynomials were derived.
As we finish the numerical studies corresponding to our first qualitative observations, we are starting the systematic and quantitative study of these p-adic flows, together with the associated L functions. In order to do this, we divide the work to be done into several stages. In each of these stages, we study the behavior of each element of the whole set of pairs of elliptic curves and 3-adic polynomials, and we do it against one reference pair (semi-stable elliptic curve; associated 3-adic polynomial). We assume the taxonomy that was established by J.E. Cremona (Algorithms for Modular Elliptic Curves, 2nd ed., Cambridge Univ. Press, 1997).
Some of these stages are already completed. In all of them, the 56 curves included in our first sample were classified according to the behavior of their L functions in a neighbourhood of the critical point z = 1. In parallel, we measured the topological entropy that characterizes the flow of each 3-adic polynomial that we associate with those curves. This measurement was made simultaneously for 10 flows (including the reference one), starting from a common, randomly chosen initial condition.
After spending about 500 hours of work, in computer time, we have completed the stages corresponding to curves 15A1, 21A5, 33A2, 96A1 and 96B4. These curves were arbitrarily chosen, apart from the demand that they represent different isogeny classes. There are still 17 isogeny classes to work with, in our sample. Nevertheless, the results obtained so far enable us to assert that the coherence that characterizes each isogeny class, due to the fact that all the elliptic curves that belong to it have an identical L function, is a coherence that, somehow, is reflected on the dynamical behavior of the associated 3-adic polynomials, as we are able to infer from the measurement of the topological entropy of their flows. This coherence is total, or absolute, on 71% of the isogeny classes that were put to the test (roughly 100 = 5 × 20). If we assume that there is still coherence in a group of five polynomials (coming from the same isogeny class) when only one of them shows some apparently different behavior, then, the above percentage changes to 76%. These numbers do not take into account any of those isogeny classes for which there is only one polynomial with non-trivial behavior.
These results are robust against the variation of some parameters whose fine-tunning we are still adjusting (number of iterates of the maps, dimension of the symbolic blocks, etc).
(Pierre ROY, 1931)