# Linking L function dynamics with behavior close to s = 1 ?

A division, found on a sample set of semi-stable elliptic curves, calls for interpretation regarding the Birch and Swinnerton-Dyer conjecture and the dynamic behavior of the L functions involved.

Actually, taken a sample of 30 semi-stable elliptic curves, their L functions can be generated, say, up to the 1000th term of each, respective, series expansion.

Afterward, the values taken by these truncated L series expansions at s = 1 (the point of interest in the Birch and Swinnerton-Dyer Conjecture) can all be calculated. In parallel, a statistical quantification of how fast the iterates of a function explode to infinity, namely, the escape rates can also be numerically evaluated for each one of these truncated series expansions, and the ordinal correlation between the correspondent results and the above mentioned values, at s = 1, established.

An ordinal correlation coefficient (Spearman rank correlation coefficient, rs) of about minus 0.76 was found (rs=-0.76), with statistical significance on a level of confidence alfa (aa) 0.001 (i.e. aa=0.001).

Now, in what concerns escape rates, L functions show up clearly divided, or separated in two groups: those with high escape rates and those with low escape rates, the difference coming from the behavior on the right side of a critical, narrow strip, centered at about s = 1.

Following J.E. Cremona notation, the curves whose L functions have high escape rates in the sample are: 48A1, 327A1, 507B1, 552A1, 588B1, 606B1, 861C1, 921A1 and 996B1. The curves whose L functions have low escape rates are: 84A1, 168A1, 267B1, 366A1, 462D1, 492A1, 537A1, 546A1, 600C1, 645C1, 735A1, 768C1, 780A1, 849A1, 912A1, 930A1, 933A1, 975C1, 978B1 and 984B1. The sampling was made in the population of all (semi-stable) curves with conductors, N, up to 1000. In each isogeny class, the strong Weyl curve was chosen.

The first such study was made by December, 2010.

Incidentally, during September, 2012, a different sample composed of 70 curves was also chosen and studied, following the same procedure, except for the fact that these new curves were taken from the population of all semi-stable elliptic curves with conductors, N, up to 10000.

It came as a surprise that the same value for the rank correlation coefficient was found, moreover, on the same level of confidence, but with a quite remarkable p-value, p = 1.51762E-14 (not determined for the first sample that was studied). This equality may well be just a coincidence, as an aditional computation, by October, 2012, has shown: the new sample includes 8 curves with conductor 6090, ten curves with conductor 7296 and eight curves with conductor 8160. Then, inside each of these isogeny classes, a different set of numbers was obtained, as follows:

For N=6090: rs=-0.93 and aa=0.001 (p=0.00086)

For N=7296: rs=-0.78 and aa=0.01 (p=0.00755)

For N=8160: rs=-0.91 and aa=0.005 (p=0.00201)

By the middle of July, 2013, another similar, numerical study was concluded. This time, the sample size was 325. The curves were chosen in a similar way, but among those with conductors up to 60 000. The result, in what concerns the rank correlation coeficiente, was similar: rs=-0.78. The p-value associated to this last test was p=7.9098E-69.

(I would appreciate some advise about where to publish this last finding.)Still,

Hopefully, there will be an elementary answer for the following question:

Given the statistical significance of the correlation measured, what does this division, or separation mean, and what consequence can it have to the comprehension of L function's behavior near s = 1 ?

• I'm finding this hard to understand. Is "division" the right word? What do you mean by "escape rates" here? – David Feldman Sep 3 '12 at 2:46
• Thank you for your comment, David Feldman. "Division", above, is not meant in any technical sense, so, I added "separation". I also included an informal definition of an "escape rate". – Luis Borges Sep 3 '12 at 10:10
• Maybe it would help if you mentioned which 30 elliptic curves you examined, and what category each of them fell into. Perhaps those in one category have root number +1 and the other half have root number -1 (Either globally, or perhaps just at the infinite place). Without any data I'm taking a stab in the dark though. Is there any particular reason you restricted your attention to semi-stable elliptic curves? – Jamie Weigandt Sep 3 '12 at 16:21
• Thank you for your comment and suggestion, Jamie Weigandt. I will edit my question and include the list of all curves in the sample. As to my choice to work with semi-stable elliptic curves only, it is quite a long story; it comes from the fact that they are isomorphic to one dimensional analytic tori and, therefore, that they are rigid analytic varieties. All curves in my sample have bad reduction over p = 3 (more important, in my studies, they all have bad reduction at the same place). – Luis Borges Sep 3 '12 at 20:32