Computing
** Some details below of the definition of are wrong.**
The Definition
Let be a non-CM elliptic curve over .
Let be an odd positive integer.
Let vary over Dirichlet characters of order whose conductor is coprime to the conductor of .
We have where is the sign of , is the Gauss sum, and is the period mapping.
To make things very concrete we assume that is prime. This is not necessary.
Let denote the quotient of of order , so we have an exact sequence where is the unique subgroup of of order .
Choose a generator , i.e., any element of of order , so .
Choose of order , and write
Since , we have
Fix so that is an algebraic multiple of (so, e.g., might just be the least real or imaginary period of ). The rational period mapping is
For , let
The distribution is the distribution of real numbers , where we vary over all and . More concretely, for each integer coprime to such that , compute the real numbers and add them to our set of values. The distribution is then the result of doing this as goes to .
NOTE: The term is the sensitive theta coefficient.
Regarding complexity the work in doing this computation is the work of computing the rational numbers for all . It's the same bottlekneck that goes into approximating -adic -series using the classical Riemann sums algorithm. The code in Sage for this is fairly slow, but I have some fast code in psage, which I used for some papers on -adic -series.
Example 11a
In terms of Sage, the rational_period_mapping
method on a modular symbols space computes a choice of :