Sato-Tate for E/K, [K:QQ]>1

""" Author: Stefan Keil, 2010-02-21. This is an extended Version of http://sage.math.washington.edu/sato-tate/ by William Stein. Let E/K be an elliptic curve over a number field K, which is not the rational numbers QQ. For each finite place v of K, for which E has good reduction, one can calculate the number a_v := q_v+1-|E(FF_v)|, where q_v=Nv is the usual norm of v and |E(FF_v)| is the number of points of the mod v reduced curve over the finite residuo field FF_v := O_K / v, which is the ring of integers of K modulo v. Hasse says |a_v| <= 2 \sqrt q_v, so one can normalize a_v/(2 \sqrt q_v) = cos theta_v, with an unique angle theta_v in [0,pi]. The Sato-Tate-Conjecture says, that if one runs over all finite places v of good reduction, then the theta_v (ordered by increasing norm Nv) are equidistributed with respect to the so called Sato-Tate-Measure 2/pi sin^2 theta d theta. This Worksheet calculates the distribution of the angles theta_v of a given elliptic curve defined over a number field K up to all finite places v of good reduction of E, with norm Nv=q_v <= N, whereas the intervall [0,pi] will be divided into B bins. Additionally it draws the curve of the Sato-Tate-Measure for comparison. """ 

# returns all theta_v of a given elliptic curve E over a number field K up to Nv=q_v<N def sato_tate(E, K, N): all_theta_v = [] for v in primes_of_K_up_to_norm_N(K,N): if E.has_good_reduction(v): all_theta_v.append(theta_of(E, v)) return all_theta_v 

# calculates theta_v of a given elliptic curve E and prime v def theta_of(E, v): E_red=E.reduction(v) q_v=v.norm() cardinality_of_E_red=E_red.cardinality() a_v=q_v+1-cardinality_of_E_red theta_v = acos(a_v/(2*sqrt(q_v))) return theta_v 

# returns all primes v of a given number field K with Nv=q_v<N def primes_of_K_up_to_norm_N(K, N): primes =[] for p in prime_range(N+1): principle_ideal_p = K.ideal(p) primfactors_of_principle_ideal_p = principle_ideal_p.prime_factors() # take only primes with norm < N+1 for v in primfactors_of_principle_ideal_p: if v.norm()<N+1: primes.append(v) return primes 

# this function divides the intervall [0,pi] into B bins and then counts how many of the theta_v # are in which bin, and then returns this distribution and the number of all theta_v def distribution_of_all_theta_v_in_B_bins(all_theta_v, B): normalize = float(B/pi) vals = {} distribution = dict([(i,0) for i in range(B)]) for theta_v in all_theta_v: # n is the number of the bin n = int(normalize*float(theta_v)) # in case of theta_v=pi if n > B-1: n=B-1 distribution[n] += 1 return distribution, len(all_theta_v) 

# this will make the blue recangles according to the given distribution, the number of bins B, # and the total number of thetas def graph(distribution, B, number_of_thetas): s = Graphics() left = float(0); right = float(pi) length = right - left w = length/B k = 0 for i, n in distribution.iteritems(): k += n # ith bin has n objects in it. s += polygon([(w*i+left,0), (w*(i+1)+left,0), \ (w*(i+1)+left, n/(number_of_thetas*w)), (w*i+left, n/(number_of_thetas*w))], \ rgbcolor=(0,0,0.5)) return s 

# plots curve of Sato-Tate measure def sato_tate_measure(): PI = float(pi) return plot(lambda x: (2/PI)*math.sin(x)^2, 0,pi, plot_points=200, \ rgbcolor=(0.3,0.1,0.1), thickness=2) 

# returns the plotted distribution of all theta_v with Nv=q_v<N of an elliptic curve E over K # divided in B bins and adds the curve of the Sato-Tate measure for comparison def graph_ellcurve(E, K, B, N): all_theta_v = sato_tate(E, K, N) distribution, number_of_thetas = distribution_of_all_theta_v_in_B_bins(all_theta_v,B) return graph(distribution, B, number_of_thetas) + sato_tate_measure() 

# defines a number field K and an elliptic curve E over K # CAUTION: # K cannot be QQ K.<i> = NumberField(x^2 + 1) E = EllipticCurve(K,[2*i,3]) E 

Elliptic Curve defined by y^2 = x^3 + 2*i*x + 3 over Number Field in
i with defining polynomial x^2 + 1

# starts the program and shows the distribution of all theta_v with Nv=q_v<N # of an elliptic curve E over a number field K divided in B bins g = graph_ellcurve(E,K,B=40,N=10000) show(g) 

# starts the program and shows the distribution of all theta_v with Nv=q_v<N # of an elliptic curve E over a number field K divided in B bins g = graph_ellcurve(E,K,B=50,N=10000) show(g) 

K.<t> = NumberField(x^2 + 3) E = EllipticCurve(K,[-5,-3*t]) E 

Elliptic Curve defined by y^2 = x^3 + (-5)*x + (-3*t) over Number
Field in t with defining polynomial x^2 + 3

E.global_integral_model() 

Elliptic Curve defined by y^2 = x^3 + (-5)*x + (-3*t) over Number
Field in t with defining polynomial x^2 + 3

# starts the program and shows the distribution of all theta_v with Nv=q_v<N # of an elliptic curve E over a number field K divided in B bins g = graph_ellcurve(E,K,B=50,N=10000) show(g) 

K.<t> = NumberField(x^2 + 167) E = EllipticCurve(K,[t,2]) E 

Elliptic Curve defined by y^2 = x^3 + t*x + 2 over Number Field in t
with defining polynomial x^2 + 167

E.global_integral_model() 

Elliptic Curve defined by y^2 = x^3 + t*x + 2 over Number Field in t
with defining polynomial x^2 + 167

# starts the program and shows the distribution of all theta_v with Nv=q_v<N # of an elliptic curve E over a number field K divided in B bins g = graph_ellcurve(E,K,B=50,N=10000) show(g) 

K.<t> = NumberField(x^5 + 4*x^3 -17) E = EllipticCurve(K,[t,1]) E 

Elliptic Curve defined by y^2 = x^3 + t*x + 1 over Number Field in t
with defining polynomial x^5 + 4*x^3 - 17

E.global_integral_model() 

Elliptic Curve defined by y^2 = x^3 + t*x + 1 over Number Field in t
with defining polynomial x^5 + 4*x^3 - 17

# starts the program and shows the distribution of all theta_v with Nv=q_v<N # of an elliptic curve E over a number field K divided in B bins g = graph_ellcurve(E,K,B=50,N=10000) show(g)