tate conference graphics

506 days ago by wstein

All the pictures below have pdf files below them that give vector art versions of the figures (in pdf format).  You can right click and save them, then import them to art programs such as Adobe Illustrator. 

All the diagrams below involve the elliptic curve with label 389a,  given by the equation y^2 + y = x^3 + x^2 - 2x.
E = EllipticCurve('389a') print latex(E) 
        
y^2 + y = x^3 + x^2 - 2x 
y^2 + y = x^3 + x^2 - 2x
show(E) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}y^2 + y = x^3 + x^2 - 2x 
\newcommand{\Bold}[1]{\mathbf{#1}}y^2 + y = x^3 + x^2 - 2x

This elliptic curve has rank 2 with generators (-1,1) and (0,0):
P, Q = E.gens(); Q=-Q; P,Q 
        
((-1 : 1 : 1), (0 : 0 : 1))
((-1 : 1 : 1), (0 : 0 : 1))
P + Q 
        
(1 : 0 : 1)
(1 : 0 : 1)

The diagram below illustrates the "group law on the elliptic curve" by adding P and Q to get (1,0).  This is a precise actual example of this, sort of like what is on the cover over Silverman-Tate.  I think it is better than that cover, since this precise, where as that is simply "artistic" but not genuine. 
G = E.plot(thickness=3,plot_points=500,alpha=.7) ps = 125 G += point([P[0],P[1]], pointsize=ps, color='red') G += point([Q[0],Q[1]], pointsize=ps, color='darkgreen') R = P+Q; mR = -R G += point([mR[0],mR[1]], pointsize=ps, color='#ffa80a') G += point([R[0],R[1]], pointsize=ps, color='#ff2b0a') G += line([(1,-2),(1,1.5)], color='grey') G += text("$P$", (-1,1.3), color='black', fontsize=18) G += text("$Q$", (0.15,0.2), color='black', fontsize=18) G += text("$P+Q$", (1.4,0.25), color='black', fontsize=18) G += text("$P*Q$", (1.4,-1), color='black', fontsize=18) G += line([(-1.5,1.5), (1.5,-1.5)], color='grey') G.show(xmax=1.5,ymin=-2,ymax=1.5, gridlines=True) G.save('elliptic.pdf',xmax=1.5,ymin=-2,ymax=1.5, gridlines=True) 
       

Next we move to the Sato-Tate distribution. First we precompute some stuff which takes about 1.5 minutes.
time pr = [2*math.sqrt(float(p)) for p in prime_range(10^7)] 
        
Time: CPU 7.12 s, Wall: 7.63 s
Time: CPU 7.12 s, Wall: 7.63 s
time v = E.aplist(10^7) 
        
Time: CPU 52.51 s, Wall: 60.79 s
Time: CPU 52.51 s, Wall: 60.79 s

The following sequence of images illustrates the Sato-Tate conjecture for the above-plotted elliptic curve.  The convergence is terrible at first, because the curve has "high rank", but it gets much better. (This is a proven conjecture, due to work of Richard Taylor.)
t = stats.TimeSeries([float(v[i])/pr[i] for i in range(prime_pi(10^2))]) a = t.plot_histogram(bins=20)+ plot(sqrt(1-x^2)*(2/pi), (x,-1,1), color='red') a.show(frame=True, aspect_ratio=1) a.save('st2.pdf', frame=True, aspect_ratio=1) 
       
t = stats.TimeSeries([float(v[i])/pr[i] for i in range(prime_pi(10^3))]) a = t.plot_histogram(bins=30)+ plot(sqrt(1-x^2)*(2/pi), (x,-1,1), color='red') a.show(frame=True, aspect_ratio=1) a.save('st3.pdf',frame=True, aspect_ratio=1) 
       
t = stats.TimeSeries([float(v[i])/pr[i] for i in range(prime_pi(10^4))]) a = t.plot_histogram(bins=50)+ plot(sqrt(1-x^2)*(2/pi), (x,-1,1), color='red') a.show(frame=True, aspect_ratio=1) a.save('st4.pdf',frame=True, aspect_ratio=1) 
       
t = stats.TimeSeries([float(v[i])/pr[i] for i in range(prime_pi(10^5))]) a = t.plot_histogram(bins=100)+ plot(sqrt(1-x^2)*(2/pi), (x,-1,1), color='red') a.show(frame=True, aspect_ratio=1) a.save('st5.pdf',frame=True, aspect_ratio=1) 
       
t = stats.TimeSeries([float(v[i])/pr[i] for i in range(prime_pi(10^6))]) a = t.plot_histogram(bins=300) + plot(sqrt(1-x^2)*(2/pi), (x,-1,1), color='red') a.show(frame=True, aspect_ratio=1) a.save('st6.pdf',frame=True, aspect_ratio=1) 
       
t = stats.TimeSeries([float(v[i])/pr[i] for i in range(prime_pi(10^7))]) a = t.plot_histogram(bins=300) +plot(sqrt(1-x^2)*(2/pi), (x,-1,1), color='red') a.show(frame=True, aspect_ratio=1) a.save('st7.pdf',frame=True, aspect_ratio=1) 
       
 
       
 
       
 
       

Much (most) of Tate's work was in the p-adic context.  The following are diagrams that are supposed to illustrate the p-adic numbers graphically. Reference: Cuoco, A. ‘’Visualizing the p -adic Integers’‘, The American Mathematical Monthly, Vol. 98, No. 4 (Apr., 1991), pp. 355-364
Z = Zp(7) a = Z.plot() a.show() a.save('padic.pdf')