Curves over C and Abel at 26

Date: January 26, 2024

In this seminar, I discussed some interesting things I learned while preparing for my comprehensive exam.

The focus was on the analytic theory of curves we encounter in number theory, closely related to my lecture on the Poncelet Closure Theorem. Niels Abel, who died of tuberculosis at 26 in 1829, was the hero of my talk. In 1824, Abel sent a paper on the unsolvability of the quintic equation to Gauss, who proceeded to discard without a glance what he believed to be the worthless work of a crank. Later, in 1825, Abel traveled to Paris, where he presented his paper revealing the double periodicity of the elliptic functions. However, Cauchy proceeded to misplace the manuscript. Finally, two days after his death, a letter from Crelle arrived informing him that he had been offered a professorship at the University of Berlin. :/

Outline

Elliptic curve

Three Fermat Trails to Elliptic Curves - Ezra Brown: Local copy and Stable link
Why Ellipses are not elliptic curves - Adrian Rice and Ezra Brown: Local copy and Stable link
Plotting elliptic curves over $\mathbb{C}$ - Ricardo Acuna: Local copy and Stable link
Visualizing complex points of elliptic curves - Sebastian Bozlee and Silviana Violet Amethyst: Local copy and Stable link
Visualizing Elliptic Curves and Their Tori - Galen Ballew and James Duncan: Local copy and Stable link
Abel’s Theorem on the Lemniscate - Michael Rosen: Local copy and Stable link
The lemniscate and Abel’s discovery of complex multiplication for elliptic curves - Christian Skau: Local copy and Stable link
The first part of Abel’s “Recherches sur les fonctions elliptiques” - Marcus Emmanuel Barnes: Local copy and Stable link.
Elliptic functions and elliptic curves - Jan Nekovář: Local copy and Stable link

Modular curve

The modular curves $X_0(11)$ and $X_1(11)$ - Tom Weston: Local copy and Stable link
Modular Curves, Hecke Correspondences, and L-Functions - David Rohrlich: Local copy and Stable link
Elliptic and modular curves over finite fields and related computational issues - Noam Elkies: Local copy and Stable link

Plotting the modular polynomial $\Phi_2(X,Y)$ over $\mathbb{C}$ - Ricardo Acuna: Local copy and Stable link

Whole code in Sage: https://sagecell.sagemath.org/

P2.<X,Y,Z> = ProjectiveSpace(QQ, 2)
Phi2 = -X^2*Y^2 + X^3 + 1488*X^2*Y + 1488*X*Y^2 + Y^3 - 162000*X^2 + 40773375*X*Y - 162000*Y^2 + 8748000000*X + 8748000000*Y - 157464000000000
CC = Curve(Phi2.homogenize('Z'), P2)
#CC.is_singular()
#AA = CC.affine_patch(2) 
#AA.is_singular()
#plot(AA, (X,-10,20000),(Y,-10,20000), axes = True)
#AA.is_ordinary_singularity([-3375,-3375])
#CC.genus()
CC.rational_parameterization() 
s = var('s')
X = (s^3 + 768*s^2 + 196608*s + 16777216)/(s^2)
Y = (s^4 + 48*s^3 + 768*s^2 + 4096*s)/(s^2)
X = X.numerator().factor()/X.denominator()
Y = Y.numerator().factor()/Y.denominator()
s = lambda u,v: (u+i*v)
X = lambda u,v: ((s(u,v) + 256)^3/s(u,v)^2)
Y = lambda u,v: ((s(u,v) + 16)^3/s(u,v))
G0 = (lambda u,v: X(u,v).real(), lambda u,v: X(u,v).imag(), lambda u,v: Y(u,v).real())
cf0 = lambda u,v: Y(u,v).imag() - floor(Y(u,v).imag())
cm0 = colormaps.hsv
pp1 = parametric_plot3d(G0,(-1000,-0.1),(-1000,-0.1),color = (cf0,cm0))
pp1 = pp1.add_condition(lambda X,Y,Z: abs(X)^2+abs(Y)^2+abs(Z)^2 < (20000)^2)
pp2 = parametric_plot3d(G0,(0.1,1000),(0.1,1000),color = (cf0,cm0))
pp2 = pp2.add_condition(lambda X,Y,Z: abs(X)^2+abs(Y)^2+abs(Z)^2< (20000)^2)
show(pp1+pp2, frame=false, viewpoint=[[-0.9521,-0.2128,-0.2196],91.01])

Abelian variety

Abelian Varieties and Moduli - Donu Arapura: Local copy and Stable link
What is Abel’s Theorem Anyway? - Steven Kleiman: Local copy and Stable link
Abel’s Memoir - Harold Edwards: Local copy and Stable link

Jacobian variety

Curves and their Jacobians - David Mumford: Local copy and Stable link
Motivation for the Jacobian Variety - Damien Robert: Local copy and Stable link
What does the Jacobian of a curve tell us about the curve? - Donu Arapura: Local copy and Stable link
For what algebraic curves do rational points form a group? - Franz Lemmermeyer: Local copy and Stable link
- Conics, a Poor Man’s Elliptic Curves - Franz Lemmermeyer: Local copy and Stable link
- Generalized Jacobians in cryptography - Isabelle Déchène: Local copy and Stable link

Hyperelliptic curve

An elementary introduction to hyperelliptic curves - Alfred Menezes, Yi-Hong Wu, and Robert Zuccherato: Local copy and Stable link
Hyperelliptic curves - Steven Galbraith: Local copy and Stable link
On Abel’s hyperelliptic curves - Torsten Ekedahl: Local copy and Stable link
Visualizing complex points of hyperelliptic curves - Donu Arapura: Local copy and Stable link
- For example, $X_0(23)$ is a hyperelliptic curve - René Schoof: Local copy and Stable link
- One way to get two dimensional abelian varieties is as Jacobians of genus two curves. That is, $J_0(23)$.
- However, for genus 2 curves we can’t have a non-singular model in the complex projective plane. Therefore, we generally use a plane hyperelliptic equation which represents the image of a projection $X_0(23)$ with a single singularity at infinity.
```
R.<x> = QQ[] #polynomial ring
H = HyperellipticCurve(x^6 - 8*x^5 + 2*x^4 + 2*x^3 - 11*x^2 + 10*x - 7)
from sage.schemes.curves.projective_curve import ProjectivePlaneCurve
ProjectivePlaneCurve.is_singular(H)
from sage.schemes.curves.affine_curve import AffineCurve
AffineCurve.is_singular(H.affine_patch(2))
plot(H) #affine plot over real numbers instead of rationals
```

Gaurish Korpal