Complex multiplication

Elliptic curves have an endomorphism ring larger than the integers, which are higher dimensions of abelian varieties.

C/ΘZ[i]

C is complex multiplication, and Z[i] is the Gaussian integer ring, and θ is any non-zero complex number.

You can see always the Gaussian integer as module in Elliptic curves.

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Y^2=X^3-X+1

59 is the Gaussian prime.

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The Gaussian prime must be symmetry in any Elliptic curves.

2=(1-i)(1+i)=(-1-i)(-1+i)

This is the norm. 1±i,-1±i are Gaussian primes.

±3,±3i,5=(2-i)(2+i)=(1-2i)(1+2i),±7,±7i,13=(2+3i)(2-3i)=(3+2i)(3-2i),17=(4-i)(4+i)=(1+4i)(1-4i)

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Y^2=4X^3-X

59 is the Gaussian prime.

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I think that it is fractal.

Gaussian primes

This is quite tricky. Modulo primes in Y^2=X^3-X are sometimes primes.

p-2sqrt(p) < N < p+2sqrt(p)

p is prime, and sqrt is √

When p is 19, N is 19.
p=23 - N=23
p=29 - N=39

Carl Friedrich Gauss proved it. Moreover, p=N is Gaussian primes such as 19 and 23.

This is Gaussian integers which are complex numbers. Elliptic curve is rational, so this is contradictory but prime is real.

5=(2+i)(2-i)

You can divide 5 in Z[i], so 5 isn't prime. However, 5 is prime in the real world.

This is Fermat's theorem on sums of two squares.

p≡1 (mod 4)

ex. 5,13=(2+3i)(2-3i),17=(1+4i)(1-4i),29=(2+5i)(2-5i)

When 3=αβ, this isn't also prime in Z[i].

N is the norm.

N(a+bi):=a^2+b^2=(a+bi)(a-bi)
N(αβ)=N(α)N(β)

3≠(a+bi)(a-bi)

Therefore, 3 is prime in Z[i].

4n+3 is Gaussian primes such as 3,7,11,19,23,31,43,47,59,67,71,79,83. This is in Y^2=X^3-X.

E(N)=G(p)≒4n+3≠(a+bi)(a-bi)

E(N) is the sum of the same modules in Y^2=X^3-X. G(p) is Gaussian primes.



You see COUNTIF(mod(y^2),mod(x^3-x)) which is p=N.

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