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Take a separable cubic polynomial $4x^3-ax-b = 4 (x-e_1)(x-e_2)(x-e_3)$, let $h'(x) = (4x^3-ax-b)^{-1/2}$ and define its elliptic integral $h(x)= \int h'(x)dx$. Let $P(z) = h^{-1}(z)$ its inverse function. Then $\displaystyle P'(z) = \frac{1}{h'(P(z))}$ and $$P'(z)^2 = P(z)^3-a P(z)-b \tag{1}$$

  • For $z,x \in \mathbb{C}$ this is the definition of the Weierstrass function $\wp$ of the complex elliptic curve $$E(\mathbb{C}) = \{ (x,y) \in \mathbb{C}^2, y^2 = 4x^3-ax-b\}$$

    Question 1 : How to show easily that $P$ is doubly periodic ?

    Take a $c \in\mathbb{C}$ with $P(c) \ne 0$ and a closed-curve $\gamma : P(c) \to P(c)$ enclosing one of the root $e_i$. Then $h \circ \gamma$ is a non-closed curve $c \to c+\omega$ and we find $$0 = \int_\gamma dx = \int_{h \,\circ\, \gamma} P'(z)dz = P(c+\omega)-P(c)$$ Thanks to $(1)$ it implies $P'(c+\omega) = \pm P'(c)$. We can show the sign is $+$ (if it was not we could double $\omega$) and the differential equation shows that $P$ is $\omega$ periodic.

    Applying the same process with a curve enclosing a different root will produce a different period $\omega_2$ which is $\mathbb{Z}$-linearly independent to $\omega$ (why ?)

    So $P$ is doubly periodic and we obtain the (Riemann surface and abelian group) isomorphism with a complex torus $$\varphi : \mathbb{C}/(\omega \mathbb{Z}+ \omega_2\mathbb{Z}) \to E(\mathbb{C}), \qquad \varphi(z) = (P(z),P'(z))$$

  • Question 2 : Can we do the same with another (algebraically closed) field $K$ not contained in $\mathbb{C}$ and the corresponding elliptic curve over $K$ ? $K=\overline{\mathbb{F}}_p$ seems out of reach because it doesn't have an absolute value for making sense to analytic functions.

    What about the case $K=\overline{\mathbb{Q}}_p$ ? Algebraically, will $h$ be in some field of formal series, being the anti-derivative of $h' \in \overline{K(x)}$ ? And will its inverse function $P$ be well-defined ? In that case, does it tell us the structure of $E(\overline{\mathbb{Q}}_p)$ ?

reuns
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2 Answers2

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One way to do this is to prove there is a lattice $\Lambda$ in $\Bbb C$ whose $\wp$-function satisfies $$\wp'(z)^2=4\wp(z)^3-a\wp(z)-b.$$ Cox gives a proof in Primes of the form $x^2+ny^2$. This involves the $j$ modular function. It is a fact that $j$ is surjective, so there is $\tau$ with $$j(\tau)=1728\frac{a^3}{a^3-27b^2}.$$ Then a lattice of the form $\alpha\Bbb Z+\alpha\tau\Bbb Z$ works.

As $P$ and $\wp$ satisfy the same differential equation and their Laurent series at $0$ start off the same, then their Laurent series are the same, so $P=\wp$ by analytic continuation. As $\wp$ is periodic, so is $P$.

Angina Seng
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  • Yes, but the problem is this doesn't generalize in any other field. – reuns Aug 17 '17 at 10:23
  • The problem in finite fields is that Laurent series like $\wp$ really want to have big factorials in their denominators... @reuns – Angina Seng Aug 17 '17 at 10:25
  • I tried an argument for the periodicity in my question, which seems consistent with the last lines of this. Also are you saying (possibly) the Laurent series of $P$ won't depend on the field (so it can't exist in some characteristics) ? – reuns Aug 17 '17 at 14:04
  • Hi do you know the chapter V of Silverman's "advanced topics of elliptic curves" about the Tate curve over $\mathbb{Q}_p$ ? – reuns Aug 27 '17 at 23:18
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See chapter V of Silverman's "advanced topics of elliptic curves" about the Tate curve over $\mathbb{Q}_p$.

One of the steps is to use $z \mapsto e^{2i \pi z}$ to replace $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$ by $\mathbb{C}^* / e^{2i \pi \tau \mathbb{Z}}$ which generalizes to $K^*/q^\mathbb{Z}$ for complete fields $K$. When $K$ is a finite extension of $\mathbb{Q}_p$ it works well and we obtain a $p$-adic Weierstrass function.

reuns
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