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I need help with the following problem.

"Let $C : y^2 = x^3 − 5x^2 + 6x$ be a cubic curve with the standard group law. Find a meromorphic function on $C$ having the pole of order two at $B=(1,\sqrt{2})$ and one of the zeros at $A=(0,0)$."

If $C$ is given as $\mathbb{C}/\Lambda$, I can construct the associated Weierstrass's $\wp$ function and use Abel's theorem to construct a meromorphic function with prescribed poles and zeroes. Unfortunately, I couldn't use that in the problem above because I cannot calculate two things I would additionally need: the periods of the $\wp$ function, and the Abel-Jacobi map.

user145836
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2 Answers2

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How about $x/(x-1)^2$ or $y/(x-1)^2$? If you want a simple zero at $(0,0)$, then you have to be careful. Similarly, you need to calculate the divisor carefully to make sure that you have a double pole at $(1,\sqrt{2})$ and not a pole of higher order. See this answer for a calculation of the divisors of the coordinate functions.

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Álvaro Lozano-Robledo
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  • Shouldn't the function be doubly periodic (because it is a function on a torus)? Also, by Abel's theorem, it should have a zero at $C$ such that $2B=A+C$ (addition here is the group law on the curve) which is in this case $(\frac{48}{25},−\frac{72}{125\sqrt{2}})$. – user145836 Apr 26 '14 at 19:07
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Tangent at $B$ is $t_B: y-\frac{5}{2\sqrt{2}}+x\frac{1}{2\sqrt{2}}=0$. The other point of intersection of $t_B$ and the elliptic curve is $C(\frac{25}{8},-5\sqrt{2})$. Line connecting $A$ and $C$ is $l:y+x\frac{8\sqrt{2}}{5}=0$. The desired meromorphic function will be the quotient of the LHS of $l$ and LHS of $t_B$:

$f(x,y)=\frac{32x+10\sqrt{2}y}{5x+10\sqrt{2}y-25}$

user145836
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