Papers › Two-Torsion Subgroups of some Modular Jacobians
Two-Torsion Subgroups of some Modular Jacobians
Elvira Lupoian
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We give a practical method to compute the 2-torsion subgroup of the Jacobian of a non-hyperelliptic curve of genus $3$, $4$ or $5$. The method is based on the correspondence between the 2-torsion subgroup and the theta hyperplanes to the curve. The correspondence is used to explicitly write down a zero-dimensional scheme whose points correspond to elements of the $2$-torsion subgroup. Using p-adic or complex approximations (obtained via Hensel lifting or homotopy continuation and Newton-Raphson) and lattice reduction we are then able to determine the points of our zero-dimensional scheme and hence the $2$-torsion points. We demonstrate the practicality of our method by computing the $2$-torsion of the modular Jacobians J₀( N ) for N = 42, 55, 63, 72, 75. As a result of this we are able to verify the generalised Ogg conjecture for these values.
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