论文标题
一些模块化雅各布人的两个扭转亚组
Two-Torsion Subgroups of some Modular Jacobians
论文作者
论文摘要
我们提供了一种实用方法来计算雅各比族的2个扭转子群，该属的属属属属属于$ 3 $，$ 4 $或$ 5 $。该方法基于2个扭矩亚组与曲线的Theta超平面之间的对应关系。该通信用于明确写下一个零维方案，该方案对应于$ 2 $ - torsion子组的元素。使用$ p $ - adic或复杂的近似值（通过Hensel提升或同型连续性以及Newton-Raphson获得）和降低晶格，然后可以确定零维方案的点，从而确定$ 2 $ torsion的点。我们通过计算模块化jacobians $ j_ {0} \ left（n \ right）$的$ 2 $ torsion来证明我们方法的实用性。结果，我们能够为这些值验证广义OGG猜想。
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 $J_{0}\left( N \right)$ for $N = 42, 55, 63, 72, 75$. As a result of this we are able to verify the generalised Ogg conjecture for these values.
