Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves
Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves
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对椭圆曲线的秩、Selmer 群和 Shafarevich-Tate 群的分布进行建模
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发表时间：
2013
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通讯作者：
E. Rains
中科院分区：
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作者：
M. Bhargava;D. Kane;H. Lenstra;B. Poonen;E. Rains
利用Z_p上二次模中的极大迷向子模，证明了余有限型Z_p-模的短正合序列的同构类集合上的自然离散概率分布的存在性，进而猜想当E在固定整体域k上的椭圆曲线上变化时，
0 --> E（k）张量Q_p/Z_p --> Sel_{p^infty} E --> Sha[p^infty] --> 0就是那个。我们表明，这个单一的猜想将解释许多已知的定理和代数的行列，塞尔默群，和Shafarevich-泰特群的椭圆曲线。我们还证明了有限交换p-群的同构类集合的离散概率分布的存在性，该有限交换p-群的同构类集合具有非退化交替配对，定义为Z_p上随机交替矩阵的上核，并且我们证明了这两个概率分布彼此相容，并且与Sha的Delaunay预测分布相容。最后，我们证明了椭圆曲线的fppf上同调的新定理，为我们的猜想提供了进一步的证据。
Using maximal isotropic submodules in a quadratic module over Z_p, we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short exact sequences of co-finite type Z_p-modules, and then conjecture that as E varies over elliptic curves over a fixed global field k, the distribution of
0 --> E(k) tensor Q_p/Z_p --> Sel_{p^infty} E --> Sha[p^infty] --> 0 is that one. We show that this single conjecture would explain many of the known theorems and conjectures on ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves. We also prove the existence of a discrete probability distribution of the set of isomorphism classes of finite abelian p-groups equipped with a nondegenerate alternating pairing, defined in terms of the cokernel of a random alternating matrix over Z_p, and we prove that the two probability distributions are compatible with each other and with Delaunay's predicted distribution for Sha. Finally, we prove new theorems on the fppf cohomology of elliptic curves in order to give further evidence for our conjecture.