AI 中文总结
本文为短Weierstrass形式的椭圆曲线推导了编码Tamagawa积的Euler积生成函数，计算了四个统计量的极限分布与矩界，并证明多数曲线Tamagawa积平凡，结果由AxiomProver在Lean中自动形式化。
AI 中文摘要
对于短Weierstrass形式 $E=E(a_4,a_6): y^2=x^3+a_4x+a_6$ 的椭圆曲线 $E/\mathbb{Q}$，我们推导出一个多变量Euler积生成函数，该函数编码了Tamagawa积 $Tam(E)=\prod_p c_p(E)$。利用此生成函数，我们计算了Tamagawa数上四个重要统计量的极限分布、精确协方差以及矩和尾部界；例如，我们证明在此高度排序中，超过一半的曲线具有平凡的Tamagawa积，约$42.2\\%$的曲线恰好有一个素数具有非平凡的局部Tamagawa数，而仅有约$6.8\\%$的曲线恰好有两个这样的素数。该乘积是通过特化一个由Tate算法导出的、以局部约化数据为索引的Euler积而得到的。本文的结果由AxiomProver在Lean中自动形式化。
英文摘要
For elliptic curves $E/\mathbb{Q}$ in short Weierstrass form \[ E=E(a_4,a_6): y^2=x^3+a_4x+a_6, \] we derive a multivariable Euler product generating function which encodes the Tamagawa product $Tam(E)=\prod_p c_p(E)$. Using this generating function, we compute limiting distributions, exact covariances, and moment and tail bounds for four important statistics on the Tamagawa number; for example we show that more than half of all curves in this height ordering have trivial Tamagawa product, about $42.2\%$ have exactly one prime with nontrivial local Tamagawa number, and only about $6.8\%$ have exactly two such primes. The product is obtained by specializing an Euler product indexed by local reduction data that we derive from Tate's algorithm. The results in this paper were autoformalized in Lean by AxiomProver.
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