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arXiv 2108.13625math.NT
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数域上椭圆曲线的玉河积

Tamagawa Products for Elliptic Curves Over Number Fields

Yunseo Choi, Sean Li, Apoorva Panidapu, Casia Siegel

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40 pages, 9 figures, 17 tables
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AI总结：

本文将Griffin等人的玉河积L-级数推广至任意数域，通过构造马尔可夫链精确计算玉河积分布，并给出了分布比例与平均值的界及渐近序列。

摘要：

在近期的工作中，Griffin, Ono和Tsai构造了一个$L-$级数，证明了$\mathbb{Q}$上具有平凡玉河积的短Weierstrass椭圆曲线的比例为$0.5054\dots$，且平均玉河积为$1.8183\dots$。继其工作之后，我们将其$L-$级数推广到任意数域$K$上，定义为\[L_{\mathrm{Tam}}(K; s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(K; m)}{m^s},\]其中$P_{\mathrm{Tam}}(K;m)$是$K$上玉河积为$m$的短Weierstrass椭圆曲线的比例。随后，我们构造了马尔可夫链来计算所有数域$K$和正整数$m$对应的$P_{\mathrm{Tam}}(K;m)$的精确值。作为推论，我们还计算了平均玉河积$L_{\mathrm{Tam}}(K;-1)$。接着，我们利用这些结果，用$K$的次数对$P_{\mathrm{Tam}}(K;1)$和$L_{\mathrm{Tam}}(K,-1)$进行了一致界定。最后，我们证明了存在使得$P_{\mathrm{Tam}}(K;1)$趋于$0$且$L_{\mathrm{Tam}}(K;-1)$趋于$\infty$的$K$的序列，同时也存在使得$P_{\mathrm{Tam}}(K;1)$和$L_{\mathrm{Tam}}(K;-1)$均趋于$1$的$K$的序列。

英文摘要：

In recent work, Griffin, Ono, and Tsai constructs an $L-$series to prove that the proportion of short Weierstrass elliptic curves over $\mathbb{Q}$ with trivial Tamagawa product is $0.5054\dots$ and that the average Tamagawa product is $1.8183\dots$. Following their work, we generalize their $L-$series over arbitrary number fields $K$ to be \[L_{\mathrm{Tam}}(K; s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(K; m)}{m^s},\] where $P_{\mathrm{Tam}}(K;m)$ is the proportion of short Weierstrass elliptic curves over $K$ with Tamagawa product $m$. We then construct Markov chains to compute the exact values of $P_{\mathrm{Tam}}(K;m)$ for all number fields $K$ and positive integers $m$. As a corollary, we also compute the average Tamagawa product $L_{\mathrm{Tam}}(K;-1)$. We then use these results to uniformly bound $P_{\mathrm{Tam}}(K;1)$ and $L_{\mathrm{Tam}}(K,-1)$ in terms of the degree of $K$. Finally, we show that there exist sequences of $K$ for which $P_{\mathrm{Tam}}(K;1)$ tends to $0$ and $L_{\mathrm{Tam}}(K;-1)$ to $\infty$, as well as sequences of $K$ for which $P_{\mathrm{Tam}}(K;1)$ and $L_{\mathrm{Tam}}(K;-1)$ tend to $1$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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