Tamagawa Products of Elliptic Curves Over ℚ

IF 0.6 4区 数学 Q3 MATHEMATICS
Michael Griffin;Ken Onowei-Lun Tsai;Wei-Lun Tsai
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引用次数: 1

Abstract

We explicitly construct the Dirichlet series $$\begin{equation*}L_{\mathrm{Tam}}(s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(m)}{m^s},\end{equation*}$$ where $P_{\mathrm{Tam}}(m)$ is the proportion of elliptic curves $E/\mathbb{Q}$ in short Weierstrass form with Tamagawa product m. Although there are no $E/\mathbb{Q}$ with everywhere good reduction, we prove that the proportion with trivial Tamagawa product is $P_{\mathrm{Tam}}(1)={0.5053\dots}$ . As a corollary, we find that $L_{\mathrm{Tam}}(-1)={1.8193\dots}$ is the average Tamagawa product for elliptic curves over $\mathbb{Q}$ . We give an application of these results to canonical and Weil heights.
椭圆曲线上的Tamagawa积
我们显式构造了Dirichlet级数$$\begin{equation*}L_{\mathrm{Tam}}(s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(m)}{m^s},\end{equation*}$$,其中$P_{\mathrm{Tam}}(m)$是短Weierstrass形式的椭圆曲线$E/\mathbb{Q}$与Tamagawa积m的比例。虽然没有处处都好的约简$E/\mathbb{Q}$,但我们证明了与平凡Tamagawa积的比例为$P_{\mathrm{Tam}}(1)={0.5053\dots}$。作为推论,我们发现$L_{\mathrm{Tam}}(-1)={1.8193\dots}$是$\mathbb{Q}$上椭圆曲线的平均Tamagawa积。我们给出了这些结果在正则高度和韦尔高度上的应用。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
36
审稿时长
6-12 weeks
期刊介绍: The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. All major areas of pure mathematics are represented on the editorial board.
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