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Creators/Authors contains: "Onowei-Lun Tsai, Ken"

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  1. ; ; (, The Quarterly Journal of Mathematics)
    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. 
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