1. Tamagawa Products of Elliptic Curves Over ℚ.
- Author
-
Griffin, Michael, Tsai, Ken Onowei-Lun, and Tsai, Wei-Lun
- Subjects
ELLIPTIC curves ,DIRICHLET series - 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. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF