Tamagawa Products for Elliptic Curves Over Number Fields
Yunseo Choi, Sean Li, Apoorva Panidapu, Casia Siegel

TL;DR
This paper generalizes the study of Tamagawa products for elliptic curves from rational numbers to arbitrary number fields, providing exact computations, bounds, and asymptotic behaviors of Tamagawa-related quantities.
Contribution
It extends the construction of Tamagawa-related $L$-series to all number fields and develops Markov chains for exact computation of Tamagawa proportions.
Findings
Exact values of Tamagawa proportions for all number fields.
Bounds on Tamagawa proportions and averages in terms of degree.
Existence of sequences of number fields with extreme Tamagawa behaviors.
Abstract
In recent work, Griffin, Ono, and Tsai constructs an series to prove that the proportion of short Weierstrass elliptic curves over with trivial Tamagawa product is and that the average Tamagawa product is . Following their work, we generalize their series over arbitrary number fields to be \[L_{\mathrm{Tam}}(K; s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(K; m)}{m^s},\] where is the proportion of short Weierstrass elliptic curves over with Tamagawa product . We then construct Markov chains to compute the exact values of for all number fields and positive integers . As a corollary, we also compute the average Tamagawa product . We then use these results to uniformly bound and in terms of the degree of…
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Taxonomy
TopicsAnalytic Number Theory Research · Algebraic Geometry and Number Theory · Vietnamese History and Culture Studies
