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 $1$.
Categories: math.NT