Tamagawa products for elliptic curves over number fields (Q6642854)

Let \(E\) be an elliptic curve defined over a number field \(K\) and, for each prime \(\mathfrak{p}\) in \(K\), let \(c_{\mathfrak{p}}\) be the Tamagawa number at \(\mathfrak{p}\). Define the Tamagawa product for \(E\) as \(\operatorname{Tam}(E/K)=\prod_{\mathfrak{p}} c_{\mathfrak{p}}\). Let \(P_{\operatorname{Tam}}(K,m)\) be the proportion of elliptic curves in short Weierstrass form defined over \(K\) (and ordered by naive height) with Tamagawa product equal to \(m\), and let \(L_{\operatorname{Tam}}(K,s)=\sum_m P_{\operatorname{Tam}}(K,m)s^{-m}\). The value of \(P_{\operatorname{Tam}}(K,1)\), that is the probability that an elliptic curve has Tamagawa product equal to \(1\), and the value of \(L_{\operatorname{Tam}}(K,-1)\), that is the average of the Tamagawa products, were already known in the case \(K=\mathbb{Q}\) and the authors generalize that result to all number fields. In particular, they construct Markov chains to compute the exact values of \(P_{\operatorname{Tam}}(K,m)\) for all number fields \(K\) and positive integers \(m\). They also uniformly bound \(P_{\operatorname{Tam}}(K,1)\) and \(L_{\operatorname{Tam}}(K,-1)\) in terms of the degree of \(K\). Finally, they confirm a conjecture by Ono, showing that there exist sequences of \(K\) for which \(P_{\operatorname{Tam}}(K,1)\) tends to \(0\) or to \(1\), and for which \(L_{\operatorname{Tam}}(K,-1)\) tends to \(0\) or to \(\infty\).