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Aug 24, 2016 at 6:42 comment added user37663 Thanks for the clarification. I just need some time to understand. I will try one more time... :)
Aug 23, 2016 at 21:18 comment added Joe Silverman @PraphullaKoushik Sorry for the ambiguity, it's the usual Brauer group $\text{Br}(K)=H^2(G_{\overline K}/K,\overline K^*)$. One gets a pairing from on $III(E/K)$ that takes values in $\text{Br}(K)$. Hmmm...,no actually, that's not quite right, either, one gets values in $\mathbb Q/\mathbb Z$ by getting an element of the local Brauer groups and adding up their invariants. Anyway, it's not an elliptic curve Brauer group.
Aug 23, 2016 at 19:08 comment added user37663 I am happy that my question has got your interest.. I am not familiar with elliptic curves brauer groups... I will come back and rtry to understand this once I am ready... Thank you
Aug 23, 2016 at 17:22 history answered Joe Silverman CC BY-SA 3.0