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I have mainly concentrated on Part b) on exercise 9.13. What does it mean for a point to be not on the identity component (essentially nowhere else in the book - Silverman's - is this term mentioned). And how does exercise 9.12 (I guess part a)) conclude the hint in 9.13 b)?

Any help appreciated!

DesmondMiles
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  • $E({\mathbb R})$ will not be connected. Silverman is saying that $P$ is not on the same component as $O$ (the point at infinity). – peter a g Mar 24 '20 at 00:25
  • @peterag What precisely is the component of the point at infinity? I see on a sketch that $E(\mathbb{R})$ consists of a closed and an ''open'' curve. – DesmondMiles Mar 24 '20 at 00:29
  • And how does one find all integer points on the non-identity component? – DesmondMiles Mar 24 '20 at 00:31
  • The identity component consists of the open curve of your graph, along with the point at infinity. On a real Lie (not necessarily connected) group, the component with the identity is again a (connected) Lie group [and is a normal subgroup]. For instance, the disconnected ${\rm GL}_2({\mathbb R})$ has identity component consisting of the matrices of (strictly) positive det. (Actually, people sometimes require in the definition of real Lie group that it be connected, but...) As for your 2nd question (in comments), - e.g. complete square on the LHS, etc?, to see that $x =-1$ or $x=0$. – peter a g Mar 24 '20 at 01:25

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