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Here the following is stated:

It's a basic fact in $p$-adic Hodge theory that any 2-dim. absolutely irreducible $G_{\mathbb Q_p}$-representation with distinct Hodge-Tate weights is uniquely determined by $a_p$.

Could somebody explain in maximum detail why is this true? Taking inspiration from the famous ELI5 communication style, explain like I am a generic first-year grad student.

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  • $\begingroup$ math.stanford.edu/~conrad/papers/notes.pdf $\endgroup$
    – Stanley Yao Xiao
    Commented May 11, 2019 at 19:41
  • $\begingroup$ @StanleyYaoXiao which part, exactly? $\endgroup$
    – user138661
    Commented May 11, 2019 at 19:42
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    $\begingroup$ Specifically, Theorem 8.3.6 and its proof. They really spell it out in full detail, to the point where I, as someone who was completely unfamiliar with $p$-adic Hodge theory, could understand after a few hours worth of work. $\endgroup$
    – Stanley Yao Xiao
    Commented May 11, 2019 at 19:43
  • $\begingroup$ @StanleyYaoXiao oh well, problem solved then. Thanks. You can post it as an answer, so this is not in the unanswered queue. $\endgroup$
    – user138661
    Commented May 11, 2019 at 19:44

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See these notes on Hodge theory, Th. 8.3.6.

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