# 2-dimensional absolutely irreducible $p$-adic Galois reps

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.

• math.stanford.edu/~conrad/papers/notes.pdf – Stanley Yao Xiao May 11 '19 at 19:41
• @StanleyYaoXiao which part, exactly? – user138661 May 11 '19 at 19:42
• 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. – Stanley Yao Xiao May 11 '19 at 19:43
• @StanleyYaoXiao oh well, problem solved then. Thanks. You can post it as an answer, so this is not in the unanswered queue. – user138661 May 11 '19 at 19:44

## 1 Answer

See these notes on Hodge theory, Th. 8.3.6.