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I am in my final year of my undergraduate degree, and I'm doing a project on p-adic numbers, and in particular, trying to find Galois groups of simple extensions of $\mathbb{Q}_p$ (this is a Galois theory course). I have developed and become familiar with the basic properties of valuations, local rings, and the topology/analysis/algebra of the p-adic rings & fields. I am considering pursuing this field further in graduate studies, but I've had no luck searching for open problem in this field, so I'm unsure if this is a useful field to pursue.

For research I am using the books by Koblitz and Gouvea, and then Local Fields by Cassels. I have not seen any mention of any specific points of modern research, other than a mention that $\mathbb{C}_p$ is much less well understood than $\mathbb{C}$ so I am wondering if someone here can help me out.

Thanks a lot. -Robbie

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The area of continuous representations of locally $p$-adic analytic groups (examples being $\mathbf{Z}_p$, $\mathbf{Q}_p$, $\mathrm{GL}_n(\mathbf{Q}_p)$, the unit group of a finite-dimensional division algebra over $\mathbf{Q}_p$, etc..) on locally convex topological vector spaces over $\mathbf{Q}_p$ has been the subject of great research interest in recent years (and this continues today). This theory was initiated by Peter Schneider and Jeremy Teitelbaum in the early 2000's, who introduced and studied various classes of nice representations. Aside from its own intrinsic interest, the theory is also interesting to number theorists for its usefulness in formulating the $p$-adic (local) Langlands correspondence for $\mathrm{GL}_2(\mathbf{Q}_p)$, one of the great achievements in arithmetic-algebraic geometry and representation theory of the last few years.

This area, like many areas of modern number theory, involves a really cool mix of ideas and tools from topology, representation theory, $p$-adic functional analysis, and non-Archimedean analytic and algebraic geometry.

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A nice source of open problems can be found by reading about preperiodic points.

For example, here is a search in Google scholar.

One of the places in which p-adic analysis and p-adic dynamics comes up is in the various conjectures dating back to Morton and Silverman's (1994) boundedness conjecture based on an earlier theorem of Northcott. (p-adic analysis is also a key feature in Bernard Dwork's proof of the rationality part of the Weil conjectures; you are probably familiar with this fact if you are reading Koblitz - if not, see an earlier post here.)

More precisely, one can generalize fixed points to periodic points, periodic points to preperiodic points, and then generalize to a canonical height function on the rationals that gives zero whenever the point is preperiodic.

Noting that Northcott's result concerns a finite number of rational preperiodic points for a polynomial, one could (as Morton and Silverman do) conjecture that this finite number can be bounded based on the degree of the polynomial. In keeping with the generalization above, one might offer similar conjectures about lower bounds on the so-called "canonical height function," also in terms of a polynomial's degree.

For some specifics and more references, check out the start of this paper on canonical heights, local canonical heights, and computations done with cubic polynomials to provide "evidence" regarding conjectures related to the material touched upon here.

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