What are some interesting problems in the intersection of Diophantine Approx and Algebraic Geometry?

Question

I am a first year graduate student and I am eager to work on irrationality/transcendental proofs of specific numbers like Euler's constant gamma. Because backgrounds for Elliptic Curves include very much of Algebraic Geometry but there is no professor working on Transcendental NT, Analytic NT or Elliptic Curves, so I was wondering is there a subject in the intersection of irrationality/transcendental proofs of specific numbers (or more generally Diophantine approx) with Algebraic Geometry so that I can write a proposal for the professor who works in Algebraic Geometry? I couldn't find any on the Internet.

Answer 1

Such questions may be quite hard in general.

But since you are asking, and you couldn't find anything on the internet, here is something. There is an open conjecture whether or not $\zeta(2n+1)$ is irrational for $n\ge 2$. Here we have results of Zudilin in this direction, which you could start looking at. This has also intersections with algebraic and arithmetic geometry.

You might perhaps also be interested in Schikhof's conjecture, which is less known. Consider the $p$-adic series $\sum_{n=1}^{\infty}n!$. It converges in $\Bbb Q_p$ since $|n!|_p\to 0$ for $n\to \infty$. The conjecture is, that the limit is a $p$-adic irrational number in $\Bbb Q_p$ for all primes $p$.