I'm sure you know this, the third example isn't that much different, as changing the variety to GLn/B and the multiplicative group to $\mathbb{F}^\times _q {}^n$ gives a case of the first example

Ah, I was too quick to pull the trigger. My sequence is the cumulative product of Fermat numbers, which are pairwise coprime, and for some reason I thought they are all square free. I don't know how to prove that actually :(

If we replace everything by $a(n+1)=a(n)^2+2a(n),a(0)=1$, then I can prove it's square free.

Class field theory answers this question only for characters of 1-dimensional representations. Otherwise, we expect there to be an associated automorphic form, and whether it's a Maas form or a modular form depends on properties of the character (even or odd). The first major result in this direction was the Langlands-Tunnel theorem on the automorphicity of galois representations into $S_3$, and Wiles' theorem heavily relies on it. Since then the biggest result is Khare-Winterberger's theorem, aka Serre's modularity conjecture.

Just in case you haven't seen them, I'll quickly add that Song has a few nice YouTube videos from earlier this year about this research and paper, and there's also Ermon's IAS talk from two years ago.

$p_{data} $ just means the "empirical distribution", it's the finite uniform distribution over the samples we have. This the same as in normal MLE up to wording; you're probably familiar with the more common wording "find $\theta$ that maximizes $p(\text{data}|\theta)$". And yes to first and second questions, where "for all $x$" means all the samples in our data.

Yes, the idea is to maximize $\mathbb{E}_{x\sim p_{data}} \log(p_\theta(x)) $. So what we're really doing with the gradient ascent is taking the mean over gradients at all $x$ in our data

Given approximations to the gradient of log likelihood you can get local maxima of the log likelihood as a function of $\theta$ - start anywhere, and then do gradient accent. At no point do you actually know what the log likelihood is, but you know you're getting better (stochastically).

@Aurel Once 3 goes to infinity you can't dismiss the subsequent multiplications of the mod a/b/c inverses over mod n as Timothy Chow has, and then you see the complexity is still only a constant away. Not to mention there's a quasi linear version of Euclid's algorithm which is better applied as 3 goes to infinity and used in the OP's idea

For hyperelliptic curves (and their jacobians) there's an analogous statement about the Igusa invariants

From all the implicit usages, it's most explicit in Cartwright's proof, appearing on Wikipedia: en.m.wikipedia.org/wiki/…

@HenriCohen : nah, try increasing truncation degree and you'll see that zero disappear

@TomGoodwillie : that's what I would hope for. Can this be proved?

@Conrad : the compositional inverse can be defined from the power series, and I don't think it's a multivalued function. This is similar to log being multivalued but exp is not.