I've recently been thinking a bit about $L$ and $0 \sharp$.
As is well known, the existence of $0 \sharp$ is equivalent to the existence of a non-trivial elementary embedding $j: L \longrightarrow L$.
My question, does the critical point of said embedding have any large cardinal properties (e.g. is it inaccessible? Mahlo? It can't be measurable, Jonsson, or $\omega_1$-Erdos by Godel's Second.)?
I would guess not, if $0 \sharp$ exists then there are many indiscernibles in $L$, many of them countable. So we should be able to just move quite a small indiscernible as the critical point of the embedding. However, I'm also relatively new to the details here, and sensitive to the fact that my intuitions may be unreliable!
I'd also be interested to hear if $0 \sharp$ implies the existence (not just the consistency) of any large cardinals in $V$.
If there's any references that discuss this (other than the Jech, Kanamori, and Drake), I'd love to hear about them. Thanks for any pointers.
First, let's clear up the easy part.
Suppose $0^\#$ exists, then we can work in $L[0^\#]$, where it also exists. If there are any large cardinals let in that model (weak compact, inaccessible, even worldly cardinals) we can chop the universe at that cardinal, to have a large cardinals free universe where $0^\#$ exists just fine.
For the first question, recall Silver indiscernibles which are exactly the critical points of these embeddings, and you can pretty much show that if $\alpha$ is a Silver indiscernible then $L\models\alpha\text{ as large as it can be}$.
By indiscernability it suffices to prove to just one indiscernible, of course. So for example $\omega_1^V$ is a regular cardinal in $L$, and it is a Silver indiscernible, so all indiscernibles are $L$-regular. On the other hand, $\omega_\omega$ is a limit cardinal, but it's also indiscernible, so every other indiscernible is a limit cardinal. Therefore all of them are inaccessible cardinals.
Similarly you can show that they are as large as they get. You can find more information on Jech's "Set Theory" and even more likely on Kanamori's "The Higher Infinite".