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A theorem which can be extracted from Theorem V.1.1 of Silverman's "advanced topics in the theory of elliptic curves" is the following. Here $\mathbb{Q}(u)$ denotes rational functions in a variable $u …

No there are not any mistakes in these papers of any interest. In the 1990s there were a bazillion study groups and seminars across the world devoted to these papers; I personally read all three of th …

What you say sounds fine. The absolute Galois group of $F_q$ is $\widehat{\mathbb{Z}}$ and if you take some infinite quotient of this like $\mathbb{Z}_2$ (corresponding to a closed subgroup) then that …

Here is a completely different kind of answer to this question. A perfectoid space is a term of type PerfectoidSpace in the Lean theorem prover. Here's a quote from the source code: structure perfe …

This is not a complete answer, but perhaps it's a roadmap to a counterexample. My strategy is to consider some non-Galois $K/\mathbf{Q}_p$ for which the result is true, and let's make some deductions …

No. How about $a=9$ and $b=16$? Then $c=25$ so $\Omega(a)=\Omega(c)=2\leq\Omega(b)$, the gcd's are $3,16,5$ so the left hand side is 6 and the right hand side only 5. Edit: if $a=316$ and $b=27$ then …

I don't know how "classical" you find these values, but here's perhaps something. Define $E=\sum_{m,n\in\mathbb{Z}}q^{m^2+n^2}$, which is known to be a weight 1 level 4 modular form. In fact $E$ is a …

Firstly, I don't think you should start to learn about adic spaces by thinking about perfectoid spaces. The sensible examples of adic spaces for a beginner to think about are sane Noetherian things li …

On the contrary, some conjectures suggest that the answer is NO! It follows from the Bombieri-Lang conjecture (sometimes known as Lang's conjectures) that a uniform bound should exist. More precisel …

In general one can say very little. There are some positive results (as indicated in the comments) in special cases, but the below example kills any hope that one can say something in general. NB "the …

The case $n=4$ and $m=35$ given in both the question and the original German paper (link now removed from question) is a little misleading. In this case the sets are disjoint. The case $n=2$ and $m=35 …

I think I can construct an explicit counterexample with $a\in\mathbb{Q}_p$. Choose a compatible sequence $\zeta_{p^m}$ of $p^m$th roots of unity in $\overline{\mathbb{Q}}_p$. Write $q=p^n$ with $n\g …

Not an answer, but too long for a comment. I can't see any tricks to do this other than a brute force computational approach. The naive approach would be to loop from $2^{96}$ to $2^{128}$ squaring e …

Things are perhaps a bit messier than you hope. In particular it is not true that the unitary group is non-quasi-split if and only if $v$ ramifies. Disclaimer: I did not know the answer to this questi …

Let $M$ be the set of finite products of Dirichlet $L$-functions. These surely form a class of $L$-functions as in the question. Now take some prime $p$ congruent to 1 mod 4 and let $\chi$ be one of t …