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Let $j: \mathbf{C} - \mathbf{R} \rightarrow \mathbf{C}$ denote the classical $j$-function from the theory of elliptic functions. That is, $j(\tau)$ is the $j$-invariant of the elliptic curve $\mathbf{ …

For concepts related to algebraic geometry when the base is not a field, it can be difficult for a beginner to reconcile the approach in Silverman with the approach via schemes. I wasted a lot of time …

It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, but …

Some time ago I mentioned a certain open question in an MO answer, and Pete Clark suggesting posting the question on its own. OK, so here it is: First, the setup. Let $X$ be a projective scheme over …

Let $k$ be a field complete with respect to a non-archimedean absolute value, and $X$ a separated algebraic space of finite type over $k$. If $X$ is a scheme then $X(k)$ inherits a natural (Hausdor …

I assume $G$ is affine. The quick answer is that in the simply connected case it says $1 = 1/1$ by various hard ingredients, and then it is a kind of (not easy) game with Galois cohomology and structu …

Sit at a table with the books of Borel, Humphreys, and Springer. Bounce around between them: if a proof in one makes no sense, it may be clearer in the other. For example, Springer's book develops ev …

I will give an intrinsic characterization below for what this unit class modulo 12th powers means, which may be viewed as an answer of sorts: it expresses the obstruction to extracting the 12th root …

As Hunter and Sean noted, since the inflation map ${\rm{H}}^1(L/F,G(L)) \rightarrow {\rm{H}}^1(F,G)$ is injective and ${\rm{H}}^1(F,G)$ is always finite (Borel-Serre), such an $L$ always exists. Below …

Rob, I am doubtful that in such generality (with $\phi$ presumably meant to act on $E$ by some unspecified endomorphism) there is a reasonable answer. The reason why Jordan canonical form "works" eve …

As I mentioned in connection with an answer to another question, it is not generally true for elliptic curves $f:E \rightarrow S$ over a base $S$ that there is a global embedding of $E$ into $\mathbf{ …

Since the case of interest is $W_2(k)$ with perfect $k$ of characteristic $p > 2$, the answer is given by Ioan Berbec's 2009 paper "Group schemes over artinian rings and applications. In that paper ( …

To allow all characteristics, the semisimplicity requirement on the Lie algebras should be replaced with the requirement that the $G_i$ are semisimple as $F$-groups. (In positive characteristic the L …

The correspondence requires the $(\phi,\Gamma)$-module to have the \'etale property for its underlying $\phi$-module, and this plays an essential role in the proof of the correspondence (see Fontaine' …

I don't think the discriminant being square is an issue. This seems best understood by avoiding the "quadratic formula" expression and identifying the cohomological explanation for the appearance of …