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I've been trying a learn a little more about group schemes by working through a set of exercises on Brian Conrad's website. Exercise 8.3 of http://math.stanford.edu/~conrad/papers/gpschemehw1.pdf read …

An apology first: This is more a supplement to Charles' answer than an answer itself. This was originally a set of comments, but I was not able to format the comments so as to be readable. "Arithmeti …

Thinking about $X_0(N)$ as a bunch of enhanced elliptic curves is a red herring as far as a description of the modular parametrization goes. For instance, you can obtain a similar "modular parametriza …

Background The following question was first asked by Alex Rice, who was thinking about small subsets $A\subset [1,\ldots , N]$ with lots of square differences. Certainly for any set $A$ the maximum nu …

Perhaps this doesn't count as "modern" but stacks are ubiquitous in the 1972 Antwerp paper of Deligne and Rapoport. Recall that the $\Gamma_0(N)$ moduli problem is not representable, and so they must …

inkspot is indeed correct that the component graphs are indeed not generally trees. As you seem to have deduced for yourself, Cerednik–Drinfeld uniformization is a highly nontrivial concept, and it re …

Fact 1 (The Hurwitz Bound): If $X$ is a smooth projective connected curve of genus $g\ge 2$ over $\mathbf{C}$ then $$| Aut_{\mathbf C }(X)| \le 84(g-1)$$ Fact 2: $Aut_\mathbf{C}(X) = Aut_{\overline …

One example: lay your $g$-holed torus $T$ out flat and draw a line the long way through each hole. It hits the torus in $2g + 2$ points. Consider the 180 degree rotation $w$ through that line. Now con …

Even on the level of sets, the idea that any compact Riemann surface gives rise to an algebraic curve over $\mathbf{Q}$ should feel resoundingly false. There are uncountably many compact Riemann surfa …

Well, there's a lower bound as odd theta characteristics on a canonical curve are effective, so there are at least $2^{g-1}(2^g -1)$ of them. Even thetas are trickier. Please consult Dolgachev's boo …

Continuing with Martin Bright's comment: if $F(X,Y,Z)$ is a ternary cubic form, say with integer coefficients and $M\in GL_3(\mathbf{Z})$ then $M$ acts on the variables $X,Y,$ and $Z$ in an obvious wa …

It's well-known that quadratic forms over the rational numbers $\mathbf{Q}$ satisfy the Hasse-Minkowski theorem. This is to say that they are isotropic over $\mathbf{Q}$ if and only if they are isotro …

There are a number of things floating around here. First among them is the first excellent point that Marino made that the finite generation of group of rational points of an abelian variety over a f …

Yes. Please see Theorem 7.1.3 of Katz-Mazur.

Dino Lorenzini has a preprint in which he considers "Riemann-Roch structures", roughly as you've laid them out. http://www.math.uga.edu/~lorenz/RRNovember11.dvi Namely, he considers the situation of …