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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 …

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 …

Alternately, see Faltings, Chai, "Degeneration of Abelian Varieties" Chapter I, especially Theorem 1.9 The general idea is: it can be shown that the Picard functor of a scheme $X/S$ is represented by …

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 …

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 …

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 …

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 …

Question 1(what is the group for the Shimura datum): Well, remember that $H^\times$ is just a bare group. A Shimura datum requires an algebraic group over $\mathbf{Q}$: that is, a functor from $\mat …

Here's an example: Suppose you'd like to know about the divisors on $X = \overline{M_{g,n}}$. Say for instance that you have a divisor $D$ and you'd like to know whether $D$ is ample or nef, that is, …

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 …

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 …

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 …

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 …

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 …