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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 …

Hi Barinder! As far as I know there is not an algorithm to do so. See for instance the following paper of Harris and Silverman: http://www.ams.org/journals/proc/1991-112-02/S0002-9939-1991-1055774-0 …

Your question reminds me of a current strain of research whose starting point is Serre's conjecture, now the Khare-Wintenberger Theorem: any continuous odd irreducible two-dimensional Galois repre …

We say that an elliptic curve $E$ over a number field $K$ is an elliptic $\mathbf{Q}$-curve if it is is isogenous to its Galois conjugates $E^\sigma$. These were first studied by Benedict Gross, but w …

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 …

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 …

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 …

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 …

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 …

For your titular question, Beats me. Personally I'm not aware of anyone who's studied the distribution of CM points with respect to height in the way you describe. What I have seen papers that study …

Dear Barinder, Are you familiar with Fumiyuki Momose's "Isogenies of prime degrees over number fields?" If not, you may find it here on NUMDAM In it he performs an analysis of the isogeny character a …

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 …