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Hey, I probably should have answered this one some time ago. It was proved in 1989 by J.L. Parish that the order of an $H$-rational torsion point is 1,2,3,4 or 6, and this also can be deduced from wor …

Yes. Please see Theorem 7.1.3 of Katz-Mazur.

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 …

I exceedingly concur with Emerton on the nontriviality of this problem. The ideas for its solution take up the bulk of chapter 7 in Diamond and Shurman's book on modular forms.

As far as I know, the best relation between the two is the following: the field generated by the hilbert class polynomial $h_\mathcal{O} (X)$ contains the field generated by $h_K(X)$. This is implied …

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 …

Given that I don't know exactly which relation you're talking about, I'll give you something old and something new: A priori, asking for a formula for the number of fixed points of Atkin-Lehner is as …

If in (2) you are asking whether the conductor entirely determines the number of points on all reductions, the answer is most assuredly not. If that were the case then there would only be one cusp for …

You can't prove it because it is untrue. Let $E$ be an elliptic curve with CM by $\mathbf{Z}[\sqrt{-p}]$ defined over a number field $K$ which Contains $\mathbf{Q}(\sqrt{-p})$ so that the action of …

Color me surprised if people no longer believe that the distribution of elliptic curves is half rank zero, half rank one, and a density zero subset of higher rank curves. To my knowledge this is still …

1)If an elliptic curve has integral $j$-invariant it absolutely DOES NOT NEED to have CM. The class of curves with integral $j$-invariant (let's call that the class of IM Elliptic curves for Integral …

It should be noted that Murabayashi determined that there should be finitely many (whose moduli lie in the rational numbers) for $g=2$ over the complex numbers and (mostly) explicitly determined them. …

Yes, $A(X) = cX/\sqrt{\log(X)} + O(X/\log^{3/2}(X))$ for a positive real number $c$ which I think is 1 (edit: This remark on the constant was just a vague recollection which is wrongly remembered as i …

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 …

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 …