Search Results

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 …

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. …

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 …

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 …

Yes. Please see Theorem 7.1.3 of Katz-Mazur.

No papers because it's not proven for elliptic curves of rank zero or one. The work of Kolyvagin, together with the work of Gross and Zagier almost proves that if $E_/{\mathbf{Q}}$ is an elliptic cur …

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 …

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 …

The generalization of the phenomena you see is genus theory. If $K = \mathbf{Q}(\sqrt{-d})$ and $H = K(j_d)$ then $H$ contains the Genus field $G$. If $d = \prod_{i=1}^n p_i$ is squarefree (and odd f …

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 …

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 …

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 …

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 …