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The polynomial defines a genus 2 curve. Your curve is birational to the genus two curve $C$ given by $y^2=x^5+7x^4+18x^3+24x^2+16x+4$. This curves has two rational Weierstrass points. These points cor …

I believe that the question whether $(p-1)! \bmod p^2$ is computable in polynomial time is still open. The best I am aware of is $O(p^{1/2+\epsilon})$, but I am not a specialist on this issue. The co …

I checked the proof of Granville. The proof only yields the bound $\max \{ \deg(r),\deg(s)\}(\deg(G)-2)\}+1$ which covers your counterexample. To be more detailed: The polynomials $r,s$ yield a morph …

The answer to the question depends on the parity of $n$. Suppose that $n=2k$ is even. Let $(x,y,z)$ be a nonzero solution to your equation, with $y\neq 0$. Then $(x/y,1,z/y^k)$ is also a solution of …

I do not believe you have equality in general. I sketch a counterexample below, which is a geometric version of the fact that if $E/K$ is an elliptic curves such that the quadratic twist $E^{(d)}/K$ h …

I will sketch a counterexample for the modified question. The idea behind the construction is similar to the counterexample for the original question. Only the geometric details of this construction a …

Let $\varphi:A\to A^t$ be a polarization. Then $\varphi$ is an isogeny. In order to study the difference between the Selmer groups of $A$ and of $A^t$ you need to study the torsion subgroups of $A(K)$ …

Let $S$ be a smooth projective surface and $m:=\gcd(\deg \lambda)$ where $\lambda$ runs through the ample cone. Let $m'$ be the gcd of the entries of the intersection matrix of $S$ Then $m$ equals ei …

First you use $c$ as a parameter, i.e., consider your equation as elliptic curve over $\mathbb{Q}(c)$. You can also consider this equation as an equation of an elliptic surface $S$. Now one easily pr …

I am not aware of much evidence for arbitrary high rank elliptic curves. Silverman in his book (Arithmetic of elliptic curves) gives as evidence the lack of evidence for the opposite statement. I gues …

I am not that sure that your question can be answered positively, but the following is merely speculation, so you should not pin me down on it. The basic idea is that there might be "more" weight 4 fo …

If you take the projective closure of your surface in $\mathbb{P}^3$ you find a singular quartic. This quartic has six singular points. Namely the two points found by Daniel and four points of the for …