All Questions
4 questions
4
votes
1
answer
199
views
Jacobians $\mathbb{F}_q$-isogenous to the direct square of an ordinary elliptic $\mathbb{F}_q$-curve of $j$-invariant $0$
Consider an ordinary elliptic curve $E_b\!: y^2 = x^3 + b$, of $j$-invariant $0$ over a finite field $\mathbb{F}_q$, such that $\sqrt{b} \not\in \mathbb{F}_q$.
Question. What are some examples of ...
- 1,833
2
votes
0
answers
65
views
What conditions are sufficient for two points to be independent in the Mordell-Weil group?
Consider a finite field $\mathbb{F}_{q}$ and an elliptic surface
$$
\mathcal{E}\!: y^2 + a_1(t)xy + a_3(t)y = x^3 + a_2(t)x^2 + a_4(t)x + a_6(t),
$$
where $a_i(t) \in \mathbb{F}_{q}[t]$. I am mainly ...
- 1,833
5
votes
3
answers
497
views
Is there a way to find any non-trivial $\mathbb{F}_p(t)$-point on the given elliptic curve?
Consider a finite field $\mathbb{F}_p$ (where $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$) and the elliptic curve
$$
E\!:y^2 = x^3 + (t^6 + 1)^2
$$
over the univariate ...
- 1,833
3
votes
0
answers
298
views
What does Hodge theory tell us about simply connected surfaces of general type
Let $X$ be a smooth complex projective variety. We know that $\Omega^1_X$ has a non-zero section if and only if the abelianization of the fundamental group of X is infinite. This follows from Hodge ...
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