9
votes
Is there a way to find any non-trivial $\mathbb{F}_p(t)$-point on the given elliptic curve?
You can observe that the elliptic curve $E_2: y^2=x^3+(t^3+1)^2$ is a generic fibre of a rational elliptic surface. Over the algebraically closed field $k$ the group of $k(t)$ points on the curve has ...
- 303
9
votes
Accepted
Discriminant locus of elliptic K3 surfaces
The minimal $s$ is $3$.
It is attained by several elliptic K3's,
including $y^2 = x^3 + (t^2-t)^4$ which has IV* fibers at
$t = 0, 1, \infty$ and no other singular fibers.
The comment by Ariyan ...
- 79.9k
9
votes
Is there a way to find any non-trivial $\mathbb{F}_p(t)$-point on the given elliptic curve?
If $2$ is a cube mod $p$ then you can take
$(x,y) = (ct^2, t^6-1)$ where $c^3 = -4$.
This works for every odd $p \equiv -1 \bmod 3$,
but since you specified $p \equiv +1 \bmod 3$
the first case is $p=...
- 79.9k
8
votes
Accepted
Two definitions of the narrow Mordell-Weil group
Where you say "trivial action on $\Gamma_{t0}$" near the end, you mean "carries $\Gamma_{t0}$ into itself" (i.e., doesn't move it), not that the effect on $\Gamma_{t0}$ is the identity.
The global ...
7
votes
Accepted
Foliation by Asymptotic lines
If a surface is foliated by geodesics (in the induced metric) that are also asymptotic lines, then these curves are also geodesics in the ambient manifold (just look at the definitions). Conversely, ...
- 108k
6
votes
Accepted
elliptic fibration over $\mathbb{P}^1$ with exactly two fibres with monodromy of unipotency rank 1
If you look at the global monodromy action $\pi_1 ( \mathbb P^1 -\{0,1\}) \to SL_2(\mathbb Z)$, you see that $\pi_1 ( \mathbb P^1 -\{0,1\}) = \mathbb Z$ so the image is abelian and therefore has ...
- 148k
6
votes
Accepted
Properly elliptic surface with no multiple fibers and without a section
Please confer Corollary 2.2 of the following with $d$ equal to $3$ and with $n$ equal to $2$.
Jason Starr
A pencil of Enriques surfaces of index 1 with no section
https://arxiv.org/pdf/math/0602639....
6
votes
Do singular fibers determine the elliptic K3 surface, generically?
I am expanding naf's comments to make a self-contained community wiki answer. By an elliptic fibration we mean a smooth projective relatively minimal surface $f: X \to C$ with general fiber given by ...
4
votes
Accepted
Is there a way to find any $\mathbb{F}_2(t)$-point on the elliptic curve $\mathcal{E}$?
Any $\mathbb F_2(t)$-point of $K_t$ would give, upon pullback to $E \times E'$, a $\mathbb F_2(E)$-point of $E'$. Because $E$ is ordinary, $a_2(E)\neq 0$, hence $E$ is not isogenous to its quadratic ...
- 148k
4
votes
Is there a method to check if two sections of an elliptic surface are dependent over the endomorphism ring or not?
Let $\mathcal O=\mathbb Z[\tau]=\mathbb Z+\mathbb Z\tau$. Then you're asking if the four points
$$
P_0(t),\; \tau\bigl(P_0(t)\bigr),\; P_1(t),\; \tau\bigl(P_1(t)\bigr)
$$
are $\mathbb Z$-linearly ...
- 47.4k
4
votes
Accepted
Mordell–Weil rank of some elliptic $K3$ surface
Let $E_0$ be the elliptic curve $y^2 = x^3 + 1$,
and choose $\beta$ in $k = {\bf F}_q$ so that $b = \beta^2$.
Then $W = W_b$ has $\rho=18$ unless the elliptic curve
$$
E_\beta : Y^2 = X^3 + \beta \, (...
- 79.9k
3
votes
Is there a way to find any non-trivial $\mathbb{F}_p(t)$-point on the given elliptic curve?
This is too long for a comment, so I'm making it an answer. Magma now has machinery for computing ranks and generators over function fields (see here) which only work for rational elliptic surfaces (...
- 20.4k
3
votes
Is there an isotrivial elliptic surface of positive rank having a section of order $3$?
Let $k$ be a field and let $n\geq 3$ be an integer which is invertible on $k$.
The following lemma will allow you to conclude that an elliptic curve over $k(t)$ which is isotrivial with $j$-invariant ...
- 9,497
2
votes
Accepted
On the degeneration of the elliptic surface $E(n)$
You can degenerate the covering $z^n \colon \mathbb{P}^1 \to \mathbb{P}^1$ to a morphism
$$
\mathbb{P}^1 \cup \mathbb{P}^1 \to \mathbb{P}^1
$$
equal to $z^{n-2}$ and $z^2$ on the first and second ...
- 39.3k
2
votes
Existence of elliptic surface on Riemann surface with marked points
Let us start from any elliptic surface $X_0 \longrightarrow \Sigma$ with $\chi(X_0) >0$ and whose
fibre at the point $x_i$ is of type $I_0$ (i.e., smooth) for all $i \in \{1, \ldots, p\}$. For ...
- 66.3k
2
votes
Accepted
Analogue of Kodaira surfaces
You could follow Suwa's construction of the Kodaira surfaces from his paper Compact quotients of $C^2$ by affine transformation groups, and define various surfaces which are quotients of $k^2$ by ...
- 26.3k
2
votes
Accepted
Calculate blowup of a pencil of cubics "by hand"
If the two cubics $C_1=V(F_1)$ and $C_2=V(F_2)$ do not share a common component, then the ideal $I = (F_1,F_2)$ defines a $0$-dimensional subscheme $V(I)\subset \mathbb{P}^2$ length 9, which is a ...
- 1,306
2
votes
Accepted
Blow-up of a pencil of cubic curves (from Miranda's basic theory of elliptic surfaces)
As explained in abx's comment, the canonical divisor of your surface is given by $$K_X=-3 \pi^*L + \sum E_i,$$ and this is precisely the class of $-\widetilde{C}_1$ (note that $C_1$ is a curve in $\...
- 66.3k
1
vote
Accepted
Coefficients of elliptic curves over function fields
Are you asking if, for all tuples $p_1,\dots, p_8$, there exists such a $C'$ with $A_9$ of degree one? This is false, assuming $A_9=0$ does not count as degree one.
We can for example choose one of ...
- 148k
1
vote
Do singular fibers determine the elliptic K3 surface, generically?
Edit notice: As Evgeny Shinder pointed out in the comment,
it is unclear why we have $S = J^{-1}(\{0,1,\infty\})$
where $J : C \to \mathbf{P}^1$ is the $j$-invariant map.
The problem is that $X \to S$ ...
- 1,829
1
vote
Specializing p-torsion in a family of elliptic surfaces
As Will Sawin points out, there is no $p$-torsion in the Picard in the first place. Here is my attempt of an answer.
Let $f_{\bar{K}}:Y_{\bar{K}}\longrightarrow\mathbb{P}_{\bar{K}}^1$ be the ...
- 599
Only top scored, non community-wiki answers of a minimum length are eligible