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 ...

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=...

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 ...

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, ...

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 ...

5
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 ...

Community wiki

5
votes

Accepted

### Elliptic fibrations with few singular fibers

Consider first an elliptic fibration with a section over $\mathbb{P}^1$. (In this case none of the singular fibers are multiples of smooth curves.)
Assume that the minimal discriminant has degree $...

5
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....

Community wiki

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 ...

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 \, (...

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 ...

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 ...

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 (...

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 ...

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 ...

2
votes

Accepted

### Looking for Schmickler-Hirzebruch' monograph on elliptic surfaces

This is the journal that is now called the Münster Journal of Mathematics. Only recent volumes, from 2008, are online. The complete journal, including the volume 33 you are looking for, has actually ...

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 ...

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 $\...

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 ...

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 ...

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 $X \to S$ ...

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