All Questions
74
questions
3
votes
0
answers
146
views
Examples of curves $C$ with $\operatorname{Jac}(C) \cong E^3$, $E$ a CM elliptic curve
Let $k$ be a field of your choice— I'm particularly interested in algebraically closed fields. Are there explicit examples of curves over $k$ whose Jacobian is isogenous to the product of three copies ...
- 371
0
votes
1
answer
260
views
Is there an isotrivial elliptic surface of positive rank having a section of order $3$?
Let $k$ be a field of characteristic $p > 3$. I cannot find any example of ordinary isotrivial elliptic $k$-surface $E$ (i.e., elliptic $k(t)$-curve, where $t$ is a variable) whose Mordell-Weil ...
- 1,813
3
votes
0
answers
176
views
Does every (ordinary) elliptic curve over a quite large prime field $\mathbb{F}_{p}$ have a lift to the function field with huge Mordell-Weil rank?
Ulmer (and then other authors) showed existence of elliptic curves $E$ over the function field $\mathbb{F}_{p}(t)$ (where $p$ is a prime) with huge Mordell-Weil ranks (i.e., depending on $p$). The ...
- 1,813
6
votes
2
answers
664
views
Could the Weil zeroes of curves be evenly distributed?
If $X$ is a smooth, geometrically connected, projective curve of genus $g$
over $\mathbb{F}_q$, then the zeta function of $X$ is of the form $P(s)/(1 - s)(1 - qs)$, where $P(s)$ is a polynomial of ...
- 9,441
1
vote
0
answers
190
views
Brauer-Manin obstruction and affine curves
I'm looking for references that can justify to what extent is the following statement true:
Statement. Let $X$ be a smooth geometrically integral curve over a number field $k$. Then the Brauer-Manin ...
- 885
2
votes
1
answer
300
views
Computing $H^1$ with coefficients in a torsion-free abelian group
Let $k$ be a number field and denote by $H^i(k,-)$ the Galois cohomology functor $H^i(\mathrm{Gal}(\bar{k}/k),-)$. Let $X$ be a smooth geometrically integral curve over $k$. One can easily show that ...
- 885
2
votes
0
answers
83
views
Are there non-trivial $\mathbb{F}_q$-covers of the j-invariant 0 elliptic curve by a hyperelliptic or cyclic trigonal curve?
Consider the ordinary elliptic curve $E\!: y^2 = x^3 + b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_q$ such that $\sqrt{b}, \sqrt[3]{b} \not\in \mathbb{F}_q$. Also, for any $n \in \mathbb{...
- 1,813
2
votes
0
answers
225
views
Cartier operator and logarithmic differentials
Let $k$ be an algebraically closed field of characteristic $p$, let $C$ be a curve over $k$ and let $\omega$ be a meromorphic differential form on $C$. If $\omega$ gets mapped to itself by the Cartier ...
- 75
3
votes
2
answers
382
views
Galois stable elements of the Picard group of a curve and the rational divisors
Let $C$ be a (smooth,proper) curve over a field $k$. Let $\operatorname{Div}_C(k)$ be the free abelian group generated by the closed points of $C/k$ and $k(C)^\times$ be the group of rational ...
- 7,716
2
votes
1
answer
291
views
Is the reduction of an absolutely irreducible plane curve still irreducible except for the finite number of cases?
Help me please.
Let $k$ be an algebraically closed field (I am mainly interested in $k = \overline{\mathbb{Q}}, \overline{\mathbb{F}_q}$). Consider a plane curve $C \subset \mathbb{A}^2$ of degree $d$ ...
- 1,813
9
votes
0
answers
206
views
Belyi-like results over function fields of characteristic zero
In the paper Unifying themes suggested by Belyi's theorem from 2011, the following question is raised:
Let $X$ be a projective non-singular curve over the function field $K:=\overline{\Bbb{Q}}(t)$. ...
- 3,554
0
votes
1
answer
350
views
Deformations of the Fermat curve
We work over an algebraically closed field $k$, say of characteristic $0$, just in case, and we let $C$ be a smooth curve over $k$. First-order deformations of $C$ (or of any smooth variety for that ...
- 119
3
votes
1
answer
236
views
Field of definition/moduli through the monodromy
Let $Y$ be a smooth projective curve defined over a number field $K$, and let $B$ be a subset of $Y(K)$. It is known that the isomorphism class of a branched cover $f:X(\Bbb{C})\rightarrow Y(\Bbb{C})$ ...
- 3,554
2
votes
0
answers
167
views
Complex Geometric Interpretation of Mordell conjecture
The Mordell conjecture/Falting's Theorem says that any smooth projective curve $X$ of genus $g\geq 2$ over $\mathbb{Q}$ has finitely many integer points (using the valuatlive criterion).
We can of ...
- 4,286
4
votes
0
answers
124
views
What does hyperbolicity of curves buy us in the arithmetic context?
This is going to be a fairly vague question but hopefully it will have concrete answers:
There is this recurrent phenomenon in the arithmetic of curves (possibly stacky,affine) where there is a "...
- 7,716