All Questions
13 questions from the last 30 days
3
votes
1
answer
367
views
Variants of Grothendieck section conjecture
Let $X$ be a smooth projective variety defined over a field $k$. We fix the following notations : $\overline{k}$ denotes the algebraic closure of the field $k$, $X_{\overline{k}}$ denotes the variety $...
- 443
4
votes
1
answer
183
views
About the reduction type of the Kodaira symbol of elliptic curves defined over p-adic local fields
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be an elliptic curve defined over $K$. Tate's algorithm can be used to compute the Kodaira symbol of the reduction type of $E$.
However, I ...
- 77
3
votes
1
answer
196
views
Surjectivity of specialization map
Let $S$ be a henselian DVR and $X/S$ be a flat and proper curve with $X$ being regular. Under what conditions the specialization map $Pic^0_{X/S}(S)\to Pic^0_{X/S}(Spec(k(s)))$ is surjective? Here $s\...
- 918
1
vote
1
answer
121
views
Rational functions on elliptic curves over global fields with given support
Let $E$ be an elliptic curve over a global field $k$. Let $x_1, \dots, x_r$ be a set of generators of $E(k) / E(k)_{tor}$ (or more generally, a $\mathbb Q$-basis of $E(k)_{\mathbb Q}$), and let $x_0$ ...
- 77
2
votes
1
answer
125
views
Questions about elliptic curves with level-$n$ structure
Let $n$ be a positive integer, which is $4$ or a prime number $l$.
Let $E$ be an elliptic curve defined over a number field $K$.
Assume that all the $n$-torsion points of $E$ are defined over $K$, i.e....
- 77
3
votes
1
answer
114
views
Selmer complex and total complex
Thanks for your reading. I'm studying Selmer complex book by Jan Nekovar. For the definition of Selmer complex I meet a problem.
In the introduction(page 9, 0.8.0) the author gives us a definition of ...
- 319
4
votes
0
answers
110
views
Euler factors from bad primes and the Beilinson-Bloch vanishing conjecture
The vanishing part of the Beilinson-Bloch conjecture asserts that for a smooth projective variety $X$ over a number field $K$, $\dim_{\mathbb{Q}} \operatorname{CH}^i(X) \otimes_{\mathbb{Z}} \mathbb{Q} ...
- 531
2
votes
1
answer
127
views
Changing the weight space for an eigenvariety
Let $G$ be an algebraic group (like $\operatorname{GL}_2$ or $\operatorname{GL}_2 \times\operatorname{GL}_2$ for example). Assume that there exists an eigenvariety $\mathcal{E}^G$ parameterizing ...
- 93
2
votes
0
answers
139
views
Effective Bombieri-Lang conjecture
The Bombieri-Lang conjecture is the following well-known conjecture:
Let $X$ be a projective variety defined over a number field $K$. Suppose that $X$ is general type. Then $X(K)$, the set of $K$-...
- 27k
1
vote
0
answers
89
views
Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 5,986
3
votes
0
answers
77
views
primes that ramify in division fields for hyperelliptic jacobians
Let $C$ be a hyperelliptic curve $y^2=f(x)$ of genus $g\geq 2$ and $\Delta$ the discriminant of $f(x)$. Let $\ell>2$ be a prime that divides $\Delta$ to the order $e:=\operatorname{ord}_\ell(\Delta)...
- 685
1
vote
0
answers
84
views
Descent of isogenies between p-divisible groups
Let $\mathcal{G}$ be a $p$-divisible group over $K$, which is a finite extension of $\mathbb{Q}_p$. Let $\rho: \text{Gal}(\bar{K}/K)\rightarrow \text{GL}(T_p\mathcal{G})$ be the associated Galois ...
- 11
3
votes
0
answers
56
views
p-torsion in the Tate-Shafarevich group of supersingular elliptic curves
Let $E$ be a supersingular elliptic curve over $\mathbb{F}_p(t)$. Is something known on the $p$-torsion of the Tate–Shafarevich group in this case? In particular, I would like to know if (or if known ...
- 103