7
votes
Accepted
About the reduction type of the Kodaira symbol of elliptic curves defined over p-adic local fields
The Kodaira symbol characterises the geometry of the special fibre, that is the structure over the algebraically closed field $\bar{\mathbb{F}}_p$. Good reduction means type I${}_0$, multiplicative ...
- 8,945
5
votes
Accepted
Questions about elliptic curves with level-$n$ structure
Let $K$ be a finite extension of $\mathbb{Q}_p$ with $p\neq 2$. Suppose that $E/K$ is an elliptic curve with additive reduction and such that $E$ has full $4$-torsion over $K$. By the Kodaira ...
- 8,945
4
votes
Accepted
Variants of Grothendieck section conjecture
A comment on Q1. which is too long for an answer. Let $k$ be a number field. What you formulated is called the "weak section conjecture" for $X/k$. This is not known in general.
However ...
- 156
4
votes
Accepted
Surjectivity of specialization map
Since $\dim X_s = 1$, we have $H^2(X_s, \mathcal{O}_{X_s}) = 0$. Therefore, by deformation theory every line bundle on $X_s$ extends to a line bundle on the formal scheme $\widehat{X}$ (completion ...
- 16.1k
3
votes
Known cases of Tate conjecture for varieties which are smooth over a curve
Assume first that $X$ is a product of curves $C_1 \times C_2$. Then the Künneth formula expresses $H^1(X,\mathbb Q_\ell)$ as a sum of three pieces. Two of the pieces are generated by the classes of ...
- 148k
3
votes
Accepted
Can the number of elements of order 4 in the Tate–Shafarevich group grow arbitrarily large?
This follows from Theorem 1.5 of Alex Smith's paper "The distribution of $\ell^\infty$-Selmer groups in degree $\ell$ twist families I" which states
Suppose $A/\mathbb Q$ is an elliptic ...
- 148k
2
votes
Accepted
Selmer complex and total complex
You should view the morphisms of complexes of the notation with Tot as complexes of complexes concentrated in only two degrees with differential given by the morphism, which is a particularly simple ...
- 10.9k
1
vote
Changing the weight space for an eigenvariety
I think this question is based on a misconception. “Being an eigenvariety” isn’t a rigorously defined property of a space (or map of spaces) which you could prove to hold or to not hold. It’s more ...
- 37k
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