All Questions
58 questions
11
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1
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1k
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Finiteness of Tate-Shafarevich
Does anyone happen to know who conjectured the finiteness of the Tate-Shafarevich group?
We recall the conjecture. Let $E/K$ be an elliptic curve where $K$ is a number field. Then $Ш(E/K)$ is finite.
- 1,458
2
votes
1
answer
569
views
Generalization of singular moduli
$j$-invariants of CM curves $E$ over (say) the complex numbers are known as singular moduli. As the theory of complex multiplication explains, singular moduli are algebraic integers of great ...
- 2,406
3
votes
2
answers
737
views
What's known about complete split primes in Q(E[p])?
Let E be an elliptic curve over $\mathbf{Q}$, and p be a prime of good reduction for E such that the Galois representation $\bar\rho_p$ of $\mathbf{Q}$ on the p-torsion of E surjects onto Aut(E[p]). ...
- 2,406
7
votes
1
answer
1k
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Beilinson's height pairing vs. Néron–Tate
In the literature there are several different definitions of what is often referred to as Beilinson's height pairing (see for example section 4.3.8 of Gillet and Soulé's paper Arithmetic intersection ...
- 5,637
9
votes
3
answers
3k
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Elliptic Curves over Global Function Fields
I am currently thinking about a problem, and I feel that by knowing more about elliptic curves over extensions of $\mathbb{F}_q(T)$, for $q$ a power of $p$ say, might lead to insight. I am also ...
- 831
29
votes
0
answers
3k
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What are the possible singular fibers of an elliptic fibration over a higher dimensional base?
An elliptic fibration is a proper morphism $Y\rightarrow B$ between varieties such that the fiber over a general point of the base $B$ is a smooth curve of genus one.
It is often required for the ...
- 3,022
12
votes
3
answers
2k
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What is the etymology for the term conductor?
This is related to the previous question of how to define a conductor of an elliptic curve or a Galois representation.
What motivated the use of the word "conductor" in the first place?
A friend ...
- 3,268
16
votes
1
answer
2k
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Reference for the `standard' Tate curve argument.
I'd like a reference (e.g. something published somewhere that I can cite in a paper) for the proof of the following:
Let $E$ be an elliptic curve over $\mathbb Q$ with minimal discriminant $\Delta$...
- 10.5k