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Results tagged with nt.number-theory
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user 6121
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
3
votes
Question about ring of integers of cyclotomic field
Some more can be said here, though. For instance, writing ${\bf{Q}}(\zeta_{p^n})$ to denote the extension of ${\bf{Q}}$ obtained by adjoining a primitive $p^n$-th root of unity $\zeta_{p^n}$ (for $p > …
- 1,141
3
votes
What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}...
The case of Kummer extensions is not yet completely understood, though partial results are obtained by Darmon-Tian in "Heegner points over towers of Kummer Extensions", Canad. J. Math. 62 (5) 2010, 10 …
- 1,141
5
votes
1
answer
561
views
Selmer of an abelian variety versus that of its dual.
What is the precise relationship between the Selmer group of an abelian variety and that of its dual? For instance, does the vanishing of one not imply the same for the other?
To fix ideas, let $A$ …
- 1,141
5
votes
1
answer
897
views
Tamagawa numbers of abelian varieties and torsion.
Let $A$ be an abelian variety defined over a number field $K$. Fix a prime $v \subset \mathcal{O}_K$, with underlying rational prime $p$. What relationship, known or conjectural (if any), should there …
- 1,141
17
votes
1
answer
2k
views
Construction of abelian varieties from Hilbert modular forms?
Some experts tell me that the construction of abelian varieties from
Hilbert modular forms is an (apparently difficult) open problem. However,
in view of the construction of $l$-adic Galois representa …
- 1,141
4
votes
0
answers
676
views
Soft proof of multiplicity one for character groups of Shimura curves?
Is it not possible to prove mutiplicity one type statements for character groups of quaternionic Shimura curves by simply using Raynaud's description for character groups at primes dividing the underl …
- 1,141
7
votes
Nonvanishing of central L-values of quadratic twists?
Though it is perhaps not an "answer" as such, let me try to explain some intuition.
In certain settings, it is possible to formulate subtle analogues of Mazur's conjecture
for nonvanishing of central …
- 1,141
4
votes
A route towards understanding Shimura varieties?
I think the general wisdom is that Deligne's Travaux de Shimura and Milne's Introduction to Shimura Varieties are the most comprehensive references, with the latter being somewhat lighter on prerequis …
- 1,141
8
votes
A non-technical account of the Birch—Swinnerton-Dyer Conjecture
In my opinion, the best non-technical overview has to be in the Clay Millenium Problems description of Wiles:
http://www.claymath.org/millennium/Birch_and_Swinnerton-Dyer_Conjecture/birchswin.pdf
Th …
- 1,141
6
votes
1
answer
814
views
Characterization of algebraic points on Shimura varieties?
Is there any (conjectural) characterization of $\overline{\bf{Q}}$-points
on Shimura varieties?
The question of course does not always make sense
for ${\bf{Q}}$-points: a theorem of Shimura shows th …
- 1,141