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Results tagged with arithmetic-geometry
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user 6121
Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.
3
votes
Accepted
Status of Ihara's lemma for Shimura curves over totally real fields?
Okay, so the answer to the naive question is most likely no for several reasons. For instance, the identifications explained in the second paragraph show that the injection in Rajaei is really an anal …
- 1,141
7
votes
1
answer
1k
views
Status of Ihara's lemma for Shimura curves over totally real fields?
What is the status of Ihara's lemma for Shimura curves over totally real fields?
In particular, why is it not implicit in the exact sequence of Rajaei, "On the levels of mod $l$ Hilbert modular forms" …
- 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
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
6
votes
2
answers
2k
views
"Bad" reduction of Shimura curves via dual graphs
I have the following naive (and inexpert) question about the
reduction of Shimura curves at primes dividing the discriminant
of the underlying quaternion algebra. It requires some background
to state. …
- 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
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