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Results tagged with abelian-varieties
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Abelian varieties are projective algebraic varieties endowed with an Abelian group structure. Over the complex numbers, they can be described as quotients of a vector space by a lattice of full rank. They are analogs in higher dimensions of elliptic curves, and play an important role in algebraic geometry and number theory.
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Failure of injectiveness of maps between cotangent spaces of abelian varieties
Let $p$ be a prime and $K$ a finite extension of $\mathbb Q_p$ with ramification index $e$. Let $\mathcal O_K$ be the ring of integers of $K$ and $k$ its residue field and the unique maximal ideal. Le …
- 2,224
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Compute the principal polarization on $J_0(N)$ in terms of modular symbols
If we consider the modular curve $X = X_0(N)$ as a curve over $\mathbb C$ then one can describe the jacobian $J(X)$ as $H^0(X,\Omega^1_X)^\vee/H_1(X,\mathbb Z)$ as one can do for any curve $X$. Howeve …
- 2,224
4
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How often is the rank of J_0(p)^- zero
As mentioned in this answer there is a conjecture by
Kimball Martin that, formulated slightly informally, has the following special case.
Conjecture:
On average $J_0(p)$ has 2 simple components when …
- 2,224
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Simple abelian varieties of $\mathrm{GL}_2$ type with positive rank and large dimension
$\DeclareMathOperator\GL{GL}$I would like know if there are known constructions of simple abelian varieties of $\GL_2$ type of arbitrarily large dimension and positive Mordell-Weil rank, whose rank is …
- 63.9k
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answer
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Do all simple factors of jacobians of curves come from correspondences?
For this question I will let the overly ambiguous word curve mean: smooth projective and connected curve over $\mathbb C$ (or equivalently a smooth compact Riemann-Surface).
Let $C$ be a curve over …
- 28.8k
5
votes
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answer
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Simple abelian varieties over non algebraically closed fields.
I was wondering what people would normally mean by a simple abelian variety $A$ where $A$ is defined over a field $k$ that is not algebraically closed.
The definition I found in for example on the bo …
- 65.4k