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Results tagged with abelian-varieties
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user 2284
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.
11
votes
Accepted
Tamagawa numbers
Denote by $\Phi$ the quotient of $\mathcal A^\vee$, the special fiber of the smooth (but not necessarily proper) model of the dual abelian variety $A^\vee$, by the connected component of $0$ of $\math …
- 10.9k
8
votes
Why study CM abelian varieties?
Why did mathematicians begin to study CM abelian varieties?
Because they could. The study of CM abelian varieties arguably starts with Fagnano's work on the length of the lemniscate. In 1799, Gauss l …
- 10.9k
7
votes
Is Galois representation induced by semistable elliptic curve semistable?
A Galois representation $\rho_\ell:\operatorname{Gal}(\bar{\mathbb Q}_{\ell}/\mathbb Q_{\ell})\longrightarrow\operatorname{GL}_2(\mathbb Q_{\ell})$ can be semistable (technically $B_{st}$-admissible i …
- 10.9k
6
votes
Accepted
Atkin-Lehner involution on the modular abelian varieties
Since an algebraic number is zero if and only if any of its conjugates is zero, $I_f J_1$ is stable under $W_N$ and so indeed $W_N$ descends to an automorphism of $A_f$.
Now, the important thing to re …
- 10.9k
4
votes
Accepted
CM abelian varieties and potential good reduction
No, absolutely not
In fact, the hypotheses you discuss are rather weak. Take $F$ a totally real number field. If $A/F$ is the abelian variety attached to an eigenform $f$ of weight $2$ and level $N$, …
- 10.9k