# Tagged Questions

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 ...

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### Automorphism group of a class of abelian varieties

Given two abelian varieties $V$ and $V'$ sharing the same Hasse-Weil L-function, is there a well known, 'canonical' notion of automorphism groups thereof such that $Aut(V)\cong Aut(V')$? If so, does ...
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### Fields generated by torsion points of CM elliptic curves

I'm using the same setup as Corollary 1.7 on p. 44 of de Shalit manuscript (Iwasawa theory of elliptic curves with complex multiplication). I think there is a mistake in his Corollary 1.7 and I'm ...
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### Does complex multiplication for higher dimensional abelian varieties give some generalization of class field theory?

I am currently learning some aspects of the theory of complex multiplication for elliptic curves, and the relationship with class field theory. As I understand it, there is a very special class of ...
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### n-th root of unity in n-th division field of abelian variety?

Let $K$ be a number field and $A/K$ an abelian variety over it. Can it be that $K(A[n])$ does not contain a primitive $n$-th rooth of unity? If the answer is yes is it always possible to ...
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### Semisimplicity of Frobenius on *integral* Tate module

Let $K$ be a number field and $A/K$ an Abelian variety; let $l$ be a (rational) prime. Do there exist infinitely many primes $\mathfrak{p}$ of $K$ such that the Frobenius at $\mathfrak{p}$ acts ...
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### Singular curve on an abelian surface

Let $C_2$ be a smooth genus $2$ curve and $J(C_2)$ its Jacobian. It is well known that the blow-up of $J(C_2)$ at the origin $o$ is isomorphic to the second symmetric product $\textrm{Sym}^2(C_2)$, ...
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### How to prove that $A$ is supersingular iff the Picard number $\rho(A)$ is equal to the second $l$-adic Betti number $b_2(A) = 6$?

Let $A$ be an abelian surface over algebraically closed field $k$ of characteristic $p > 2$. How to prove that $A$ is supersingular (in other words, there is an isogeny between $A$ and $E^2$, where ...
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### Bilinearity of the Cassels-Tate pairing

Let $K$ be a number field and let $A$ be an abelian variety over $K$ (I'm mostly interested in the case that $A$ is an elliptic curve). We use $v$ to denote places of $K$ and we write $H^i(k, A)$ for ...
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### What is the Theorem of the Cube?

What is the "theorem of the cube" for abelian varieties? What is the statement and how should I think about it?
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### Reduction “modulo $p$” of $\mathfrak{p}$-torsion points of CM elliptic curves

Let $E/L$ be an elliptic curve defined over a number field $L$. Assume moreover that $E$ has complex multiplication by an imaginary quadratic field $K$. Let $\mathfrak{p}$ be a prime ideal of $K$. ...
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### Action of a lattice on abelian varieties

Let $\pi\colon Y\to\mathbb{P}_\mathbb{C}^1$ be a ramified cover of degree two of $\mathbb{P}_{\mathbb{C}}^1$ such that $Y$ is smooth. I fix a point $x$ on $\mathbb{P}^1$, over which the cover is étale,...
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### Is a Kummer surface over an finite field $\mathbb{F}_q$ supersingular iff $\mathbb{F}_q$-unirational?

Let $A$ be an abelian surface over an finite field $\mathbb{F}_q$. In particular, I am interested in the case when $A$ is a Jacobian variety. Is the Kummer surface $K_A/\mathbb{F}_q$ Shioda-...
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### Characters on lattices and isogenies of Abelian varieties

Let $V:=\mathbb{C}^g$ and $\Lambda \subset V$ be a lattice, i.e. a discrete subgroup of rank $2g$. Then $A:=V/ \Lambda$ is a complex torus of dimension $g$. We moreover assume that $A$ is algebraic, ...
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### Elements of arbitrary large order in the first Galois cohomology of an elliptic curve

Let $E$ be an elliptic curve over $k=\mathbb{Q}$. Consider $H^1(k,E)$. In this answer Daniel Loughran writes: "I'm pretty sure that this cohomology group has elements of arbitrarily large order". I ...
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### Lifting of Frobenius on semi-abelian varieties

Let $A$ be a semi-abelian variety over a field $k$($char\, k=p$). Namely, there is an exact sequence of group schemes $$0\to T\to A\to B\to 0$$ where $T$ is a torus, $B$ an abelian variety. Assume ...
### Pullback of line bundles and divisors from $Kum(C)$ to $Jac(C)$
Let $C$ be a genus 2 curve over $\mathbb{C}$. Let $X=J(C)$. Consider the involution $i$ on $X$, $x\mapsto -x$. Let $Y=\frac{X}{(i)}$. This is a singular surface with 16 points of singularity - these ...