# Tag Info

34

I would say the answer to both questions is no. In fact, abelian varieties should be an "easy" case. For example, it is known that for abelian varieties (but not other varieties), the variational Hodge conjecture implies the Hodge conjecture. It is disconcerting that we can't prove the Hodge conjecture even for abelian varieties, even for abelian varieties ...

31

There are several different ways to see this. Here is one: Let $G$ be our irreducible projective algebraic group variety over the field $k$, with identity element $e$. The group $G$ acts on itself by conjugation, and this action fixes $e$. Thus this induces a $k$-linear action of $G$ on the local ring $\mathcal O_e,$ and hence on the ...

30

It's a result related in spirit to Minkowski's theorem that $\mathbb Q$ admits no non-trivial unramified extensions. If $A$ is an abelian variety over $\mathbb Q$ with everywhere good reduction, then for any integer $n$ the $n$-torsion scheme $A[n]$ is a finite flat group scheme over $\mathbb Z$. Although this group scheme will be ramified at primes $p$ ...

22

There are no rational curves in an abelian variety, this is much stronger than not being uniruled. If there is a map $P^1 \to A$, $A$ abelian, the map would factor through the Albanese variety of $P^1$, by definition. However, for curves, the Albanese is the Jacobian (from general theory of the Jacobian) and the Jacobian of $P^1$ is a point.

20

Here is another construction, followed by some comments on how to solve the existence problem in general. If $A$ is a $g$-dimensional principally polarized abelian variety over $\mathbf{C}$ with $\operatorname{End} A = \mathbf{Z}$, and $G$ is a finite subgroup whose order $n$ is not a $g$-th power, then $B:=A/G$ is an abelian variety that admits no ...

20

This is false even for elliptic curves over $\mathbb{C}$. This was proved by T. Shioda in "Some remarks on abelian varieties" J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 11-21, http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6164/1/jfs240102.pdf.

19

There's a more down to earth way to deal with this, which is already explained in Mumford's GIT: make an fppf (or even etale) surjective base change to acquire a section, use that to define the principal polarization, and then show it is independent of the choice. (Short reason: varying the choice amounts to a morphism from the smooth proper curve to a Hom ...

18

A result of Kawamata (Kawamata, Yujiro, Characterization of abelian varieties. Compositio Mathematica, 43 no. 2 (1981), p. 253-276) implies that, under your assumptions, $X$ is birational to an abelian variety (in fact you just need the Kodaira dimension of $X$ to be zero and the irregularity to be equal to the dimension of $X$). Once you know that $X$ is ...

17

Over $\mathbb C$ you can argue as follows. Suppose you have a morphism $\mathbb P^1(\mathbb C) \to A$ ($A$= abelian variety ). Since $\mathbb P^1(\mathbb C)$ is simply connected , the morphism lifts to the universal cover of $A$, affine space $\mathbb C^n$. But since $\mathbb P^1(\mathbb C)$ is complete and connected, the lift to affine space must be ...

17

Here is Ribet's proof (expanding on Ulrich's comment): Let $G_K:=Gal(\bar{K} / K)$ and $V_l:=T_l(E)\otimes \mathbb{Q}_l$. The image $\rho_{l,E}(G)$ is a closed subgroup of the $l$-adic Lie group $\text{Aut}(V_l(E)) \cong \text{GL}_{2}(\mathbb{Q}_l)$ and is therefore a Lie subgroup of $\text{Aut}(V_l(E))$. Its Lie algebra $\mathfrak{g}_l$ is a subalgebra of ...

16

I've always meant to sit down and figure out some examples. OK, got it. I think the following works over any field (including finite fields and numbers fields) and so must be standard (unless I've overlooked something). Let $E$ and $E'$ be non-isogenous elliptic curves over a field $k$ (i.e., no nonzero maps between them over $k$), and $G$ a group scheme of ...

15

Yes. Take $f\in O_K$ a uniformizing element of some prime $\mathfrak p$. Consider the hyperelliptic curve defined by the equation $$y^2=x^{2g+1}+f.$$ Then this curve doesn't have semi-stable reduction at $\mathfrak p$. In fact, this equation defines a proper regular model of the curve over the localization $O_{K, \mathfrak p}$ and this model is minimal ...

14

In fact, it is no for completely elementary reasons. If $A$ is simple and $B\subset A^n$ is an abelian variety with $\dim B < g$, then $Hom(B,A^n)=Hom(B,A)^n$ is necessarily zero. So $B=0$.

14

In general, if an abelian variety $A$ contains an abelian subvariety $B\subseteq A$, then $A$ contains another abelian subvariety $B'\subseteq A$ such that $A$ is isogenous to $B\times B'$. This is Poincaré's reducibility theorem. (See also Poincaré's complete reducibility theorem, same book, next page). An abelian variety is called simple if it does not ...

14

Let $\chi$ be a one-dimensional geometric (in the sense of FM) $p$-adic Galois representation of $G_K$ and let $\psi$ be the Hecke character of $K$ associated to $\chi$ by class field theory. The fact that $\chi$ is de Rham (=pst) at all primes above $p$ imples that $\psi$ is an algebraic Hecke character. Generally, the only algebraic Hecke characters of $K$ ...

14

Incidentally, as I posted this question someone who knew the answer wandered into my office. The map $M_g \to A_g$ factors through the moduli space $\tau_g$ of pairs (A,P,L) where A is an abelian variety, P is an A torsor, and L is an ample line bundle on P which is geometrically a principal polarization. The map $M_g \to \tau_g$ is given by $C \mapsto ... 13 Here's a related question on which there has been much work. If you actually need an answer to your question for some other reason and you follow this up, I'd be interested to see where it goes. Let$X$be some moduli space of abelian varieties (so, for$g>1$, there will be extra structure beyond what you have mentioned, for example we'll also be ... 13 I believe for$l\ne p$the theorem is 'always' false, in the sense that for every positive-dimensional abelian variety$A$one can find many$B$s for which your map is not onto, independently of the reduction type: Fix$A$and take any non-constant family of abelian varieties in which$A$is a fibre, e.g. some neighbourhood of$A$in the moduli space ... 13 I think the statement can fail in the case of elliptic curves of good reduction even when$l = p$. But then your comment on Serre-Tate theory confused me for a little while! (A discussion with Jared Weinstein helped me clear it up.) The post ended up getting long as a result; it's partly for my own reference. Recently FC pointed out the following to me: if ... 13 Let us work over$\mathbb{C}$. The inclusion$u \colon B \to A$induces a surjection$\hat{u} \colon A^{\vee} \to B^{\vee}$. By general facts on Abelian varieties, the kernels of$u$and$\hat{u}$have the same number of connected components. Since$u$is injective, its kernel is trivial, so it follows$\ker \hat{u}=(\ker \hat{u})_0$; in other words$\ker ...

13

Another proof that $L = \,\overline{\bf \!Q\!}\,$: Clearly $L$ is contained in $\,\overline{\bf \!Q\!}\,$, so we need only show $L$ contains every algebraic number $x \notin \bf Q$. Let $P(X)$ be the minimal polynomial of $x$. If $\deg P$ is odd, then the class of $((x,0)) - (\infty)$ is a $2$-torsion point on the Jacobian of the elliptic or hyperelliptic ...

12

If $\lambda\in\overline{\mathbb{Q}}$, the elliptic curve $$E_\lambda\colon y^2=x(x-1)(x-\lambda)$$ has $(\lambda,0)$ as $2$-torsion point and is defined over (a subfield of) $L=\mathbb{Q}(\lambda)$. Its Weil restriction $A_\lambda:=\operatorname{Res}_{L/\mathbb{Q}}(E_\lambda)$ is an abelian variety defined over $\mathbb{Q}$ and shares the same points of ...

12

Weil introduced the term "polarization" in connection with his study of abelian varieties with complex multiplication. His definition is slightly different from what one sees today; one might call it a polarization up to isogeny instead of a polarization. One can find a discussion in Weil's article "On the theory of complex multiplication" ([1955d] in volume ...

12

Let me put it this way, Tate's conjecture for abelian varieties is known to imply the Hodge conjecture for abelian varieties, and the last is very much open for this class. For the implication, see the article by Deligne and Milne on "Hodge cycles on abelian varieties" (you can get a copy off of Milne's website). There are lot's of interesting cases known ...

12

The lattice $L_{K3}=H^2(K3,\mathbb Z)$ is $2E_8+3U$, with $E_8$ negative definite and $U$ the hyperbolic lattice for the bilinear form $xy$. It is unimodular and has signature $(3,19)$. The 16 (-2)-curves $E_i$ form a sublattice $16A_1$ of determinant $2^{16}$. It is not primitive in $L_{K3}$. The primitive lattice $K$ containing it is computed as follows. ...

12

Yes. In zero characteristic the image of an isogeny of elliptic curves is determined up to isomorphism by its kernel. Your isogeny has the same kernel as the doubling map from $E$ to itself.

12

Theorem If $m$ is relatively prime to $30$, then $N=3$ works. I'll use additive notation. The cyclic group of order $n$ is denoted $C(n)$, the generators of this cyclic group are denoted $U(n)$. Lemma Let $p \geq 7$ be prime and let $X$, $Y$ and $Z$ be subsets of $C(p)$ of size at least $(p-1)/2$. Then $0 \in X+Y+Z$. Proof By the Cauchy-Davenport theorem ...

12

An argument due to T. Saito goes as follows: Let p be a prime of good reduction for the abelian variety $A$ over a number field $K$. Consider the p-adic logarithm on $A(\bar{K_p})$; this vanishes precisely on the torsion points. Since $A(\bar{K})$ is dense in $A(\bar{K}_p)$ and the p-adic logarithm is not identically zero, $A(\bar{K})$ contains non-torsion ...

12

Yes. This follows from the main result of the following paper of Zarhin. MR0885780 (88h:14046) Zarhin, Yu. G. Endomorphisms and torsion of abelian varieties. Duke Math. J. 54 (1987), no. 1, 131–145. His result, specialized to the $K$-simple case, is the following (fantastic) theorem. Let $A$ be a $K$-simple abelian variety defined over a number ...

12

In general it could happen that the Albanese variety does not admit a principal polarization at all. For instance the Albanese variety of an abelian variety is the Abelian variety itself. So choose $X$ to be some abelian variety that has no principal polarization and you will get an example. On the other hand it can happen that the Albanese variety is ...

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