2
votes
1answer
65 views

Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication

Let $(A,a)$ be a principally polarised (with indecomposable polarisation) Abelian variety over $\mathbb C$. Assume that End(A) contains an order $R$ of a totally real number field of degree $>1$ ...
5
votes
2answers
159 views

can all CM types be realized by Jacobians?

The question is kind of self contained, but let me develop a bit further. Assume K is a CM field of degree $2g$, that is, a quadratic imaginary extension of a totally real field. A CM type of $K$ is ...
0
votes
1answer
134 views

Isogeny of abelian varieties

Suppose we have a curve $X$ (of genus $\geq 3$), and we know that $\{\phi_i : X \to E_i\ \textrm{ for } i = 1, ..., r\}$ are covers of degrees $d_i$ (with the $d_i$'s not necessarily all equal), ...
0
votes
1answer
177 views

Could we construct the Jacobian variety of a smooth curve $C$ with genus $>2$ from its derived category $D(C)$?

Let's consider a smooth curve $C$ over $\mathbb{C}$. We know that the Jacobian variety $Jac(C)$ of $C$ is the moduli space of the degree $0$ line bundles on $C$. $Jac(C)$ is an abelian variety of ...
10
votes
0answers
172 views

Infinitely many curves with isogenous Jacobians

Let $g\geq 4$. Are there infinitely many compact genus $g$ Riemann surfaces with (mutually) isogenous Jacobians? Does the situation change in positive characteristic?
2
votes
0answers
186 views

Which curves cut the Hyperelliptic locus?

Consider the moduli space $\mathcal{A}_{g,n}$ of abelian varieties with some level $n \geq 3$ structure. For simplicity, we just denote it by $\mathcal{A}_{g}$ and drop $n$. Denote the locus of ...
4
votes
1answer
308 views

Properties of subvarieties of a simple abelian variety

Let $A$ be a simple abelian variety over a field $k$. (For simplicity, we assume char $k =0$.) Let $X$ be a smooth projective geometrically connected variety over $k$ of positive dimension. Suppose ...
3
votes
2answers
210 views

Induced maps of an automorphism of a curve on the tangent ot its jacobian and on its differential forms

Let $C$ be a smooth projective curve of genus $g$ over a field $k$ and $J$ be its jacobian (defined over $k$). Let $\sigma: C \rightarrow C$ be a $k$-automorphsm of $C$. This automorphism $\sigma$ ...
2
votes
0answers
226 views

deRham cohomoloy of CM liftings of Jacobians

Let $k$ be field of characteristic $p>0$ and $W=W(k)$ the ring of Witt vectors of $k$. We call a smooth curve over $k$, ordinary, when the Jacobian of $J(X)$ of $X$ is an ordinary abelian variety. ...
8
votes
5answers
643 views

Schottky locus in genus 2

Let $\phi_g : \mathcal{M}_g \rightarrow \mathcal{A}_g$ be the period mapping from the open moduli space of genus $g$ Riemann surfaces to the moduli space of $g$-dimensional principally polarized ...
3
votes
2answers
459 views

Polarizations on intermediate Jacobians

Let $X$ be a Kahler variety of dimension $n$. For each odd number $2k-1 \leq n$ one can consider the $k$-th intermediate Jacobian, that is, the complex torus $$J^{k}X := \frac{H^{2k+1}(X, ...
14
votes
4answers
1k views

Which curves can be found on Abelian varieties ?

We know tha each genus 2 curve is embedded into its degree 1 Jacobian. Under which conditions on $C$, $A$, $g$ and $n$ is it possible for a genus $g$ smooth curve $C$ to be embedded in an Abelian ...
12
votes
2answers
863 views

Are Jacobians principally polarized over non-algebraically closed fields?

How does one define the Torelli map $M_g \to A_g$ of moduli stacks? On geometric points a curve maps to its principally polarized Jacobian. So what I am asking is: if I have a curve $C$ over a ...