# Tagged Questions

326 views

### Is there a largest prime p such that J_0(p) completely splits into elliptic curves

The question in the title is related to a more general question. Namely does there exist an integer $N$ such that for all curves $C/\mathbb C$ of genus $> N$ one has that not all simple isogeny ...
366 views

### Equation for simple Jacobian of a genus two curve

Let $X$ be a curve of genus two over a field $k$ with a $k$-rational point. Let $J$ be the Jacobian of $X$. Can we write down an explicit equation for the abelian surface $J$? I know $X$ can be ...
312 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 ...
267 views

### Are ranks of Jacobians over number fields unbounded?

Fix a number field $K$. Is the rank of $J(K)$ unbounded, where $J$ ranges over the Jacobians of all smooth, projective, geometrically connected curves over $K$? Does there exist an integer $g$ such ...
306 views

### Does a curve over a number field have a finite etale cover of given degree

Let $X$ be a (smooth projective geometrically connected) curve over a number field $K$ of genus $g\geq 2$. Let $d\geq 2$ be an integer. Does there exist a curve $Y$ over $K$ with a finite etale ...
244 views

### What is the reduction of this hyperelliptic curve

Let $K$ be a number field and $E/K$ an elliptic curve with equation $Y^2Z = X^3 +AXZ^2+BZ^3$ in $\mathbf{P}^2_K$, where $A,B\in K$. Let $S$ be non-empty finite set of finite places of $K$ and suppose ...
228 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. ...
843 views

### The use of embedding a curve into its Jacobian

I'm looking for as many examples/applications as possible of the use of embedding a smooth projective geometrically connected curve $X$ over a number field $k$ with $X(k)\neq \emptyset$ into its ...
Let $X_p$ be a projective curve over the finite field $\mathbf{F}_p$ (i.e. a projective $\mathbf{F}_p$-scheme pure of dimension 1) for every prime number $p$. Let $X_\mathbf{Q}$ be a projective curve ...
Dear MO Community, Let $X = X_0(N)_{/\mathbb{Q}}$, and $J$ its jacobian. Mazur defines the Eisenstein quotient of $J$, denoted $\widetilde{J}$, as \[ 0 \rightarrow \gamma_IJ \rightarrow J ...