# Questions tagged [jacobians]

The jacobians tag has no usage guidance.

167
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### Atkin-Lehner involution on the modular abelian varieties

Let $f$ be a CM modular forms with coefficients in the imaginary qudaratic field $K=Q(\sqrt{-3}))$ which correspond to an elliptic curve $E$ defined over $K$. Then Shimura constructed an abelian ...

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1
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118
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### Numerical partial differentiation of a convolution product with FFT

How can one numerically calculate the partial derivatives of a convolution function, particularly when the closed-form or analytical expressions of the derivatives are not readily available? I am ...

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282
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### Numerical estimation of partial derivatives of convolved functions when closed forms do not exist

Summary: Some peak functions are convolutions which may not have a closed form solution. A classical example can that of a Voigt which is a convolution of a Lorentzian and a Gaussian, followed by ...

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### Involution of the symmetric square of a smooth plane quartic

Let $C$ be a smooth plane quartic defined over a field $K$. Denote by $J$ its Jacobian, and by $C^{(2)}$ its symmetric square. Since $C$ is a smooth plane quartic, it is non-hyperelliptic, and hence ...

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### Endomorphism ring of the Jacobian of a generic smooth plane quartic

Let $C$ be an arbitrary smooth plane quartic defined over a number field $K$. Assume $C$ is not hyperelliptic, and denote by $J$ the Jacobian of $C$. How does $\text{End}(J)$ look like for a generic ...

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### Is there an $\mathbb{F}_{\!q}$-curve of geometric genus 3 and $\mathbb{F}_{\!q^3}$-cover to an elliptic $\mathbb{F}_{\!q}$-curve of $j$-invariant 0?

Let $E$ be an elliptic curve $y^2 = x^3 + b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_{\!q}$ such that $3 \mid (q-1)$. Is there an absolutely irreducible $\mathbb{F}_{\!q}$-curve $C$ of ...

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353
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### Upper bound on number of integral solutions of elliptic curves

I was studying M. Bhargava Et al's seminal paper titled "Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves"
And came across a very fascinating ...

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207
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### The ideal in $\mathbb{Z}[x]$ of all vanishing polynomials of a curve automorphism

Let $C$ be a projective irreducible algebraic curve of genus $g$ over an algebraically closed field of characteristic $0$ (for simplicity). Given an automorphism $\alpha \in \mathrm{Aut}(C)$ of order $...

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### One unexpected observation related to algebraic curves and their Jacobians

Let $C$ be a projective irreducible algebraic curve over an algebraically closed field of characteristic $0$ (for simplicity). Assume that there is also a cover $\varphi\!: C \to E$ to an elliptic ...

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### Lifting an automorphism of a curve to an automorphism of its Jacobian

Let $C: Y^3Z = f(X,Z)$, with $f(X,Z)\in K[X,Z]$, a degree 4 homogeneous polynomial, and $K$ a field.
The curve $C$ has an order $3$ automorphism, given by sending $(x,y,z)$ to $(x,\omega y, z)$, where ...

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### Embedding of symmetric square in Jacobian

Let $C$ be a projective curve defined over a field $K$, and let $C^{(2)}$ and $J$ be its symmetric square and Jacobian, respectively.
There is a natural map $C^{(2)}\hookrightarrow J$, defined as ...

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### Complete curves in $\mathcal{M}_g$ all of whose Jacobians have trivial endomorpism ring

I'm trying to construct complete smooth curves $C$ in $\overline{M}_g$ such that for all points $S \in C \cap \mathcal{M}_g$, its Jacobian $\text{Jac}(S)$ satisfies $\text{End}(\text{Jac}(S)) = \...

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### Cokernel of the Jacobian Matrix

For an algebraic variety $V = \mathcal{V}(f_1,\dots,f_m)\subset \mathbb{C}^n$, in smooth points $p$ there is a nice geometric interpretation of the Jacobian $(\partial f_i/\partial x_j)_{ij}\lvert_p$'...

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### Why an elliptic curve can be defined as an abelian variety of dimension 1?

Now we define an elliptic curve as "a smooth projective curve of genus one with a specified base point". (A little question by the way: Is the requirement "with a specified base point&...

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### Does the Jacobian functor respect deformations?

I am trying to understand the relationship between deformations of curves and deformations of their Jacobians and would greatly appreciate a sanity check. Let $C_0$ be a smooth projective curve over a ...

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### What do we know about the $\mathbb{F}_{q^2}$-curve $X^{q-1} + Y^{q-1} + Z^{q-1}$?

People have thoroughly studied the Hermitian $\mathbb{F}_{q^2}$-curve $H: X^{q+1} + Y^{q+1} + Z^{q+1} = 0$. What do we know about the similar $\mathbb{F}_{q^2}$-curve $C: X^{q-1} + Y^{q-1} + Z^{q-1} = ...

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### Intermediate Jacobian under group action

Let $X$ be a smooth Fano threefold with a finite group $G$ action. Assume that the orbit space $X/G$ is smooth. Is it true that $J(X/G)\cong J(X)^G$ As an abelian variety? Here, $J(X)^G$ is the $G$-...

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### How to prove a concentration isoperimetric inequality for a non-Lipschitz function

Definition $1$. A probability measure $\mu$ on $\mathbb{R}^{d}$ satisfies c-isoperimetry if for any bounded L-Lipschitz $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$, and any $t \geq 0$,
\begin{align}
\...

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### Compactified Jacobian of a rational curve whose normalization is a set-theoretic bijection

Suppose $C$ is a (singular) rational curve whose normalization $p: \mathbb P^1 \to C$ is a set-theoretic bijection.
Can one understand how the compactified Jacobian of $C$ looks like?
For example, the ...

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### Under what conditions is the least-squares approximation bounded with the same Lipschitz gradient constants?

Let $f(x):\mathbb{R}^K\Longrightarrow \mathbb{R}^L$ denote a multivariate continuously differentiable function. All the partial derivatives of $f$ (all its Jacobian elements) are bounded from above ...

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### Full-rank matrix

I have a sparse square matrix and want to see if it is full rank (so that I can apply the implicit function theorem).
$$\left[\begin{array}{cccccccccc}
0 & 1 & 1 & 1 & 0 & 0 & ...

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### Elements of order two in a Prym variety

Let $Y_0$ be a genus two projective smooth complex curve,
let $Y_1$ be an étale cover of degree 2 of $Y_0$, and let
$\sigma$ be the involution of $Y_1$ over $Y_0$.
If $J_1$ is the jacobian of $Y_1$, ...

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### Degree-2 étale covers of curves in characteristic 2 vs torsion points on the Jacobian

It can be found, at Hartshorne exercise 4.2.7 for example, that in the case where $\operatorname{char}(k) \neq 2$ we have a nice correspondence between etale degree 2 covers of a curve $C$ and 2-...

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### Is there a variational interpretation for the equation $\operatorname{div}(\star \circ \bigwedge^k df\circ \star )=0$?

$\newcommand{\id}{\operatorname{Id}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\TM}{\operatorname{TM}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Cof}{\operatorname{Cof}}
\newcommand{\Det}{\...

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### Derivative of dual isogeny is pullback on $H^1$

Apologies if this is not quite at the level of MathOverflow, but I have already asked at MSE and received no answer after two+ weeks and a bounty.
Let $X$ and $Y$ be elliptic curves (over an ...

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### Grassmannian of line subbundle of a stable rank 2 vector bundle on a smooth projective curve

Let $X$ be a smooth projective curve of genus $g\geq 2$. Given a rank two, degree $d=0$ vector bundle $\mathcal{F}$ on $X$, we consider the grassmannian of sub-line bundles of $\mathcal{F}$ of degree $...

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### Why is the Jacobian of a curve "irreducible" as a principally polarized abelian variety?

In J.P. Murre's "Reduction of the proof of the non-rationality of a non-singular cubic threefold to a result of mumford", in the proof of Theorem 3.11 he remarks that "the Jacobian of a ...

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### Relation between the cohomology group of a curve and the cohomology group of its jacobian

Let $J_C$ be the Jacobian of a smooth projective curve $C$ over $\mathbb{C}$. I would like understand the isomorphism between $H^1(J_C,\mathbb{C})$ and $H^1(C,\mathbb{C})$. I read in a paper that ...

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### How often is the rank of J_0(p)^- zero

As mentioned in this answer there is a conjecture by
Kimball Martin that, formulated slightly informally, has the following special case.
Conjecture:
On average $J_0(p)$ has 2 simple components when ...

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### Dual family of torsion-free rank-1 sheaves on Gorenstein curves

Let $X$ be a Gorenstein curve over a field an consider the compactified Jacobian parametrizing torsion-free, rank-1 sheaves on $X$.
Is there a chance that the dual functor $Hom(\_, \mathcal O_X)$ ...

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### Richelot isogenies in characteristic $2$

I am interested in Richelot isogenies between ordinary abelian surfaces in characteristic $2$. If I am not mistaken, the corresponding theory is developed in Article "J.-B. Bost, J.-F. Mestre, ...

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### What is the involution on the moduli space of genus 3 curves induced by the Torelli map

Let $M_g$ be the moduli space of genus $g$ curves, $A_g$ be the moduli space of principally polarized dimension $g$ abelian varieties. They have dimensions $3g-3,g(g+1)/2$ respectively. The Torelli ...

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### Generalization of torsion points on Jacobian of genus 2 over finite fields (with respect to the theta divisor)

Let $J(C)$ be the jacobian of a hyperelliptic curve $C$ of genus 2 defined over finite field $\mathbb{F}_q$. Let $\Theta$ be the image of the curve on the Jacobian under the embedding $P \mapsto P - \...

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### An explicit equation for $X_1(13)$ and a computation using MAGMA

By a general theory $X_1(13)$ is smooth over $\mathbb{Z}[1/13]$, and so is its Jacobian $J$.
And the hyperelliptic curve given by an affine model $y^2 = x^6 - 2x^5 + x^4 -2x^3 + 6x^2 -4x + 1$ is $X_1(...

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### The unit root subspace of a genus-2 odd degree hyperelliptic curve of semistable reduction

Let $K$ be a finite extension of $\mathbb{Q}_p$. If $A_K$ is a semistable Abelian variety over $K$, then we have a Frobenius endomorphism on $H_{dR}^1(A_K)$, whose definition depends on a choice of a ...

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### The intersection form on a Jacobian

$\DeclareMathOperator{\End}{End}$
Let $J=\mathop{Jac}(C)$ be a Jacobian, $\Theta$ the theta divisor. Since it is principal, it induces a bijection between the Néron-Severri group $NS(J)$ and the ...

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### Flat connection of a degree zero line bundle on curve

The question is clear from the title. Suppose we have a line bundle on a compact smooth complex curve $X$, and a line bundle $\mathcal{L}=\mathcal{O}_X(p-q)$, where $p$ and $q$ are divisors, then what ...

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### The image of a curve under the multiplication endomorphism of its Jacobian

Let $X$ be a complex smooth projective curve of genus $g\geq 2$. Embed $X$ in its Jacobian
${\rm{J}}(X)\cong{\rm{Div_0}}(X)/{\rm{Div_p}}(X)$ where ${\rm{Div_0}}(X)$ is the group of degree zero ...

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### The compactified Jacobian is birational to a $\mathbb{P}^1$-fibration over the Jacobian of normalization

Let $Y$ be an integral curve whose only singularity is one simple node at a point $y$, and
$\pi:X\rightarrow Y$ be the normalization with $\pi^{-1}(y)=\{x,z\}$. $J(X)$ is the Jacobian of $X$, and $\...

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### Using principal polarisation to "cancel" Jacobian summands in isomorphism

I'm working through the sketch proof of irrationality of cubic threefolds in Huybrechts' The geometry of cubic hypersurfaces.
Let $J(X)$ denote the intermediate Jacobian of a cubic threefold $X \...

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### Embedding of a derived category into another derived category

I am considering the following two cases:
Assume that there is an embedding: $D^b(\mathcal{A})\xrightarrow{\Phi} D^b(\mathbb{P}^2)$and the homological dimension of $\mathcal{A}$ is equal to $1$($\...

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### Continuations of derivations of Jacobian subring

Assume that the algebraically independent polynomials $f_1,\ldots, f_n\in\mathbb{C}[x_1,\ldots, x_n]$ are such that the Jacobian matrix $\text{Jac}_{x_1,\ldots, x_n}^{f_1,\ldots, f_n}\in\mathbb{C}\...

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### Functions with a Jacobian whose columns are orthogonal

I am interested in vector fields whose Jacobian has orthogonal columns; i.e. if $\mathbf{f}(\cdot):\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a function where $\mathbf{f}(\mathbf{x})=[f_1(\mathbf{x}...

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### Certain endomorphisms of $\mathbb{C}(x,y)$

Let $f: (x,y) \mapsto (p,q)$ be a $\mathbb{C}$-algebra endomorphism of $\mathbb{C}(x,y)$
satisfying the following two conditions:
(i) $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$.
(...

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### Surjectivity of the Abel-Prym map

It is well known that the Abel-Jacobi map restricted to $\text{Eff}_g(C)$ surjects onto the Jacobian $\text{Jac}(C)$, since every divisor of degree $g$ is effective.
Is there an analogous statement ...

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### Calculate the Jacobian of a particular diffeomorphism of parallelizable manifold onto itself

Let $M$ be $d$-dimensional parallelizable manifold. Let $e_k(x)$ for $k=1, \dots , d$ be the smooth vector fields forming orthonormal basis in tangent bundle of 𝑀, these vector fields exist because ...

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### A map on Jacobians coming from a correspondence explicitly

From this question, we know that every map of the form $J(C) \to J(C)$ for a curve $C$ and it's jacobian $J(C)$ comes from a correspondence between $C\times C$ and in fact we can take this ...

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### Degree of morphisms and isogenies

$\renewcommand{\J}{\mathrm{Jac}} \renewcommand{\F}{\mathbb{F}}$
I am reading this paper by B. Gross, and there is something I don't understand on p. 945. Here is the context: fix a prime $p \equiv 3 \...

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3
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### Torsion in the jacobian of a super elliptic curve

Let $y^n = f(x)$ define a smooth projective curve $C$ over some field $k$ with $\deg f \geq n$ and odd and with $f(x)$ having no repeated roots. Let $J$ be the Jacobian of $C$ and $J[n]$ it's (...

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### What is the jacobian of an image lookup function?

I posted this question on Robotics Stack Exchange (link) but thought it could be relevant here as well.
I'm trying to solve a computer vision problem whereby I wish to use Levenberg–Marquardt non-...