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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 ...
yhb's user avatar
  • 390
1 vote
1 answer
118 views

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 ...
AChem's user avatar
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1 vote
1 answer
282 views

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 ...
AChem's user avatar
  • 801
2 votes
2 answers
273 views

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 ...
kindasorta's user avatar
  • 1,963
2 votes
1 answer
129 views

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 ...
kindasorta's user avatar
  • 1,963
0 votes
0 answers
69 views

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 ...
Dimitri Koshelev's user avatar
2 votes
2 answers
353 views

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 ...
Navvye's user avatar
  • 51
1 vote
1 answer
207 views

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 $...
Dimitri Koshelev's user avatar
2 votes
1 answer
282 views

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 ...
Dimitri Koshelev's user avatar
0 votes
1 answer
111 views

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 ...
kindasorta's user avatar
  • 1,963
0 votes
1 answer
146 views

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 ...
kindasorta's user avatar
  • 1,963
7 votes
0 answers
261 views

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)) = \...
Gina's user avatar
  • 131
1 vote
0 answers
214 views

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 $(\partial f_i/\partial x_j)_{ij}\lvert_p$'...
Matthias Himmelmann's user avatar
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1 answer
377 views

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&...
HaomengXu's user avatar
5 votes
1 answer
405 views

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 ...
Catherine Ray's user avatar
3 votes
1 answer
283 views

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} = ...
Dimitri Koshelev's user avatar
1 vote
1 answer
108 views

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 $G$-...
user41650's user avatar
  • 1,952
1 vote
0 answers
202 views

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} \...
XYZ's user avatar
  • 79
4 votes
1 answer
334 views

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 ...
IntegrableSystemsEnthusiast's user avatar
2 votes
0 answers
68 views

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 ...
Yarden Levy's user avatar
2 votes
2 answers
254 views

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 & ...
MDR's user avatar
  • 188
2 votes
1 answer
137 views

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 $Y_1$, ...
user95246's user avatar
  • 237
6 votes
1 answer
441 views

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-...
TCiur's user avatar
  • 557
9 votes
0 answers
287 views

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}{\...
Asaf Shachar's user avatar
  • 6,621
4 votes
0 answers
162 views

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 ...
Hank Scorpio's user avatar
2 votes
1 answer
123 views

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 $...
hennlu's user avatar
  • 323
3 votes
0 answers
230 views

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 ...
TCiur's user avatar
  • 557
7 votes
1 answer
857 views

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 ...
Roxana's user avatar
  • 519
4 votes
0 answers
138 views

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 ...
Maarten Derickx's user avatar
3 votes
1 answer
166 views

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 $Hom(\_, \mathcal O_X)$ ...
Raffaele C's user avatar
3 votes
0 answers
138 views

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, ...
Dimitri Koshelev's user avatar
3 votes
1 answer
307 views

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 ...
Asvin's user avatar
  • 7,686
3 votes
1 answer
315 views

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 - \...
AVP82000's user avatar
  • 125
4 votes
1 answer
291 views

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(...
k.j.'s user avatar
  • 1,352
4 votes
0 answers
86 views

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 ...
E. Kaya's user avatar
  • 41
2 votes
0 answers
59 views

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 ...
RandomMathUser's user avatar
1 vote
1 answer
318 views

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 ...
MKR's user avatar
  • 93
5 votes
0 answers
152 views

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 ...
KhashF's user avatar
  • 3,230
1 vote
0 answers
129 views

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 $\...
Aoki's user avatar
  • 297
3 votes
0 answers
110 views

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 \...
mathphys's user avatar
  • 305
12 votes
1 answer
535 views

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 $1$($\...
user41650's user avatar
  • 1,952
1 vote
1 answer
54 views

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}\...
A.Skutin's user avatar
  • 319
3 votes
1 answer
415 views

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}...
oasis93's user avatar
  • 33
4 votes
1 answer
175 views

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) $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$. (...
user237522's user avatar
  • 2,783
10 votes
1 answer
369 views

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 ...
Yoav Len's user avatar
  • 147
2 votes
0 answers
72 views

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 ...
Fernanda's user avatar
1 vote
0 answers
135 views

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 ...
Asvin's user avatar
  • 7,686
2 votes
1 answer
401 views

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 \...
Watson's user avatar
  • 1,702
4 votes
3 answers
658 views

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 (...
Asvin's user avatar
  • 7,686
0 votes
0 answers
273 views

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-...
goh's user avatar
  • 101