ag.algebraic geometry – What are possible applications of ‘fast arithmetic’ in the Jacobian (degree zero Picard group) of projective curves over fields?

It is well known that there are plenty possible applications of ‘fast arithmetic’ (that is, 1. having an algorithm at hand that actually computes in…, and 2. the running time of that algorithm is known and not too bad) in the degree zero Picard group (the ‘Jacobian’) of an integral, smooth and projective curve $C$ with small genus (for instance elliptic curves and hyperelliptic curves of genus 2) over a field $k$. For instance, there are cryptosystems whose safety relies on the discrete logarithm problem in the Jacobian of elliptic curves and thus it is crucial to know how fast the arithmetic in that group can be carried out.

I wonder whether there are applications of ‘fast arithmetic’ in the degree zero Picard group of more general projective curves $C$ over $k$. Especially, I am curious about the case when $C$ is at least one of the following: reducible, singular or has large genus.

For instance, in the paper of Michael Stoll and Peter Bruin the authors use the Mordell-Weil sieve to prove the non-existence of rational points on smooth projective curves $C$ over $mathbb{Q}$ of genus $g geq 2$. They use the embedding of the rational points on $C$ into the Mordell-Weil group (degree zero Picard group) and local data of reductions at primes $p$ of $mathbb{Q}$. And there is a speed up in their computation if they do not restrict themselves to primes $p$ of good reduction, but also consider $p$ with bad reduction. In that case the authors provide a variant of the Cantor algorithm to compute in the Jacobian in the case that the reduction has genus $2$. Here a generalization to higher genus may be useful to also gather information from bad primes whose corresponding reduction has large(er) genus.

I am interested if there are further applications of ‘fast arithmetic’ in the Jacobian/degree zero Picard group in the same manner as above or even in a complete different style (for instance, it might be valuable to know a lower bound for the running time of such an algorithm to provide statements about how hard a problem related to the Jacobian is).

Many thanks in advance for your help!

ag.algebraic geometry – Zero endomorphisms of elliptic curves

The following question is related to the discussion based on the discussion in Elliptic curves by L. Washington.

Let $E$ denote an elliptic curve over a field $K$ and ${overline{K}}$ denote the algebraic closure of $K$. It is well known that $E({overline{K}})$ is an abelian group. Define $E(n)$ be the subgroup of all elements $(x,y) in {overline{K}} times {overline{K}}$ so that $(x,y)$ has order $n$. The following is Theorem 3.2 (Page 79, Washington).

$bf{Theorem:}$ If $n$ is a positive integer so that ${mathrm{Char}}~K nmid n$ or ${mathrm{Char}}~K = 0$, then $E(n) = {mathbb{Z}}_n oplus {mathbb{Z}}_n$.

If ${mathrm{Char}}~K = p > 0$ and $p mid n$, write $n = p^r n^{prime}$ so that $p nmid n^{prime}$. Then $E(n) = {mathbb{Z}}_{n^{prime}} oplus {mathbb{Z}}_{n^{prime}}$ or ${mathbb{Z}}_n oplus {mathbb{Z}}_{n^{prime}}$.

Most of the proof assumes that the endomorphism $(n) =$ multiplication by $n$ to a point is a non-zero endomorphism. Notice that $E(n)$ is the kernel of $(n)$ here.

$bf{Question:}$ Is there a classification to when $(n)$ can be a zero endomorphism?

For example if $K = {mathbb{Q}}$, a deep theorem of Mazur (Rational isogenies of prime degree, Invent. Math. 44 (1978) 129–162) proves that the torsion subgroup $E_{mathrm{tor}}$ is one of the following:

${mathbb{Z}}_m ~(m = 1, dotsc, 10, 12)$ or ${mathbb{Z}}_2 oplus {mathbb{Z}}_{nu} ~(nu = 1, 2, 3, 4)$.

ag.algebraic geometry – The “easier” case in Knudsen’s stabilization of pointed curves

My question concerns Knudsen’s (super-important) proof that $overline{M}_{g,n+1}$ is the universal curve over $overline{M}_{g,n}$ (F. Knudsen, The projectivity of the moduli space of stable curves II: the stacks $M_{g,n}$, Math. Scand. 52 (1983)), more specifically the proof of Theorem 2.4.

Using notation as in Knudsen’s paper, in Case II of the proof, I don’t understand how it follows from Corollary 1.5. that $X^mathrm{s} to S$ is flat and prestable.

I don’t know what is Corollary 1.5 applied to. If it’s $X^mathrm{s}$ in the role of $X$, $X$ in the role of $Y$, and ${mathcal O}_{X^mathrm{s}}$ in the role of ${mathcal F}$, then the issue is that the flatness of ${mathcal F}$ over the base is a hypothesis in Corollary 1.5, so we can’t use that to argue that $X^mathrm{s} to S$ is flat.

Am I missing something obvious? It’s extremely possible. If not, what would be the right way to fill in the details?

P.S. As an aside, it is known that some details are missing in Case I (which should be the “hard” case), and these are addressed in a much more recent paper by Knudsen (A closer look at the stacks of stable pointed curves, J. Pure Applied Algebra, 2012), as well as a related paper by R. Ile ( https://arxiv.org/abs/1110.3909 )

(Edit: changed $X_s$ to $X^mathrm{s}$.)

fa.functional analysis – Does anyone know if it’s possible to construct Moduli space of J holomorphic curves using Holder spaces?

let Y be a contact (3) manifold and X be its symplectization. let’s say the Reeb dynamics is at least Morse Bott. let $u: Sigma rightarrow X$ be a $J$ holomorphic curve. I know the usual model for a neighbourbood for it is the weighted sobolev space $W^{1,p,d}(u^*TX)$ plus maybe some asymptotic vectors. Do people know if we can replace this space with some Holder space (say C^{2,0}). Is it still true the linearized Cauchy Riemann operator is Fredholm of the same index? (we already the kernel and cokernel if we use the Sobolev spaces have exponential decay, can we say it doesn’t have any new kernel and cokernel?) (I would really like some way to get rid of the exponential weights..)

ag.algebraic geometry – Sheaf of elliptic curves up to isogeny

For a scheme $X$, denote by $mathcal{Ell}^{isog}(X)$ the groupoid of elliptic curves on $X$ (where we consider isogeny classes as isomorphism classes in the category of elliptic curves localized at isogenies). Consider the functor
$$mathcal{Ell}^{isog}:Sch/S^{op}rightarrow text{Gpd}, quad X rightarrow mathcal{Ell}^{isog}(X).$$
It was asked in this question if this was an algebraic stack (https://math.stackexchange.com/questions/675164/moduli-space-of-isogeny-classes-of-elliptic-curves). It is not, because ” “forget isomorphism class and remember only isogeny class” map from the usual moduli space ought to be algebraic, but an algebraic map of curves has finite fibers but (over C, say) isogeny classes of non-isomorphic curves are infinite” (https://math.stackexchange.com/q/675207). I’m wondering if the functor $mathcal{Ell}^{isog}$ if not represented by an algebraic stack is a $2$-sheaf. I don’t have any intuition about that, so I’d be glad for any hint.

Fulton, Algebraic curves 4.8

Let $V=mathbb{P}^1$, with corresponding coordinate ring $Gamma_h(V)=k(X,Y)$. Let $t=X/Yin k(V)$. The question is following :

  1. Show $k(V)=k(t)$.
  2. There is a 1-1 correspondence between points of $mathbb{P}^1$ and the DVR’s with quotient field $k(V)$ that contain $k$.

I could solve 1, but I have no idea for 2. I guess desired DVR is $O_p(V)$ (local ring of V at p), but I can’t show why the maximal ideal $mathfrak{m}_p(V):={f/g : g(p)neq0 , f(p)=0}$ is principal ideal, and conversely why such local rings of V at p are all discrete valuation ring with quotient field $k(V)$ containing $k$. Thank you for any helps.

java – Drawing dragon curves using Turtle graphics

This is exercise 3.2.23. from the book Computer Science An Interdisciplinary Approach by Sedgewick & Wayne:

Write a recursive Turtle client that draws dragon fractal.

The following is the data-type implementation for Turtle graphics from the book which I beautified:

public class Turtle {
    private double x;
    private double y;
    private double angle;

    public Turtle(double x, double y, double angle) {
        this.x = x;
        this.y = y;
        this.angle = angle;
    }
    public void turnLeft(double delta) {
        angle += delta;
    }
    public void goForward(double step) {
        double oldX = x, oldY = y;
        x += step * Math.cos(Math.toRadians(angle));
        y += step * Math.sin(Math.toRadians(angle));
        StdDraw.line(oldX, oldY, x, y);
    }
}

StdDraw is a simple API written by the authors of the book.

Here is my program:

public class Dragon {
    public static void drawDragonCurve(int n, double step, Turtle turtle) {
        if (n == 0) {
            turtle.goForward(step);
            return;
        }
        drawDragonCurve(n - 1, step, turtle);
        turtle.turnLeft(90);
        drawNodragCurve(n - 1, step, turtle);

    }
    public static void drawNodragCurve(int n, double step, Turtle turtle) {
        if (n == 0) {
            turtle.goForward(step);
            return;
        }
        drawDragonCurve(n - 1, step, turtle);
        turtle.turnLeft(-90);
        drawNodragCurve(n - 1, step, turtle);
    }
    public static void main(String() args) {
        int n = Integer.parseInt(args(0));
        double step = Double.parseDouble(args(1));
        Turtle turtle = new Turtle(0.67, 0.5, 0);
        drawDragonCurve(n, step, turtle);
    }
}

I checked my program and it works. Here is one instance of it:

Input: 12 0.007

Output:

enter image description here

Is there any way that I can improve my program?

Thanks for your attention.

ag.algebraic geometry – Tate-Shafarevich groups of high-rank elliptic curves over $mathbb Q$

Assume the BSD conjecture. By checking various examples, it seems that the Tate-Shafarevich groups of elliptic curves over $mathbb Q$ satisfies the following propositions:

  • If an elliptic curve E over ℚ has rank ≥ 2, then Ш(E)=1 or Ш(E)=4.
  • If an elliptic curve E over ℚ has rank ≥ 3, then Ш(E)=1.

EDIT: The second proposition is false. The elliptic curve $E:y^2 = x^3 + 1916840x$ has rank 3 and Ш(E)=4, by the following SageMath computation:

A=EllipticCurve((0,0,0,1916840,0))
A.rank()                   #=3
A.sha().an_numerical()     #=4.0000000000

Question: Are there references, heuristics, counterexamples, etc. to the first proposition above?

at.algebraic topology – Untangling two simple closed curves on a surface

Let $S$ be a smooth surface and $gamma_1, gamma_2$ be two transversal simple closed curves on it. Suppose moreover that there exists a simple closed curve $gamma_1’$ on $S$ isotopic to $gamma_1$ and such that $#(gamma_1cap gamma_2)>#(gamma_1’cap gamma_2)$.

Question. Is it true that there is a disk on $Ssetminus (gamma_1cupgamma_2)$ whose boundary is composed of one arc of $gamma_1$ and one arc of $gamma_2$?

Note that in case such a disk exists, one can construct an isotopy of $gamma_1$ that would decrease the number of intersections of $gamma_1$ with $gamma_2$ by two.

ag.algebraic geometry – Berkovich Integration on algebraic curves

Berkovich developed a theory of integrating one-forms on his analytic spaces in his book “Integration of One-forms on $P$-adic analytic spaces”. As this book is difficult to digest for me, I am wondering how this theory breaks down, if one just considers analytifications of smooth projective curves. More specifically, I am wondering if his theory can be used to translate some of the result over $mathbb C$ (which I will recall below) to the $p$-adic world.

In the complex world the moduli space $Omega M_g$ of pairs $(X, omega)$, where $X$ is a smooth projective curve of genus $g$ and $0 neq omegain H^0(X, Omega_X)$ is a holomorphic differential on $X$, is a well studied object. The space $Omega M_g$ is a complex orbifold and the points are called translation surfaces. One first result is that every translation surface can be represented as a finite union of polygons in the complex plane with edge identifications. One gets this equivalence by integrating (using $omega$) along paths between the zeros of $omega$.

$Omega M_g$ comes equipped with a natural stratification: Let $kappa$ be a partition of $2g-2$ (the number of zeros of $omega$ counting multiplicity). Then $mathcal H(kappa)$ is the subset of $Omega M_g$ containing the points $(X,omega)$ such that the order of zeros of $omega$ corresponds to the partition $kappa$. This subset $mathcal H (kappa)$ is itself a complex orbifold. Roughly speaking, charts are given by integrating the same paths with respect to different differentials.
If you want more details on this topic, I would suggest having a look at a nice overview paper by: Alex Wright

I would like to bring those two results over to the $p$-adic world ($mathbb C_p$), so let me restate my question:

  • Using Berkovich integration on the analytification of a projective smooth curve $X$, is there a nice geometric description of the pair $(X^{an},omega)$ (where $omega$ is a global section of the differentials on $X^{an}$)?
  • On the strata of $Omega M_g$ (which exists algebraically) can we find some kind of “coordinates” by integrating using the differentials?

I would appreciate any kind of feedback, whether those results are clearly unobtainable or might very well be possible.