Tagged Questions

An elliptic curve is an algebraic curve of genus one with some additional properties. Questions with this tag will often have the top-level tags nt.number-theory or ag.algebraic-geometry in addition; note also the tag arithmetic-geometry as well as some related tags such as rational-points, ...

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Weierstrass form of genus one $y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0$

Related to the n-conjecture. We are looking for Weierstrass form and map from it of the genus one curve: $$y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0$$ It is ...
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Straight line complexity of $k!a$ where $(a,p)=1$

In Qi Cheng's paper, an algorithm is provided to calculate $k!a$ at some random $a∈ℕ$ ($a$ is not input to algorithm, only $k$ is), what is the probability that given a prime $p$ such that ...
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Equidistribution of representations by a binary cubic form

Let $f(x,y)\in\mathbb{Z}[x,y]$ be a binary cubic form with nonzero discriminant, and for a positive integer $m$ consider the integral representations $f(x,y)=m$. Assume that the number of ...
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Why there are only finitely many $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves with CM by $\mathcal{O}$?

For someone who does not have a very extensive knowledge of number theory, what is a good intuitive explanation as to why there are only finitely many $\overline{\mathbb{Q}}$ isomorphism classes of ...
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Result of Deuring, intuitive way to see it's true/quickest way to prove?

There is the following result of Deuring that goes as follows: Let $E/L$ be an elliptic curve defined over a number field $L$ with complex multiplication by an order $\mathcal{O}$ in an imaginary ...
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Question on paper of Stewart and Top about ranks of elliptic curves over Q(t)

I'm reading "On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms" by Stewart and Top, and struggling to understand the argument on pg 962 which shows that the rank of a ...
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isogeny clases of CM abelian varieties

Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$ and with complex multiplication by a CM field $K$. Looking at the action of $K$ on $H^0(A, \Omega^1_A)$ one gets a CM type of $K$, ...
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Conditions for splitting of short exact sequence?

Are there conditions under which the short exact sequence $$0\rightarrow E (K)/mE (K)\rightarrow H^1_{Sel}(K,E_m)\rightarrow \Sha(E|K)_m\rightarrow 0$$ splits? I assume $K$ to be a number field and ...
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References for modular curves over finite fields [closed]

I'm looking for a detailed reference for modular curves over finite fields, such as $X(N)$, $X_1(N)$, and $X_0(N)$. There seems to be a lot of literature dealing with them over $\mathbb{C}$, but I'm ...
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Special type Diophantine equations with integer solutions

The following problem on Diophantine equation is still solved or not I don't know. However, I found few solutions by trail and error method. Problem: $X^2 - X = Y^5 - Y$ has integer solutions or not? ...
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From an eigenfom with $\mathbf{Q}$-coefficients to $j$-invariants

Given a cuspidal, classical eigenform $f\in S_2(\Gamma_0(N))$ of weight $2$ and with $\mathbf{Q}$-coefficients is there a way of describing the set $J_f$ of $j$-invariants of the elliptic curves lying ...
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What is the complexity of finding a generator for the cyclic elliptic curves?

Let $E$ be an elliptic curve which is defined over a finite field $\mathbb{F}_p$, where $p$ is a prime number. If we know that $E(\mathbb{F}_p)$ is cycyclic, is there an algorithm to find its ...
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Serre's Open Image Theorem Without Shafarevich's Theorem

In Abelian l-adic Representations and Elliptic Curves (1968), J. P. Serre showed that the adelic representation $$\rho_{E}\colon G_K \to \mathrm{GL}(\hat{\mathbb{Z}}^2)$$ associated to an elliptic ...
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Secant varieties of curves in $\mathbb{P}^4$

My question is motivated by the following simple observations. By a standard dimensions count in $\mathbb{P}^4$ there should not exist neither an hypersurface of degree $3$ with multiplicity $2$ in ...
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Are the elliptic curve discrete log problem and the elliptic curve Diffie-Hellman problem equivalent?

Suppose that $G=\langle g\rangle$ is a general group of order $p$. Maurer has introduced an algorithm to reduce the discrete log problem to the Diffie-Hellman problem under a conjecture about smooth ...
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Prime divisors of the norm of the first coefficient of an elliptic newform at width-1 cusps.

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ and let $f$ be its newform. Suppose $p \geq 5$ is a prime such that $p^2 \mid N$. We assume $f$ is $p$-minimal, which is equivalent to that the ...
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On the number of 3-Selmer elements of rational elliptic curves

I am trying to understand a step in the proof of Theorem 39 in the recent work of Bhargava and Shankar, "Ternary Cubic Forms having bounded invariants, and the existence of a positive proportion of ...
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Elliptic curves and supercuspidal representations of conductor $p^2$

Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $p \geq 5$ be a prime of additive reduction for $E$. Let $f$ be the newform associated to $E$, and let $\pi$ be the irreducible admissible ...
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Example of a genus-1 degree-7 plane curve

I am wondering if anyone knows how to construct an explicit example of an irreducible plane curve of degree 7 with 14 double points. Such a curve would have genus 1. One can show that for a general ...
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n torsion groups of quadratic twists of elliptic curves

If $E$ is an elliptic curve over a number field $K$ and $E^{F}$ is a quadratic twist of $E$. Then it is stated in Ranks of twists of elliptic curves and Hilbert’s tenth problem" due to Mazur and ...
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Is the upper half plane an algebraic stack?

Here by algebraic stack I mean an algebraic stack over the etale site $\textbf{Sch}/\mathbb{C}$. So I've read from various nonrigorous sources that the upper half plane $\mathcal{H}$ is a fine moduli ...
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What are the strongest conjectured uniform versions of Serre's Open Image Theorem?

This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho_{E,\ell}$ associated to a non-CM ...
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Order of torsion group

What can one say about the order of the torsion group of an elliptic curve defined over the compositum of all quadratic extensions of $\mathbb{Q}$ ?
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Elliptic Curve Multiplication [closed]

What would happen if I performed Elliptic Curve multiplication on some random point within the FiniteField that wasn't actually on the curve? I assume that I would get a point in return but would that ...
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Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system, $$x_1^2+x_2^2+\dots+x_n^2 = y^2$$ $$x_1^3+x_2^3+\dots+x_n^3 = z^3$$ and define the function, $$F(s_m) = x_1+x_2+\dots+x_n$$ For $n\geq3$, using an ...
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Mazur's torsion theorem on elliptic curves and its generalisations

I want to study Mazur's torsion theorem for elliptic curves over $Q$ and its generalizations for number fields, i.e., papers by Kamienny, Kenku & Momose, Filip Najman. So please suggest to me what ...
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how do automorphisms of elliptic curves act on the Tate module?

Let $E/k$ be an elliptic curve over some algebraically closed field $k$ of characteristic $p\ge 0$. It's known that $Aut(E)$ acts faithfully on the Tate module $T_\ell(E)$ ($\ell\ne p$) with ...
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Interpreting Frobenius pullback as an invariant differential in the case of an elliptic curve

Let $S$ be an affine scheme of characteristic $p > 0$, let $E \rightarrow S$ be an elliptic curve over $S$, and let $F$ denote the absolute Frobenius. Since $E$ is its own $\mathrm{Pic}^0$ there is ...
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Models for the moduli space $\overline{M}_{1,n}$

Let $\overline{M}_{1,n}$ denote the coarse moduli space of $n$-pointed elliptic curves. Is there an explicit description of these spaces (a la Kapranov's construction) for low $n$? Apparently this ...