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        Questions tagged [elliptic-curves]

        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. Note also the tag arithmetic-geometry as well as some related tags such as rational-points, abelian-varieties, heights. Please do not use this tag for questions related to ellipses; instead use conic-sections.

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        202 views

        Origin of $j$-invariant

        It is often asserted that the $j$-invariant was first introduced by Felix Klein. Is there any evidence for this claim? What works of Felix Klein do deal with it? What is the origin of the symbol $j$ ...
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        votes
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        646 views
        +50

        Universal homotheties for elliptic curves

        Let $K$ be a number field and $E_1, \cdots, E_n$ elliptic curves over $K$. Let $\ell$ be a prime. Then there exists an element $\sigma \in \text{Gal}(\overline{K}/K)$ such that $\sigma$ acts on $T_\...
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        votes
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        87 views

        Regularity of the modular curves $Y(N)$, $Y_1(N)$

        I'm reading the five chapter of the book of Katz-Mazur, Arithmetic moduli of elliptic curves, concerning regularity of the moduli problems of $\Gamma(N)$-structures, $\Gamma_1(N)$-structures and $\...
        6
        votes
        1answer
        129 views

        Endomorphism rings of ordinary elliptic curves

        Let's say $p$ is a prime and $t\neq 0$ is a trace of Frobenius that occurs over $\mathbb{F}_p$. The discriminant of the Frobenius polynomial is $\Delta:=t^2-4p.$ So we obtain $4p=t^2-\Delta.$ If $E$ ...
        3
        votes
        1answer
        305 views

        Squares in the set $\{\sum_{j=1}^m j^2: m\in\mathbb{N}\}$ [closed]

        Are there infinitely many squares in the set $$\{\sum_{j=1}^m j^2: m\in\mathbb{N}\} ?$$
        3
        votes
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        78 views

        What is the right basis of solutions of the Picard-Fuchs equation of the Legendre family around 0?

        I have been trying to reconstruct some elliptic curves theory computationally and have gotten stuck on some period computations. Specifically, let $$E_\lambda:\ y^2=x(x-1)(x-\lambda)$$ be the ...
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        125 views

        Topological modular forms on cuspidal elliptic curves

        If you take the moduli stack of smooth elliptic curves, there is a sheaf of $E_{\infty}$-rings on it whose global sections are isomorphic to $TMF$. This sheaf extends to the moduli stack of smooth/...
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        126 views

        Centralizers of Cartan subgroups

        Let $E$ be an elliptic curve with CM by an order $\mathcal O$ in an imaginary quadratic field $K$. Choose a basis for $E[N]$ to get an isomorphism $\operatorname{Aut}(E[N])\cong \operatorname{GL}_2(\...
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        84 views

        $\lambda$-invariants in cyclotomic $\mathbb{Z}_p$ extensions

        The idea that Selmer groups and class groups are related is not new. More recently, we understand that the growth patterns of fine Selmer groups are very similar to that of class groups in cyclotomic $...
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        votes
        1answer
        80 views

        Cassels Pairing for Fine Selmer groups

        Let $S_n$ be the Selmer group of $E/K_n$ where $K_n/K$ is the $n$-th layer of the cyclotomic $\mathbb{Z}_p$-extension of $K$ and $C_n$ be the torsion part of $S_n$. By Cassels pairing, we know that ...
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        0answers
        120 views

        Second derivative at 1 of L function of elliptic curve

        Let $E$ be an elliptic curve over $\mathbb Q$ of conductor $N$ and rank $0$. It follows from the functional equation that $$L'(E,1)=(\log(2\pi/\sqrt{N})+\gamma)L(E,1)$$ where $\gamma$ is Euler's ...
        3
        votes
        0answers
        91 views

        Counting elliptic curves over a number field by their Faltings height

        In this paper, Hortsch gives an asymptotic formula for the number of elliptic curves over $\mathbb{Q}$, given by their minimal Weierstrass model, of bounded Faltings' height. In general, is it ...
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        votes
        0answers
        132 views

        $\mu=0$ for CM Elliptic curves?

        Let $E$ be an elliptic curve defined over $F$ with CM by $\mathcal{O}_K$ where $K$ is an imaginary quadratic field. We may assume the $F$ contains $K$ and also contains the $p$-division points, where $...
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        votes
        0answers
        68 views

        Finding Coefficients of a Pairing Friendly Elliptic Curve

        Assuming that I know the size of the base field ($q$), size of the prime order subgroup ($r$), and embedding degree of the curve ($k$) of a pairing friendly elliptic curve ($E$), I would like to ...
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        votes
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        148 views

        Geometric interpretation of the rationality of the $j$-invariant

        Consider the modular curve $X_0(N)$. Let $\Phi_N(X,Y)$ be the modular equation. Then the curve $\Phi_N(X,Y) = 0$ can be interpreted as a model for $ X_0(N)$ because the function field of $X_0(N)$ is $\...

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