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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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      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 ...
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      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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      $\mu=0$ 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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      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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      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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      188 views

      Fermat's cubic equation in quadratic extension of $\mathbb{Q}$

      Is still relevant or interesting be capable to bring a criteria in order to classifly quadratic extensions of $\mathbb{Q}$ based on the existence or not existence of non-trivial solutions of Fermat's ...
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      A mysterious number related to Hasse-Weil L-function of elliptic curve

      Let $E/K$ be a non-isotrivial elliptic curve over a function field $K$ of characteristic $p$, with field of constant $F_q$, with semistable reduction. Its Hasse-Weil L-function $L(s)$ is a polynomial ...
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      172 views

      What integer value can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$?

      Fix a positive integer $g$. What positive integer $N$ can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$ ? For example, as there is no abelian varieties over $\mathbb Z$, $N$ ...
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      2answers
      194 views

      Growth of Mordell-Weil Rank of Elliptic Curves over Field Extensions

      I'm a graduate student just checking to make sure that what he's researching isn't already known. Let $\mathbb{F}$ be a number field, and let $E$ be an elliptic curve defined over $\mathbb{F}$. Is ...
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      147 views

      How to decide the isogeny between Neron models is etale?

      Let there be an isogeny $f:A_1 \rightarrow A_2$ between two abelian varieties over a $p$-adic field $F$ and assume $f$ has degree $p^n$. By the universal property we get a moprhism $f_0: \mathcal{A}_1 ...
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      Find wrapping angle of helix on a torus

      I need some help in calculating the wrapping angle of a spiral helix wrapped on a torus with constant angle against all the meridians of the torus. The wrapping angle (or the angle measured around and/...
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      52 views

      anomalous primes and CM elliptic curves

      Let $E$ be an elliptic curve defined over a number field $F$ and suppose $E$ has CM by $\mathcal{O}_K$ where $K$ is an imaginary quadratic field. What can we say about the non-anomalous primes of such ...
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      226 views

      Deligne Pairing v.s. Weil Pairing on a Family of curves

      We have the Deligne Pairing on a family of curve $\pi:X\to S$ by using $$\langle L,M\rangle_{\mathrm{Pic}^0(X/S)}=\det R\pi_*(L\otimes M) \otimes (\det R\pi_*L)^{-1}\otimes (\det R\pi_*M)^{-1} \otimes ...
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      231 views

      What do non-principal divisors in a Picard group look like

      The way Divisors on Elliptic Curves are motivated in cryptography is to say it's a convenient way to represent rational functions by keeping track of multiplicities of the zeros and poles of a ...
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      111 views

      Increase in rank of elliptic curves

      I expect answers to these questions are known, or at least partial answers are known: Let $E$ be a rank 0 elliptic curve defined over $\mathbb{Q}$ and let $p$ be an odd prime. Is it possible that the ...

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