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Question on x coordinates of Mordell Curves where $y^2=x^3+k$ and $k^2 = 1$ mod $24$
In my ongoing search for Mordell curves of rank 8 and above I have currently identified 144,499 curves of a type where $k$ is squarefree and $k^2 = 1$ mod $24$.
In each case the x coordinates are ...
6
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1answer
344 views
Why is the gcd so large in an identity related to the $abc$ conjecture?
Consider the identity
$$ (x+z)^5+(y-z)^5 = (-3 x + 4 y)^2 (x + y)^3 + (x+y) f(x,y,z) $$
Where $f(x,y,z)=(-8*x^4 + 5*x^3*y + 24*x^2*y^2 - 9*x*y^3 - 15*y^4 + 5*x^3*z - 5*x^2*y*z + 5*x*y^2*z - 5*y^3*z + ...
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2answers
127 views
What is the exact meaning of the real period in the $p$-adic formulation of BSD?
Let $E$ be an elliptic curve over $\mathbf{Q}$ which has split multiplicative reduction at $p$ (a prime). If one chooses a global Neron model of $E$ over $\mathbf{Z}$ (unique up to unique isomorphism ...
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47 views
What is the amount of abstract algebra needed to study elliptic curves? [migrated]
To be more specific, how much is needed to understand the book 'Rational points on elliptic curves' by Silverman?
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140 views
Is it expected that every natural number is the rank of some elliptic curve over the rationals?
It is a well-known problem on the theory of elliptic curves that the rank of an elliptic curve (the number of generators of the free part of the Mordell group of the elliptic curve) cannot be ...
3
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1answer
135 views
Mordell-Weil of an elliptic surface after adjoining a nontorsion section: as small as possible?
Let $k$ be an algebraically closed field of characteristic $0$, let $C_{/k}$ be a nice (smooth, projective, geometrically integral curve), let $K = k(C)$, and let $\overline{K}$ be an algebraic ...
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59 views
Picard-Fuchs equation of elliptic curves with level $N$ structure.
Let $X_N$ be the moduli space of elliptic curves with level $N$ structure. Is the Piard Fuchs equation of the universal family over $X_N$ (take some covering if $X_N$ is not a fine moduli space)?
In ...
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0answers
165 views
Birch/Swinnerton-Dyer “Notes on Elliptic Curves II”
I would like to know if any of you know if there is a more general treatment to what Birch and Swinnerton-Dyer did in "Notes on Elliptic Curves II" (http://www.ams.org/mathscinet-getitem?mr=179168).
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3
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1answer
97 views
Division Field of a nonCM elliptic curve
This might be a ridiculous question, but please bear with me.
Let $E$ be an elliptic curve over a $p$-adic field $K$. Denote by $K(E_{p^∞}):=\bigcup_{n∈Z≥1} K(E[p^n])$ the field extension obtained by ...
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1answer
119 views
Hecke $L$-series exercise in Silverman's Advanced Topics in Arithmetic of EC
This has been posted on SE, but I haven't gotten a reply, so I thought I'll try my luck here.
I would like to refer you to 2.30 & 2.32 in Silverman's book Advanced Topics in the Arithmetic of ...
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1answer
151 views
Elliptic Curves isogenous only over an extension?
Let $l$ be a prime $\geq 5$. Does there exist a pair $E,E'$ of elliptic curves, both defined over the same number field $K$, which are not $l$-isogenous over $K$, but are $l$-isogenous over a ...
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0answers
241 views
A property of supersingular $j$-invariants (reference request)
Edit 2: For those who understandably don't want to read such a long post, I think Voloch's suggestion reduces the problem to asking whether $j$-invariants of supersingular curves are 3rd powers in ...
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1answer
89 views
A certain property of elliptic curves in a paper by Rees
In the paper "On a problem of Zariski", David Rees presents a counterexample to the following problem of Zariski.
Let $F/k$ be a f.g. field extension, $S$ a f.g. normal integral domain over $k$ ...
3
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1answer
227 views
a question on CM elliptic curves
Let $E$ be an elliptic curve over $\overline{\mathbb{Q}}$ defined by an equation
$y^2=4x^3-g_2x-g_3$
and let $\omega=\int_\gamma \frac{dx}{y}$ be the integral of the regular differential form ...
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1answer
60 views
Hecke Character vs Grossencharakter
I would like to know if there is any difference between
(1) an algebraic Hecke character
(2) a Hecke character
(3) a Grössencharakter
All of the above in the setting of ellitpic curves with complex ...
