Growth of Mordell-Weil Rank of Elliptic Curves over Field ExtensionsTwo questions on isomorphic elliptic...
Growth of Mordell-Weil Rank of Elliptic Curves over Field Extensions
Two questions on isomorphic elliptic curvesElliptic curves with Mordell-Weil group Z/2Z over QWhich level structures on elliptic curves are twist-invariant?Inequality relating rank and analytic rankFull $n$-torsion of elliptic curves and the cyclotomic field of order $n$Are elliptic Kummer extensions big?Elliptic curves with potential good reduction over a prescribed extensionArtin representations appearing in Mordell-Weil groups of elliptic curvesInfinite extensions such that every elliptic curve has finite rankIncrease in rank of elliptic curves
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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 it already known that, for any integer $rgeq1$, there exists a finite-degree extension $mathbb{K}$ of $mathbb{F}$ so that $textrm{rank}left(Eleft(mathbb{K}right)right)geq r$?
elliptic-curves
asked 2 hours ago
MCSMCS
1805
$endgroup$
add a comment |
$begingroup$
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 it already known that, for any integer $rgeq1$, there exists a finite-degree extension $mathbb{K}$ of $mathbb{F}$ so that $textrm{rank}left(Eleft(mathbb{K}right)right)geq r$?
elliptic-curves
asked 2 hours ago
MCSMCS
1805
$endgroup$
add a comment |
$begingroup$
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 it already known that, for any integer $rgeq1$, there exists a finite-degree extension $mathbb{K}$ of $mathbb{F}$ so that $textrm{rank}left(Eleft(mathbb{K}right)right)geq r$?
elliptic-curves
asked 2 hours ago
MCSMCS
1805
$endgroup$
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 it already known that, for any integer $rgeq1$, there exists a finite-degree extension $mathbb{K}$ of $mathbb{F}$ so that $textrm{rank}left(Eleft(mathbb{K}right)right)geq r$?
elliptic-curves
elliptic-curves
asked 2 hours ago
MCSMCS
1805
asked 2 hours ago
MCSMCS
1805
asked 2 hours ago
MCSMCS
1805
asked 2 hours ago
MCSMCS
1805
asked 2 hours ago
MCSMCS
1805
1805
add a comment |
add a comment |
1 Answer
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This is (unfortunately for you) well-known. Theorem 10.1 of this paper shows that over an algebraically closed field of characteristic zero the Mordell-Weil rank of any positive-dimensional abelian variety is infinite, and (as remark 1 below the theorem mentions) this implies that for any abelian variety over field of characteristic zero, there are finite extensions over which the ranks get arbitrarily large.
The result also holds for fields of positive characteristic which are not algebraic over a finite field.
answered 1 hour ago
WojowuWojowu
6,62412851
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1 Answer
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active
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This is (unfortunately for you) well-known. Theorem 10.1 of this paper shows that over an algebraically closed field of characteristic zero the Mordell-Weil rank of any positive-dimensional abelian variety is infinite, and (as remark 1 below the theorem mentions) this implies that for any abelian variety over field of characteristic zero, there are finite extensions over which the ranks get arbitrarily large.
The result also holds for fields of positive characteristic which are not algebraic over a finite field.
answered 1 hour ago
WojowuWojowu
6,62412851
$endgroup$
add a comment |
$begingroup$
This is (unfortunately for you) well-known. Theorem 10.1 of this paper shows that over an algebraically closed field of characteristic zero the Mordell-Weil rank of any positive-dimensional abelian variety is infinite, and (as remark 1 below the theorem mentions) this implies that for any abelian variety over field of characteristic zero, there are finite extensions over which the ranks get arbitrarily large.
The result also holds for fields of positive characteristic which are not algebraic over a finite field.
answered 1 hour ago
WojowuWojowu
6,62412851
$endgroup$
add a comment |
$begingroup$
This is (unfortunately for you) well-known. Theorem 10.1 of this paper shows that over an algebraically closed field of characteristic zero the Mordell-Weil rank of any positive-dimensional abelian variety is infinite, and (as remark 1 below the theorem mentions) this implies that for any abelian variety over field of characteristic zero, there are finite extensions over which the ranks get arbitrarily large.
The result also holds for fields of positive characteristic which are not algebraic over a finite field.
answered 1 hour ago
WojowuWojowu
6,62412851
$endgroup$
This is (unfortunately for you) well-known. Theorem 10.1 of this paper shows that over an algebraically closed field of characteristic zero the Mordell-Weil rank of any positive-dimensional abelian variety is infinite, and (as remark 1 below the theorem mentions) this implies that for any abelian variety over field of characteristic zero, there are finite extensions over which the ranks get arbitrarily large.
The result also holds for fields of positive characteristic which are not algebraic over a finite field.
answered 1 hour ago
WojowuWojowu
6,62412851
answered 1 hour ago
WojowuWojowu
6,62412851
answered 1 hour ago
WojowuWojowu
6,62412851
answered 1 hour ago
WojowuWojowu
6,62412851
6,62412851
add a comment |
add a comment |
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