Mixing
BSD conjecture for rank 1 elliptic curves
Clash Royale CLAN TAG#URR8PPP
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13
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Let $E/mathbbQ$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that
$$textord_s=1L(E, s)=textrank E(mathbbQ).$$
Thanks to the work of Gross-Zagier and Kolyvagin, we know that this conjecture is true if $textord_s=1L(E, s)le 1$.
What is known in the case $textrank E(mathbbQ)le 1$? Is it known that if $textrank E(mathbbQ)=1$, then $L'(E, 1)neq 0$?
Thank you!
nt.number-theory reference-request arithmetic-geometry elliptic-curves l-functions
asked Aug 25 at 10:29
baobab
685
add a comment |Â
up vote
13
down vote
favorite
Let $E/mathbbQ$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that
$$textord_s=1L(E, s)=textrank E(mathbbQ).$$
Thanks to the work of Gross-Zagier and Kolyvagin, we know that this conjecture is true if $textord_s=1L(E, s)le 1$.
What is known in the case $textrank E(mathbbQ)le 1$? Is it known that if $textrank E(mathbbQ)=1$, then $L'(E, 1)neq 0$?
Thank you!
nt.number-theory reference-request arithmetic-geometry elliptic-curves l-functions
asked Aug 25 at 10:29
baobab
685
8
In fairness, it's probably worth mentioning that in addition to the work of G-Z and Kolyvagin, one also needs the theorem of Wiles et al that every elliptic curve over $mathbb Q$ is modular, since [G-Z] and [K] start with the assumption that their elliptic curve is modular.
– Joe Silverman
Aug 25 at 11:35
...and a non-vanishing result for special values of L functions.
– Pasten
Aug 26 at 0:29
add a comment |Â
up vote
13
down vote
favorite
up vote
13
down vote
favorite
Let $E/mathbbQ$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that
$$textord_s=1L(E, s)=textrank E(mathbbQ).$$
Thanks to the work of Gross-Zagier and Kolyvagin, we know that this conjecture is true if $textord_s=1L(E, s)le 1$.
What is known in the case $textrank E(mathbbQ)le 1$? Is it known that if $textrank E(mathbbQ)=1$, then $L'(E, 1)neq 0$?
Thank you!
nt.number-theory reference-request arithmetic-geometry elliptic-curves l-functions
asked Aug 25 at 10:29
baobab
685
Let $E/mathbbQ$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that
$$textord_s=1L(E, s)=textrank E(mathbbQ).$$
Thanks to the work of Gross-Zagier and Kolyvagin, we know that this conjecture is true if $textord_s=1L(E, s)le 1$.
What is known in the case $textrank E(mathbbQ)le 1$? Is it known that if $textrank E(mathbbQ)=1$, then $L'(E, 1)neq 0$?
Thank you!
nt.number-theory reference-request arithmetic-geometry elliptic-curves l-functions
asked Aug 25 at 10:29
baobab
685
edited Aug 25 at 10:40
asked Aug 25 at 10:29
baobab
685
asked Aug 25 at 10:29
baobab
685
asked Aug 25 at 10:29
baobab
685
685
8
In fairness, it's probably worth mentioning that in addition to the work of G-Z and Kolyvagin, one also needs the theorem of Wiles et al that every elliptic curve over $mathbb Q$ is modular, since [G-Z] and [K] start with the assumption that their elliptic curve is modular.
– Joe Silverman
Aug 25 at 11:35
...and a non-vanishing result for special values of L functions.
– Pasten
Aug 26 at 0:29
add a comment |Â
8
In fairness, it's probably worth mentioning that in addition to the work of G-Z and Kolyvagin, one also needs the theorem of Wiles et al that every elliptic curve over $mathbb Q$ is modular, since [G-Z] and [K] start with the assumption that their elliptic curve is modular.
– Joe Silverman
Aug 25 at 11:35
...and a non-vanishing result for special values of L functions.
– Pasten
Aug 26 at 0:29
8
8
In fairness, it's probably worth mentioning that in addition to the work of G-Z and Kolyvagin, one also needs the theorem of Wiles et al that every elliptic curve over $mathbb Q$ is modular, since [G-Z] and [K] start with the assumption that their elliptic curve is modular.
– Joe Silverman
Aug 25 at 11:35
In fairness, it's probably worth mentioning that in addition to the work of G-Z and Kolyvagin, one also needs the theorem of Wiles et al that every elliptic curve over $mathbb Q$ is modular, since [G-Z] and [K] start with the assumption that their elliptic curve is modular.
– Joe Silverman
Aug 25 at 11:35
...and a non-vanishing result for special values of L functions.
– Pasten
Aug 26 at 0:29
...and a non-vanishing result for special values of L functions.
– Pasten
Aug 26 at 0:29
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
14
down vote
accepted
The following theorem is due to Chris Skinner, in this 2014 paper.
Let E/Q be an elliptic curve such that rank E(Q) = 1 and the
Tate-Shafarevich group Sha(E / Q) is finite, and some other technical
assumptions hold. Then $ord_s = 1 L(E, s) = 1$, and in particular
$L'(E, 1) ne 0$.
This is, as far as I know, the best one can do at the moment; if you don't know that Sha (or at least its p-primary part for some p) is finite, then you're stuck.
answered Aug 25 at 11:27
David Loeffler
18.7k146112
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
14
down vote
accepted
The following theorem is due to Chris Skinner, in this 2014 paper.
Let E/Q be an elliptic curve such that rank E(Q) = 1 and the
Tate-Shafarevich group Sha(E / Q) is finite, and some other technical
assumptions hold. Then $ord_s = 1 L(E, s) = 1$, and in particular
$L'(E, 1) ne 0$.
This is, as far as I know, the best one can do at the moment; if you don't know that Sha (or at least its p-primary part for some p) is finite, then you're stuck.
answered Aug 25 at 11:27
David Loeffler
18.7k146112
add a comment |Â
up vote
14
down vote
accepted
The following theorem is due to Chris Skinner, in this 2014 paper.
Let E/Q be an elliptic curve such that rank E(Q) = 1 and the
Tate-Shafarevich group Sha(E / Q) is finite, and some other technical
assumptions hold. Then $ord_s = 1 L(E, s) = 1$, and in particular
$L'(E, 1) ne 0$.
This is, as far as I know, the best one can do at the moment; if you don't know that Sha (or at least its p-primary part for some p) is finite, then you're stuck.
answered Aug 25 at 11:27
David Loeffler
18.7k146112
add a comment |Â
up vote
14
down vote
accepted
up vote
14
down vote
accepted
The following theorem is due to Chris Skinner, in this 2014 paper.
Let E/Q be an elliptic curve such that rank E(Q) = 1 and the
Tate-Shafarevich group Sha(E / Q) is finite, and some other technical
assumptions hold. Then $ord_s = 1 L(E, s) = 1$, and in particular
$L'(E, 1) ne 0$.
This is, as far as I know, the best one can do at the moment; if you don't know that Sha (or at least its p-primary part for some p) is finite, then you're stuck.
answered Aug 25 at 11:27
David Loeffler
18.7k146112
The following theorem is due to Chris Skinner, in this 2014 paper.
Let E/Q be an elliptic curve such that rank E(Q) = 1 and the
Tate-Shafarevich group Sha(E / Q) is finite, and some other technical
assumptions hold. Then $ord_s = 1 L(E, s) = 1$, and in particular
$L'(E, 1) ne 0$.
This is, as far as I know, the best one can do at the moment; if you don't know that Sha (or at least its p-primary part for some p) is finite, then you're stuck.
answered Aug 25 at 11:27
David Loeffler
18.7k146112
answered Aug 25 at 11:27
David Loeffler
18.7k146112
answered Aug 25 at 11:27
David Loeffler
18.7k146112
answered Aug 25 at 11:27
David Loeffler
18.7k146112
18.7k146112
add a comment |Â
add a comment |Â
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8
In fairness, it's probably worth mentioning that in addition to the work of G-Z and Kolyvagin, one also needs the theorem of Wiles et al that every elliptic curve over $mathbb Q$ is modular, since [G-Z] and [K] start with the assumption that their elliptic curve is modular.
– Joe Silverman
Aug 25 at 11:35
...and a non-vanishing result for special values of L functions.
– Pasten
Aug 26 at 0:29