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What is the universal formal deformation of a supersingular elliptic curve?
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In Katz and Mazur's book "Arithmetic moduli of elliptic curves" (available here), the previously undefined notion of "universal formal deformation" is stated and I fail to understand how it is defined (actually, what it even is). The first occurence of it is on page 130 (page 71 in the pdf). More precisely, here is the statement.
Let $k$ be an algebraically closed field of characteristic $p>0$, $E_0/k$ a supersingular elliptic curve, and $mathbbE/W(k)[[T]]$ its universal formal deformation.
My question is: how is the object $mathbbE/W(k)[[T]]$ defined? In particular, what is $W$?
If anybody could provide a definition for it or a reference, I would be really thankful.
NB: If this is any relevant, a supersingular elliptic curve was previously defined as those elliptic curves whose $p$-divisible group is, up to $k$-isomorphism, the unique $1$-parameter formal Lie group over $k$ of height $2$.
definition elliptic-curves deformation-theory
asked Jul 30 at 7:08
Suzet
2,203427
add a comment |Â
up vote
0
down vote
favorite
In Katz and Mazur's book "Arithmetic moduli of elliptic curves" (available here), the previously undefined notion of "universal formal deformation" is stated and I fail to understand how it is defined (actually, what it even is). The first occurence of it is on page 130 (page 71 in the pdf). More precisely, here is the statement.
Let $k$ be an algebraically closed field of characteristic $p>0$, $E_0/k$ a supersingular elliptic curve, and $mathbbE/W(k)[[T]]$ its universal formal deformation.
My question is: how is the object $mathbbE/W(k)[[T]]$ defined? In particular, what is $W$?
If anybody could provide a definition for it or a reference, I would be really thankful.
NB: If this is any relevant, a supersingular elliptic curve was previously defined as those elliptic curves whose $p$-divisible group is, up to $k$-isomorphism, the unique $1$-parameter formal Lie group over $k$ of height $2$.
definition elliptic-curves deformation-theory
asked Jul 30 at 7:08
Suzet
2,203427
1
I presume $W(k)$ is the ring of Witt vectors over $k$.
– Lord Shark the Unknown
Jul 30 at 7:11
@LordSharktheUnknown I see, thank you very much. I actually never heard of them before. Then, I assume that $mathbbE/W(k)[[T]]$ is obtained by base change from $E_0/k$. Would you (or anybody reading this) know a reference as for the applications of this construction in the context of elliptic curves? Namely, what is meant by "formal deformation", what information this elliptic curve encodes, and so on...
– Suzet
Jul 30 at 7:28
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In Katz and Mazur's book "Arithmetic moduli of elliptic curves" (available here), the previously undefined notion of "universal formal deformation" is stated and I fail to understand how it is defined (actually, what it even is). The first occurence of it is on page 130 (page 71 in the pdf). More precisely, here is the statement.
Let $k$ be an algebraically closed field of characteristic $p>0$, $E_0/k$ a supersingular elliptic curve, and $mathbbE/W(k)[[T]]$ its universal formal deformation.
My question is: how is the object $mathbbE/W(k)[[T]]$ defined? In particular, what is $W$?
If anybody could provide a definition for it or a reference, I would be really thankful.
NB: If this is any relevant, a supersingular elliptic curve was previously defined as those elliptic curves whose $p$-divisible group is, up to $k$-isomorphism, the unique $1$-parameter formal Lie group over $k$ of height $2$.
definition elliptic-curves deformation-theory
asked Jul 30 at 7:08
Suzet
2,203427
In Katz and Mazur's book "Arithmetic moduli of elliptic curves" (available here), the previously undefined notion of "universal formal deformation" is stated and I fail to understand how it is defined (actually, what it even is). The first occurence of it is on page 130 (page 71 in the pdf). More precisely, here is the statement.
Let $k$ be an algebraically closed field of characteristic $p>0$, $E_0/k$ a supersingular elliptic curve, and $mathbbE/W(k)[[T]]$ its universal formal deformation.
My question is: how is the object $mathbbE/W(k)[[T]]$ defined? In particular, what is $W$?
If anybody could provide a definition for it or a reference, I would be really thankful.
NB: If this is any relevant, a supersingular elliptic curve was previously defined as those elliptic curves whose $p$-divisible group is, up to $k$-isomorphism, the unique $1$-parameter formal Lie group over $k$ of height $2$.
definition elliptic-curves deformation-theory
asked Jul 30 at 7:08
Suzet
2,203427
asked Jul 30 at 7:08
Suzet
2,203427
asked Jul 30 at 7:08
Suzet
2,203427
asked Jul 30 at 7:08
Suzet
2,203427
2,203427
1
I presume $W(k)$ is the ring of Witt vectors over $k$.
– Lord Shark the Unknown
Jul 30 at 7:11
@LordSharktheUnknown I see, thank you very much. I actually never heard of them before. Then, I assume that $mathbbE/W(k)[[T]]$ is obtained by base change from $E_0/k$. Would you (or anybody reading this) know a reference as for the applications of this construction in the context of elliptic curves? Namely, what is meant by "formal deformation", what information this elliptic curve encodes, and so on...
– Suzet
Jul 30 at 7:28
add a comment |Â
1
I presume $W(k)$ is the ring of Witt vectors over $k$.
– Lord Shark the Unknown
Jul 30 at 7:11
@LordSharktheUnknown I see, thank you very much. I actually never heard of them before. Then, I assume that $mathbbE/W(k)[[T]]$ is obtained by base change from $E_0/k$. Would you (or anybody reading this) know a reference as for the applications of this construction in the context of elliptic curves? Namely, what is meant by "formal deformation", what information this elliptic curve encodes, and so on...
– Suzet
Jul 30 at 7:28
1
1
I presume $W(k)$ is the ring of Witt vectors over $k$.
– Lord Shark the Unknown
Jul 30 at 7:11
I presume $W(k)$ is the ring of Witt vectors over $k$.
– Lord Shark the Unknown
Jul 30 at 7:11
@LordSharktheUnknown I see, thank you very much. I actually never heard of them before. Then, I assume that $mathbbE/W(k)[[T]]$ is obtained by base change from $E_0/k$. Would you (or anybody reading this) know a reference as for the applications of this construction in the context of elliptic curves? Namely, what is meant by "formal deformation", what information this elliptic curve encodes, and so on...
– Suzet
Jul 30 at 7:28
@LordSharktheUnknown I see, thank you very much. I actually never heard of them before. Then, I assume that $mathbbE/W(k)[[T]]$ is obtained by base change from $E_0/k$. Would you (or anybody reading this) know a reference as for the applications of this construction in the context of elliptic curves? Namely, what is meant by "formal deformation", what information this elliptic curve encodes, and so on...
– Suzet
Jul 30 at 7:28
add a comment |Â
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1
I presume $W(k)$ is the ring of Witt vectors over $k$.
– Lord Shark the Unknown
Jul 30 at 7:11
@LordSharktheUnknown I see, thank you very much. I actually never heard of them before. Then, I assume that $mathbbE/W(k)[[T]]$ is obtained by base change from $E_0/k$. Would you (or anybody reading this) know a reference as for the applications of this construction in the context of elliptic curves? Namely, what is meant by "formal deformation", what information this elliptic curve encodes, and so on...
– Suzet
Jul 30 at 7:28