I'm puzzled by a part of the construction of the canonical subgroup of a "not too supersingular" elliptic curve. In Katz's paper, one produces a subgroup of the formal group of the elliptic curve but to me it is not clear how that subgroup "algebraize" to a subgroup of the elliptic curve.

Our set up is the following: we work over $R_0$ a complete DVR of mixed characteristic ($0,p$), $R$ is a $p$-adically complete $R_0$-algebra and $E/R$ is an elliptic curve. If it helps we can suppose that $p$ is nilpotent in $R$.

• First of all. What is the formal group is Katz considering? The completion of $E$ at its zero section or along the zero section mod $p$?

• In Tate's paper on $p$-divisible groups an equivalence of categories between formal Lie groups and connected $p$-divisible groups is stated over complete local noetherian rings. Does it holds over general $p$-adically complete rings? I don't know how to use this result, because even if I can assume that $R$ is noetherian and complete (abelian schemes are always defined over a noetherian ring) I don't think it would be enough to work over the localizations at primes to produce the canonical subgroup.

Any idea? Thanks.

I'm puzzled by a part of the construction of the canonical subgroup of a "not too supersingular" elliptic curve. In Katz's paper, one produces a subgroup of the formal group of the elliptic curve but to me it is not clear how that subgroup "algebraize" to a subgroup of the elliptic curve.

Our set up is the following: we work over $R_0$ a complete DVR of mixed characteristic ($0,p$), $R$ is a $p$-adically complete $R_0$-algebra and $E/R$ is an elliptic curve. If it helps we can suppose that $p$ is nilpotent in $R$.

• First of all. What is the formal group Katz is considering? The completion of $E$ at its zero section or along the zero section mod $p$?

• In Tate's paper on $p$-divisible groups an equivalence of categories between formal Lie groups and connected $p$-divisible groups is stated over complete local noetherian rings. Does it holds over general $p$-adically complete rings? I don't know how to use this result, because even if I can assume that $R$ is noetherian and complete (abelian schemes are always defined over a noetherian ring) I don't think it would be enough to work over the localizations at primes to produce the canonical subgroup.

Any idea? Thanks.

I'm puzzled by a part of the construction of the canonical subgroup of a "not too supersingular" ellipti curve. In Katz's paper, one produces a subgroup of the formal group of the elliptic curve but to me it is not clear how that subgroup "algebraize" to a subgroup of the elliptic curve.

Our set up is the following: we work over $R_0$ a complete DVR of mixed characteristic ($0,p$), $R$ is a $p$-adically complete $R_0$-algebra and $E/R$ is an elliptic curve. If it helps we can suppose that $p$ is nilpotent in $R$.

• First of all. What is the formal group is Katz considering? The completion of $E$ at its zero section or along the zero section mod $p$?

• In Tate's paper on $p$-divisible groups an equivalence of categories between formal Lie groups and connected $p$-divisible groups is stated over complete local noetherian rings. Does it holds over general $p$-adically complete rings? I don't know how to use this result, because even if I can assume that $R$ is noetherian and complete (abelian schemes are always defined over a noetherian ring) I don't think it would be enough to work over the localizations at primes to produce the canonical subgroup.

Any idea? Thanks.

I'm puzzled by a part of the construction of the canonical subgroup of a "not too supersingular" elliptic curve. In Katz's paper, one produces a subgroup of the formal group of the elliptic curve but to me it is not clear how that subgroup "algebraize" to a subgroup of the elliptic curve.

Our set up is the following: we work over $R_0$ a complete DVR of mixed characteristic ($0,p$), $R$ is a $p$-adically complete $R_0$-algebra and $E/R$ is an elliptic curve. If it helps we can suppose that $p$ is nilpotent in $R$.

• First of all. What is the formal group is Katz considering? The completion of $E$ at its zero section or along the zero section mod $p$?

• In Tate's paper on $p$-divisible groups an equivalence of categories between formal Lie groups and connected $p$-divisible groups is stated over complete local noetherian rings. Does it holds over general $p$-adically complete rings? I don't know how to use this result, because even if I can assume that $R$ is noetherian and complete (abelian schemes are always defined over a noetherian ring) I don't think it would be enough to work over the localizations at primes to produce the canonical subgroup.

Any idea? Thanks.