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edited Apr 13, 2017 at 12:58

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I answer the question of M. Emerton on Tate's algorithm here. I am not allowed to write long comment.

Sorry my memory is bad ! I just checked Tate's paper (Lect. Notes in Maths 476). The algorithm he describes is over a DVR with perfect residue field (note that the ring does not need to be complete actually). That the algorithms terminates is OK and is explained clearly in Tate and in Silverman (what I said in the above comment is not true).

In Tate's paper, §0, he said at two places that the result is conjectural. The first one is on the value of $\nu$ for the type $I^*_{\nu}$. Actually the algorithm will give $\nu$, but in § 7, page 51, he said ''A crude estimate gives $\nu={\rm ord} \Delta -3$ if I'm not mistaken.''. This is a typo (it would imply that the conductor $f=-1$ with the formula of page 50, line 6). Even worst, in residue characteristic 2, there can not be formula relating directly ${\rm ord} \Delta$ to $\nu$ because $f={\rm ord}\Delta-4-\nu$ does not depend only on the reduction type in this case. Silverman does not make this mistake.

The second place he said the result is conjectural is a few lines below, but I d'ont see what he was concerned with. Did he think aobut Ogg's formula $$f={\rm ord} \Delta - n +1$$ ($n=$ number of geometric irrducible components in the Kodaira-Néron type) ? In that time, Ogg's formula was not fully proved. But this becomes a little off topic, I will write what I know on this formula in a response here a response here.

I answer the question of M. Emerton on Tate's algorithm here. I am not allowed to write long comment.

Sorry my memory is bad ! I just checked Tate's paper (Lect. Notes in Maths 476). The algorithm he describes is over a DVR with perfect residue field (note that the ring does not need to be complete actually). That the algorithms terminates is OK and is explained clearly in Tate and in Silverman (what I said in the above comment is not true).

In Tate's paper, §0, he said at two places that the result is conjectural. The first one is on the value of $\nu$ for the type $I^*_{\nu}$. Actually the algorithm will give $\nu$, but in § 7, page 51, he said ''A crude estimate gives $\nu={\rm ord} \Delta -3$ if I'm not mistaken.''. This is a typo (it would imply that the conductor $f=-1$ with the formula of page 50, line 6). Even worst, in residue characteristic 2, there can not be formula relating directly ${\rm ord} \Delta$ to $\nu$ because $f={\rm ord}\Delta-4-\nu$ does not depend only on the reduction type in this case. Silverman does not make this mistake.

The second place he said the result is conjectural is a few lines below, but I d'ont see what he was concerned with. Did he think aobut Ogg's formula $$f={\rm ord} \Delta - n +1$$ ($n=$ number of geometric irrducible components in the Kodaira-Néron type) ? In that time, Ogg's formula was not fully proved. But this becomes a little off topic, I will write what I know on this formula in a response here.

I answer the question of M. Emerton on Tate's algorithm here. I am not allowed to write long comment.

Sorry my memory is bad ! I just checked Tate's paper (Lect. Notes in Maths 476). The algorithm he describes is over a DVR with perfect residue field (note that the ring does not need to be complete actually). That the algorithms terminates is OK and is explained clearly in Tate and in Silverman (what I said in the above comment is not true).

In Tate's paper, §0, he said at two places that the result is conjectural. The first one is on the value of $\nu$ for the type $I^*_{\nu}$. Actually the algorithm will give $\nu$, but in § 7, page 51, he said ''A crude estimate gives $\nu={\rm ord} \Delta -3$ if I'm not mistaken.''. This is a typo (it would imply that the conductor $f=-1$ with the formula of page 50, line 6). Even worst, in residue characteristic 2, there can not be formula relating directly ${\rm ord} \Delta$ to $\nu$ because $f={\rm ord}\Delta-4-\nu$ does not depend only on the reduction type in this case. Silverman does not make this mistake.

The second place he said the result is conjectural is a few lines below, but I d'ont see what he was concerned with. Did he think aobut Ogg's formula $$f={\rm ord} \Delta - n +1$$ ($n=$ number of geometric irrducible components in the Kodaira-Néron type) ? In that time, Ogg's formula was not fully proved. But this becomes a little off topic, I will write what I know on this formula in a response here.

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created Jan 26, 2010 at 21:41

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I answer the question of M. Emerton on Tate's algorithm here. I am not allowed to write long comment.

Sorry my memory is bad ! I just checked Tate's paper (Lect. Notes in Maths 476). The algorithm he describes is over a DVR with perfect residue field (note that the ring does not need to be complete actually). That the algorithms terminates is OK and is explained clearly in Tate and in Silverman (what I said in the above comment is not true).

In Tate's paper, §0, he said at two places that the result is conjectural. The first one is on the value of $\nu$ for the type $I^*_{\nu}$. Actually the algorithm will give $\nu$, but in § 7, page 51, he said ''A crude estimate gives $\nu={\rm ord} \Delta -3$ if I'm not mistaken.''. This is a typo (it would imply that the conductor $f=-1$ with the formula of page 50, line 6). Even worst, in residue characteristic 2, there can not be formula relating directly ${\rm ord} \Delta$ to $\nu$ because $f={\rm ord}\Delta-4-\nu$ does not depend only on the reduction type in this case. Silverman does not make this mistake.

The second place he said the result is conjectural is a few lines below, but I d'ont see what he was concerned with. Did he think aobut Ogg's formula $$f={\rm ord} \Delta - n +1$$ ($n=$ number of geometric irrducible components in the Kodaira-Néron type) ? In that time, Ogg's formula was not fully proved. But this becomes a little off topic, I will write what I know on this formula in a response here.