Applications of integral p-adic Hodge theory
$begingroup$
What are some geometric applications of integral p-adic Hodge theory (as opposed to rational p-adic Hodge theory)? Answers which understand Hodge theory as the study of Galois stable $mathbb{Z}_p$-lattices are OK but I am more interested in applications of comparison theorems (like the one recently established by Bhatt--Morrow--Scholze).
I am aware of certain applications to the question of nice reductions of varieties (e.g. this or this).
It appears that this question is not a duplicate since the paper of Berthelot et al mentioned there uses rational p-adic Hodge theory.
ag.algebraic-geometry arithmetic-geometry
asked Feb 14 at 7:28
paulpaul
1604
$endgroup$
add a comment |
$begingroup$
What are some geometric applications of integral p-adic Hodge theory (as opposed to rational p-adic Hodge theory)? Answers which understand Hodge theory as the study of Galois stable $mathbb{Z}_p$-lattices are OK but I am more interested in applications of comparison theorems (like the one recently established by Bhatt--Morrow--Scholze).
I am aware of certain applications to the question of nice reductions of varieties (e.g. this or this).
It appears that this question is not a duplicate since the paper of Berthelot et al mentioned there uses rational p-adic Hodge theory.
ag.algebraic-geometry arithmetic-geometry
asked Feb 14 at 7:28
paulpaul
1604
$endgroup$
add a comment |
$begingroup$
What are some geometric applications of integral p-adic Hodge theory (as opposed to rational p-adic Hodge theory)? Answers which understand Hodge theory as the study of Galois stable $mathbb{Z}_p$-lattices are OK but I am more interested in applications of comparison theorems (like the one recently established by Bhatt--Morrow--Scholze).
I am aware of certain applications to the question of nice reductions of varieties (e.g. this or this).
It appears that this question is not a duplicate since the paper of Berthelot et al mentioned there uses rational p-adic Hodge theory.
ag.algebraic-geometry arithmetic-geometry
asked Feb 14 at 7:28
paulpaul
1604
$endgroup$
What are some geometric applications of integral p-adic Hodge theory (as opposed to rational p-adic Hodge theory)? Answers which understand Hodge theory as the study of Galois stable $mathbb{Z}_p$-lattices are OK but I am more interested in applications of comparison theorems (like the one recently established by Bhatt--Morrow--Scholze).
I am aware of certain applications to the question of nice reductions of varieties (e.g. this or this).
It appears that this question is not a duplicate since the paper of Berthelot et al mentioned there uses rational p-adic Hodge theory.
ag.algebraic-geometry arithmetic-geometry
ag.algebraic-geometry arithmetic-geometry
asked Feb 14 at 7:28
paulpaul
1604
asked Feb 14 at 7:28
paulpaul
1604
edited Feb 14 at 10:55
paul
asked Feb 14 at 7:28
paulpaul
1604
asked Feb 14 at 7:28
paulpaul
1604
asked Feb 14 at 7:28
paulpaul
1604
1604
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
One major application of research in integral $p$-adic Hodge theory is in proving modularity results, e.g. for elliptic curves. Here one wants to understand liftings of global mod p Galois representations to $p$-adic representations which have some reasonable local properties at $p$ (e.g. being crystalline or semistable); and to do this it's important to have a good understanding of the corresponding local problem at $p$, which belongs squarely to integral $p$-adic Hodge theory.
E.g. have a look at this famous paper:
Breuil, Conrad, Diamond, Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises.
This was the paper that completed the proof of modularity of elliptic curves over $mathbb{Q}$ (finishing the job begun by Wiles in his work on Fermat's last theorem). The key ingredient in the proof is Breuil's work on classifying p-divisible groups over p-adic integer rings via semilinear objects ("Breuil modules").
This is just one of many examples where progress in integral p-adic Hodge theory has been instrumental in studying global modularity problems. You might want to look at recent works of people like Tong Liu and David Savitt for more examples of this kind.
answered Feb 14 at 9:34
David LoefflerDavid Loeffler
19.8k150120
$endgroup$
add a comment |
Your Answer
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "504"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f323195%2fapplications-of-integral-p-adic-hodge-theory%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
One major application of research in integral $p$-adic Hodge theory is in proving modularity results, e.g. for elliptic curves. Here one wants to understand liftings of global mod p Galois representations to $p$-adic representations which have some reasonable local properties at $p$ (e.g. being crystalline or semistable); and to do this it's important to have a good understanding of the corresponding local problem at $p$, which belongs squarely to integral $p$-adic Hodge theory.
E.g. have a look at this famous paper:
Breuil, Conrad, Diamond, Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises.
This was the paper that completed the proof of modularity of elliptic curves over $mathbb{Q}$ (finishing the job begun by Wiles in his work on Fermat's last theorem). The key ingredient in the proof is Breuil's work on classifying p-divisible groups over p-adic integer rings via semilinear objects ("Breuil modules").
This is just one of many examples where progress in integral p-adic Hodge theory has been instrumental in studying global modularity problems. You might want to look at recent works of people like Tong Liu and David Savitt for more examples of this kind.
answered Feb 14 at 9:34
David LoefflerDavid Loeffler
19.8k150120
$endgroup$
add a comment |
$begingroup$
One major application of research in integral $p$-adic Hodge theory is in proving modularity results, e.g. for elliptic curves. Here one wants to understand liftings of global mod p Galois representations to $p$-adic representations which have some reasonable local properties at $p$ (e.g. being crystalline or semistable); and to do this it's important to have a good understanding of the corresponding local problem at $p$, which belongs squarely to integral $p$-adic Hodge theory.
E.g. have a look at this famous paper:
Breuil, Conrad, Diamond, Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises.
This was the paper that completed the proof of modularity of elliptic curves over $mathbb{Q}$ (finishing the job begun by Wiles in his work on Fermat's last theorem). The key ingredient in the proof is Breuil's work on classifying p-divisible groups over p-adic integer rings via semilinear objects ("Breuil modules").
This is just one of many examples where progress in integral p-adic Hodge theory has been instrumental in studying global modularity problems. You might want to look at recent works of people like Tong Liu and David Savitt for more examples of this kind.
answered Feb 14 at 9:34
David LoefflerDavid Loeffler
19.8k150120
$endgroup$
add a comment |
$begingroup$
One major application of research in integral $p$-adic Hodge theory is in proving modularity results, e.g. for elliptic curves. Here one wants to understand liftings of global mod p Galois representations to $p$-adic representations which have some reasonable local properties at $p$ (e.g. being crystalline or semistable); and to do this it's important to have a good understanding of the corresponding local problem at $p$, which belongs squarely to integral $p$-adic Hodge theory.
E.g. have a look at this famous paper:
Breuil, Conrad, Diamond, Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises.
This was the paper that completed the proof of modularity of elliptic curves over $mathbb{Q}$ (finishing the job begun by Wiles in his work on Fermat's last theorem). The key ingredient in the proof is Breuil's work on classifying p-divisible groups over p-adic integer rings via semilinear objects ("Breuil modules").
This is just one of many examples where progress in integral p-adic Hodge theory has been instrumental in studying global modularity problems. You might want to look at recent works of people like Tong Liu and David Savitt for more examples of this kind.
answered Feb 14 at 9:34
David LoefflerDavid Loeffler
19.8k150120
$endgroup$
One major application of research in integral $p$-adic Hodge theory is in proving modularity results, e.g. for elliptic curves. Here one wants to understand liftings of global mod p Galois representations to $p$-adic representations which have some reasonable local properties at $p$ (e.g. being crystalline or semistable); and to do this it's important to have a good understanding of the corresponding local problem at $p$, which belongs squarely to integral $p$-adic Hodge theory.
E.g. have a look at this famous paper:
Breuil, Conrad, Diamond, Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises.
This was the paper that completed the proof of modularity of elliptic curves over $mathbb{Q}$ (finishing the job begun by Wiles in his work on Fermat's last theorem). The key ingredient in the proof is Breuil's work on classifying p-divisible groups over p-adic integer rings via semilinear objects ("Breuil modules").
This is just one of many examples where progress in integral p-adic Hodge theory has been instrumental in studying global modularity problems. You might want to look at recent works of people like Tong Liu and David Savitt for more examples of this kind.
answered Feb 14 at 9:34
David LoefflerDavid Loeffler
19.8k150120
answered Feb 14 at 9:34
David LoefflerDavid Loeffler
19.8k150120
answered Feb 14 at 9:34
David LoefflerDavid Loeffler
19.8k150120
answered Feb 14 at 9:34
David LoefflerDavid Loeffler
19.8k150120
19.8k150120
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f323195%2fapplications-of-integral-p-adic-hodge-theory%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
