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David Loeffler
David Loeffler
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The only method we knowThe only method we know of to prove analytic / meromorphic continuation of zeta-functions of alg. varieties over number fields is to go via some kind of modularity, or potential modularity, statement. So even making sense of the statement of your program already requires the key piece of technology developed to prove FLT (and modularity of these high-degree Fermat curves is likely to be vastly harder than elliptic curves).

That is, your "whole-hearted belief" makes this proposed program circular.

EDIT: Dan Loughran's comment alerted me to the fact that this is misleading. Fermat curves have lots of automorphisms, and there are enough of these to prove analytic / meromorphic continuationforce the L-function to factor as a product of zetaL-functions of alg. varieties over number fields is to go via some kindGroessencharacters of modularity, or potential modularitycyclotomic fields, statement. So even making sensefor which analytic continuation + functional equation of the statement of your program already requires the key piece of technology developed to prove FLT (and modularity of these highL-degree Fermat curves is likely to befunction are known; see vastly harder than elliptic curves)Aoki (1991).

That This is, your "wholeof course, an automorphy statement of a kind, but one that's far easier than automorphy of general higher-hearted belief" makes this proposed program circulargenus curves would be.

There are several othersremaining serious obstacles as well; for. For instance, once you go beyond genus 1, the link between the special values of the L-function of a curve of genus > 1 and the existence or otherwise of rational points on the curve is very indirect. Rather than points on $X$, the $L$-series gives you information about points on $Jac(X)$. Sometimes, with lots of extra work, you can translate this into information about points on $X$ itself (the Chabauty--Coleman method), but there are lots of cases where this does not apply.

The only method we know of to prove analytic / meromorphic continuation of zeta-functions of alg. varieties over number fields is to go via some kind of modularity, or potential modularity, statement. So even making sense of the statement of your program already requires the key piece of technology developed to prove FLT (and modularity of these high-degree Fermat curves is likely to be vastly harder than elliptic curves).

That is, your "whole-hearted belief" makes this proposed program circular.

There are several others serious obstacles as well; for instance, once you go beyond genus 1, the link between the special values of the L-function of a curve of genus > 1 and the existence or otherwise of rational points on the curve is very indirect. Rather than points on $X$, the $L$-series gives you information about points on $Jac(X)$. Sometimes, with lots of extra work, you can translate this into information about points on $X$ itself (the Chabauty--Coleman method), but there are lots of cases where this does not apply.

The only method we know of to prove analytic / meromorphic continuation of zeta-functions of alg. varieties over number fields is to go via some kind of modularity, or potential modularity, statement. So even making sense of the statement of your program already requires the key piece of technology developed to prove FLT (and modularity of these high-degree Fermat curves is likely to be vastly harder than elliptic curves).

That is, your "whole-hearted belief" makes this proposed program circular.

EDIT: Dan Loughran's comment alerted me to the fact that this is misleading. Fermat curves have lots of automorphisms, and there are enough of these to force the L-function to factor as a product of L-functions of Groessencharacters of cyclotomic fields, for which analytic continuation + functional equation of the L-function are known; see Aoki (1991). This is, of course, an automorphy statement of a kind, but one that's far easier than automorphy of general higher-genus curves would be.

There are several remaining serious obstacles. For instance, once you go beyond genus 1, the link between the special values of the L-function of a curve and the existence or otherwise of rational points on the curve is very indirect. Rather than points on $X$, the $L$-series gives you information about points on $Jac(X)$. Sometimes, with lots of extra work, you can translate this into information about points on $X$ itself (the Chabauty--Coleman method), but there are lots of cases where this does not apply.

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David Loeffler
David Loeffler
  • 37k
  • 3
  • 89
  • 194

The only method we know of to prove analytic / meromorphic continuation of zeta-functions of alg. varieties over number fields is to go via some kind of modularity, or potential modularity, statement. So even making sense of the statement of your program already requires the key piece of technology developed to prove FLT (and modularity of these high-degree Fermat curves is likely to be vastly harder than elliptic curves).

That is, your "whole-hearted belief" makes this proposed program circular.

There are several others serious obstacles as well; for instance, once you go beyond genus 1, the link between the special values of the L-function of a curve of genus > 1 and the existence or otherwise of rational points on the curve is very indirect. Rather than points on $X$, the $L$-series gives you information about points on $Jac(X)$. Sometimes, with lots of extra work, you can translate this into information about points on $X$ itself (the Chabauty--Coleman method), but there are lots of cases where this does not apply.