|
6
|
|
|
Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. This immediately imply that the coefficients are congruent (mod $p$) p^k$) for every prime $p$. k > 0$. However, the converse is also true: if the coefficients of the $L$-series of $E$ and that of $f$ are congruent (mod $p$) p^k$) for every prime $p$, k > 0$, then the $L$-series agree. One could try to get congruence of coefficients (mod $p$) p^k$) for some value of $p$ and for larger values of $p$ k$ by a method analogous to the method via Langlands and Tunnell rather than as a consequence of modularity lifting theorems. One immediately runs into a stumbling block because when $p > 3$ k$ or $p$ is big enough the group is nonsolvable and the methods of Langlands and Tunnell can't be applied (in a known way) to prove relevant cases of the strong Artin conjecture. Nevertheless, there exists an $n$ such that there is an injective representation $\rho: GL(2, \mathbb{Z}/p\mathbb{Z}) mathbb{Z}/p^k\mathbb{Z}) \to GL(n, \mathbb{C})$. If one can take this representation to be irreducible, then according to the strong Artin conjecture, its $L$-function should be automorphic. Even assuming that this is the case, it's not at all immediately clear (at least, without knowing the modularity theorem) that the $L$-function of the corresponding automorphic representation is related to that a weight $2$ holomorphic cusp eigenform for $GL(2)$. But functoriality can sometimes be used to relate $L$-functions for automorphic representations on one group to $L$-functions of automorphic representations on another group. The arrows only go one way, and in this case it looks like the wrong way, but sometimes one can characterize the arrows' images. Given that we know ex post that there is a relationship between the (mod $p$) p^k$) Galois representation attached to an elliptic curve and that of $f$ (uniform over $p$!), k$!), one can ask whether one can see the relationship ``directly'' from functoriality, without passing through the modularity lifting theorems. Moreover, if one could do this for infinitely many $p$, k$, perhaps one could show that the coefficients of the $L$-function of the elliptic curve $E$ match up with those of the associated modular form $f$ in characteristic $0$ so as to obtain a different proof of the modularity theorem (conditional, of course, on functoriality). The picture that I've sketched above is full of holes like Swiss cheese and it will take me years to understand the precise statements that I allude to above (never mind the proofs!). Nevertheless, I feel that there's a kernel of a well-defined question in what I write. I assume that the strategy that I allude to breaks down somewhere, because otherwise I would have heard about it. I would be grateful to anybody who would be willing to enlighten me as to what goes wrong. To make one closing remark, the above strategy seems unnatural insofar as one is switching between different Galois representations when one could instead be looking at $p$-adic families.Perhaps one it would be better [Edited 10/27/12 to ask if one incorporate my last paragraph.] One could prove that the (mod $p^k$) Galois representations are modular also attempt to get a result as $k$ p$ varies by the strategy rather than $k$ if there are technical problems that I outlined above.come up with varying $k$ but not with varying $p$.
|
|
|
|
|
5
|
|
|
Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. This immediately imply that the coefficients are congruent (mod $p$) for every prime $p$. However, the converse is also true: if the coefficients of the $L$-series of $E$ and that of $f$ are congruent (mod $p$) for every prime $p$, then the $L$-series agree.
The work of Langlands and Tunnell can be used to show that if the elliptic curve has irreducible (mod $3$) Galois representation, then coefficients of the $L$-function agree with those of a weight $2$ cusp form with coefficients in some algebraic number field $K$ (mod $v$), where $v$ is a prime above $3$ in $K$. This was the starting point of Wiles' proof of the modularity theorem for semi-stable elliptic curves over $\mathbb{Q}$.
One could try to get congruence of coefficients (mod $p$) for larger values of $p$ by a method analogous to the method via Langlands and Tunnell rather than as a consequence of modularity lifting theorems. One immediately runs into a stumbling block because when $p > 3$ the group is nonsolvable and the methods of Langlands and Tunnell can't be applied (in a known way) to prove relevant cases of the strong Artin conjecture.
Nevertheless, there exists an $n$ such that there is an injective representation $\rho: GL(2, \mathbb{Z}/p\mathbb{Z}) \to GL(n, \mathbb{C})$. If one can take this representation to be irreducible, then according to the strong Artin conjecture, its $L$-function should be automorphic. Even assuming that this is the case, it's not at all immediately clear (at least, without knowing the modularity theorem) that the $L$-function of the corresponding automorphic representation is related to that a weight $2$ holomorphic cusp eigenform for $GL(2)$.
But functoriality can sometimes be used to relate $L$-functions for automorphic representations on one group to $L$-functions of automorphic representations on another group. The arrows only go one way, and in this case it looks like the wrong way, but sometimes one can characterize the arrows' images. Given that we know ex post that there is a relationship between the (mod $p$) Galois representation attached to an elliptic curve and that of $f$ (uniform over $p$!), one can ask whether one can see the relationship ``directly'' from functoriality, without passing through the modularity lifting theorems.
Moreover, if one could do this for infinitely many $p$, perhaps one could show that the coefficients of the $L$-function of the elliptic curve $E$ match up with those of the associated modular form $f$ in characteristic $0$ so as to obtain a different proof of the modularity theorem (conditional, of course, on functoriality).
The picture that I've sketched above is full of holes like Swiss cheese and it will take me years to understand the precise statements that I allude to above (never mind the proofs!). Nevertheless, I feel that there's a kernel of a well-defined question in what I write. I assume that the strategy that I allude to breaks down somewhere, because otherwise I would have heard about it. I would be grateful to anybody who would be willing to enlighten me as to what goes wrong.
To make one closing remark, the above strategy seems unnatural insofar as one is switching between different Galois representations when they naturally come in one could instead be looking at $p$-adic families. Perhaps one it would be better to ask if one could prove that the (mod $p^k$) Galois representations are modular as $k$ varies by the strategy that I outlined above. However, I'm more worried about reducibility of representations when $k > 1$.
|
|
|
|
|
4
|
|
|
Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. This immediately imply that the coefficients are congruent (mod $p$) for every prime $p$. However, the converse is also true: if the coefficients of the $L$-series of $E$ and that of $f$ are congruent (mod $p$) for every prime $p$, then the $L$-series agree.
The work of Langlands and Tunnell can be used to show that if the elliptic curve has irreducible (mod $3$) Galois representation, then coefficients of the $L$-function agree with those of a weight $2$ cusp form with coefficients in some algebraic number field $K$ (mod $v$), where $v$ is a prime above $3$ in $K$. This was the starting point of Wiles' proof of the modularity theorem for semi-stable elliptic curves over $\mathbb{Q}$.
One could try to get congruence of coefficients (mod $p$) for larger values of $p$ by a method analogous to the method via Langlands and Tunnell rather than as a consequence of modularity lifting theorems. One immediately runs into a stumbling block because once $p$ gets big enough, when $GL(2, \mathbb{Z}/p\mathbb{Z})$ p > 3$ the group is not isomorphic to a subgroup nonsolvable and the methods of $GL(2, \mathbb{C})$ Langlands and so Tunnell can't be applied (in a known way) to prove relevant cases of the strong Artin conjecturefor $GL(2, \mathbb{C}$) isn't immediately relevant.
However
Nevertheless, there exists an $n$ such that there is an injective representation $\rho: GL(2, \mathbb{Z}/p\mathbb{Z}) \to GL(n, \mathbb{C})$. If one can take this representation to be irreducible, then according to the strong Artin conjecture, its $L$-function should be automorphic. Even assuming that this is the case, it's not at all immediately clear (at least, without knowing the modularity theorem) that the $L$-function of the corresponding automorphic representation is related to that a weight $2$ holomorphic cusp eigenform for $GL(2)$.
But functoriality can sometimes be used to relate $L$-functions for automorphic representations on one group to $L$-functions of automorphic representations on another group. The arrows only go one way, and in this case it looks like the wrong way, but sometimes one can characterize the arrows' images. Given that we know ex post that there is a relationship between the (mod $p$) Galois representation attached to an elliptic curve and that of $f$ (uniform over $p$!), one can ask whether one can see the relationship ``directly'' from functoriality, without passing through the modularity lifting theorems.
Moreover, if one could do this for infinitely many $p$, perhaps one could show that the coefficients of the $L$-function of the elliptic curve $E$ match up with those of the associated modular form $f$ in characteristic $0$ so as to obtain a different proof of the modularity theorem (conditional, of course, on functoriality).
The picture that I've sketched above is full of holes like Swiss cheese and it will take me years to understand the precise statements that I allude to above (never mind the proofs!). Nevertheless, I feel that there's a kernel of a well-defined question in what I write. I assume that the strategy that I allude to breaks down somewhere, because otherwise I would have heard about it. I would be grateful to anybody who would be willing to enlighten me as to what goes wrong.
To make one closing remark, the above strategy seems unnatural insofar as one is switching between different Galois representations when they naturally come in $p$-adic families. Perhaps one it would be better to ask if one could prove that the (mod $p^k$) Galois representations are modular as $k$ varies by the strategy that I outlined above. However, I'm more worried about reducibility of representations when $k > 1$.
|
|
|
|
3
|
|
|
Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. This immediately imply that the coefficients are congruent (mod $p$) for every prime $p$. However, the converse is also true: if the coefficients of the $L$-series of $E$ and that of $f$ are congruent (mod $p$) for every prime $p$, then the $L$-series agree.
The work of Langlands and Tunnell can be used to show that if the elliptic curve has irreducible (mod $3$) Galois representation, then coefficients of the $L$-function agree with those of a weight $2$ cusp form with coefficients in some algebraic number field $K$ (mod $v$), where $v$ is a prime above $3$ in $K$. This was the starting point of Wiles' proof of the modularity theorem for semi-stable elliptic curves over $\mathbb{Q}$.
One could try to get congruence of coefficients (mod $p$) for larger values of $p$ by a method analogous to the method via Langlands and Tunnell rather than as a consequence of modularity lifting theorems. One immediately runs into a stumbling block because once $p$ gets big enough, $GL(2, \mathbb{Z}/p\mathbb{Z})$ is not isomorphic to a subgroup of $GL(2, \mathbb{C})$ and so the Artin conjecture for $GL(2, \mathbb{C}$) isn't immediately relevant.
However, there exists an $n$ such that there is an injective representation $\rho: GL(2, \mathbb{Z}/p\mathbb{Z}) \to GL(n, \mathbb{C})$. If one can take this representation to be irreducible then according to the strong Artin conjecture, its $L$-function should be automorphic. Even assuming that this is the case, it's not at all immediately clear (at least, without knowing the modularity theorem) that the $L$-function of the corresponding automorphic representation is related to that a weight $2$ holomorphic cusp eigenform for $GL(2)$.
But functoriality can sometimes be used to relate $L$-functions for automorphic representations on one group to $L$-functions of automorphic representations on another group. The arrows only go one way, and in this case it looks like the wrong way, but sometimes one can characterize the arrows' images. Given that we know ex post that there is a relationship between the (mod $p$) Galois representation attached to an elliptic curve and that of $f$ (uniform over $p$!), one can ask whether one can see the relationship ``directly'' from functoriality, without passing through the modularity lifting theorems.
Moreover, if one could do this for infinitely many $p$, perhaps one could show that the coefficients of the $L$-function of the elliptic curve $E$ match up with those of the associated modular form $f$ in characteristic $0$ so as to obtain a different proof of the modularity theorem (conditional, of course, on functoriality).
The picture that I've sketched above is full of holes like Swiss cheese and it will take me years to understand the precise statements that I allude to above (never mind the proofs!). Nevertheless, I feel that there's a kernel of a well-defined question in what I write. I assume that the strategy that I allude to breaks down somewhere, because otherwise I would have heard about it. I would be grateful to anybody who would be willing to enlighten me as to what goes wrong.
To make one closing remark, the above strategy seems unnatural insofar as one is switching between different Galois representations when they naturally come in $p$-adic families. Perhaps one it would be better to ask if one could prove that the (mod $p^k$) Galois representations are modular as $k$ varies by the strategy that I outlined above. However, I'm more worried about reducibility of representations when $k > 1$.
|
|
|
|
|
2
|
|
|
Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. This immediately imply that the coefficients are congruent (mod $p$) for every prime $p$. However, the converse is also true: if the coefficients of the $L$-series of $E$ and that of $f$ are congruent (mod $p$) for every prime $p$, then the $L$-series agree.
The work of Langlands and Tunnell can be used to show that if the elliptic curve has irreducible (mod $3$) Galois representation, then coefficients of the $L$-function agree with those of a weight $2$ cusp form with coefficients in some algebraic number field $K$ (mod $v$), where $v$ is a prime above $3$ in $K$. This was the starting point of Wiles' proof of the modularity theorem for semi-stable elliptic curves over $\mathbb{Q}$.
One could try to get congruence of coefficients (mod $p$) for larger values of $p$ by a method analogous to the method via Langlands and Tunnell rather than as a consequence of modularity lifting theorems. One immediately runs into a stumbling block because once $p$ gets big enough, $GL(2, \mathbb{Z}/p\mathbb{Z})$ is not isomorphic to a subgroup of $GL(2, \mathbb{C})$ and so the Artin conjecture for $GL(2, \mathbb{C}$) isn't immediately relevant.
However, there exists an $n$ such that there is an injective representation $\rho: GL(2, \mathbb{Z}/p\mathbb{Z}) \to GL(n, \mathbb{C})$. If one can take this representation to be irreducible then according to the strong Artin conjecture, its $L$-function should be automorphic. Even assuming that this is the case, it's not at all immediately clear (at least, without knowing the modularity theorem) that the $L$-function of the corresponding automorphic representation is related to that a weight $2$ holomorphic cusp eigenform for $GL(2)$.
But functoriality can sometimes be used to relate $L$-functions for automorphic representations on one group to $L$-functions of automorphic representations on another group. Given that we know ex post that there is a relationship between the (mod $p$) Galois representation attached to an elliptic curve and that of $f$ (uniform over $p$!), one can ask whether one can see the relationship ``directly'' from functoriality, without passing through the modularity lifting theorems.
Moreover, if one could do this for infinitely many $p$, perhaps one could show that the coefficients of the $L$-function of the elliptic curve $E$ match up with those of the associated modular form $f$ in characteristic $0$ so as to obtain a different proof of the modularity theorem (conditional, of course, on functoriality).
The picture that I've sketched above is full of holes like Swiss cheese and it will take me years to understand the precise statements that I allude to above (never mind the proofs!). Nevertheless, I feel that there's a kernel of a well-defined question in what I write. I assume that the strategy that I allude to breaks down somewhere, because otherwise I would have heard about it. I would be grateful to anybody who would be willing to enlighten me as to what goes wrong.
To make one closing remark, the above strategy seems unnatural insofar as one is switching between different Galois representations when they naturally come in coherent $p$-adic families. Perhaps one it would be better to ask if one could prove that the (mod $p^k$) Galois representations are modular as $k$ varies by the strategy that I outlined above. However, I'm more worried about reducibility of representations when $k > 1$.
|
|
|
|
|
1
|
|
|
Why doesn't functoriality immediately imply the modularity theorem?
Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. This immediately imply that the coefficients are congruent (mod $p$) for every prime $p$. However, the converse is also true: if the coefficients of the $L$-series of $E$ and that of $f$ are congruent (mod $p$) for every prime $p$, then the $L$-series agree.
The work of Langlands and Tunnell can be used to show that if the elliptic curve has irreducible (mod $3$) Galois representation, then coefficients of the $L$-function agree with those of a weight $2$ cusp form with coefficients in some algebraic number field $K$ (mod $v$), where $v$ is a prime above $3$ in $K$. This was the starting point of Wiles' proof of the modularity theorem for semi-stable elliptic curves over $\mathbb{Q}$.
One could try to get congruence of coefficients (mod $p$) for larger values of $p$ by a method analogous to the method via Langlands and Tunnell rather than as a consequence of modularity lifting theorems. One immediately runs into a stumbling block because once $p$ gets big enough, $GL(2, \mathbb{Z}/p\mathbb{Z})$ is not isomorphic to a subgroup of $GL(2, \mathbb{C})$ and so the Artin conjecture for $GL(2, \mathbb{C}$) isn't immediately relevant.
However, there exists an $n$ such that there is an injective representation $\rho: GL(2, \mathbb{Z}/p\mathbb{Z}) \to GL(n, \mathbb{C})$. If one can take this representation to be irreducible then according to the strong Artin conjecture, its $L$-function should be automorphic. Even assuming that this is the case, it's not at all immediately clear (at least, without knowing the modularity theorem) that the $L$-function of the corresponding automorphic representation is related to that a weight $2$ holomorphic cusp eigenform for $GL(2)$.
But functoriality can be used to relate $L$-functions for automorphic representations on one group to $L$-functions of automorphic representations on another group. Given that we know ex post that there is a relationship between the (mod $p$) Galois representation attached to an elliptic curve and that of $f$ (uniform over $p$!), one can ask whether one can see the relationship ``directly'' from functoriality, without passing through the modularity lifting theorems.
Moreover, if one could do this for infinitely many $p$, perhaps one could show that the coefficients of the $L$-function of the elliptic curve $E$ match up with those of the associated modular form $f$ in characteristic $0$ so as to obtain a different proof of the modularity theorem.
The picture that I've sketched above is full of holes like Swiss cheese and it will take me years to understand the precise statements that I allude to above (never mind the proofs!). Nevertheless, I feel that there's a kernel of a well-defined question in what I write. I assume that the strategy that I allude to breaks down somewhere, because otherwise I would have heard about it. I would be grateful to anybody who would be willing to enlighten me as to what goes wrong.
To make one closing remark, the above strategy seems unnatural insofar as one is switching between different Galois representations when they naturally come in coherent families. Perhaps one it would be better to ask if one could prove that the (mod $p^k$) Galois representations are modular as $k$ varies by the strategy that I outlined above. However, I'm more worried about reducibility of representations when $k > 1$.
|
|
|
