The Artin conjecture for some -extensions
Let denote the absolute Galois group of , let be a number field with Galois closure , and let
be a continuous irreducible Galois representation. Attached to is an -function which is holomorphic for . Artin conjectured that had a holomorphic continuation to the entire complex plane, with the possible exception of a pole at if was the trivial representation. Subsequently, Langlands conjectured that was modular, that is, there exists an automorphic representation for such that and is cuspidal as long as is non-trivial, which in particular implies Artin’s conjecture. If is a monomial representation (that is, induced from a character) then Artin’s conjecture was proved by Artin, and if is nilpotent, then Langlands’ conjecture is a consequence of cyclic base change (Theorem 7.1 of [AC89]). Suppose that has a faithful permutation representation of degree . If is solvable, then Artin’s conjecture follows from the fact that all such are monomial. Moreover, Langlands’ conjectures are also known in these cases. If and , then results of Langlands [Lan80] and Tunnell [Tun81] imply that there exists a cuspidal representation for which is associated to a representation with projective image . The faithful representations of (which are all of dimension three) can then be realized (up to twist) by the symmetric square of Gelbart and Jacquet. If is not solvable, then either is or . If , then Langlands’ conjecture is known providing that complex conjugation is non-trivial. A theorem of Khare–Wintenberger [KW09] guarantees the existence of an automorphic form for corresponding to any odd projective -representation, and then all the faithful representations of can be reconstructed from (and for the outer automorphism of ) via functoriality (see [Kim04]). In this note, we prove some cases of the conjectures of Artin and Langlands for using functoriality, in the spirit of Tunnell [Tun81]. Recall that the faithful irreducible representations of have dimensions , , and .
Let be an extension such that . Suppose, furthermore, that
Complex conjugation in has conjugacy class .
The extension is unramified at , and the Frobenius element has conjugacy class .
If is irreducible of dimension or , then is modular. If is irreducible of dimension , there exists a tempered cuspidal for such that for a set of places of density one.
The existence of a weak correspondence between and , though perhaps an approximation to Langlands’ conjecture, is not sufficient even to deduce Artin’s conjecture for (although one can deduce from our arguments that is holomorphic for for some ineffective constant ). We may, however, remedy this lacuna under a more stringent hypothesis, which shows that the conjectures of Artin and Langlands can be established unconditionally for some -extensions.
Let be as in Theorem 1.1. Let be the quadratic subfield of , let be a subfield of of degree over , and let be the compositum of and . Suppose that does not vanish for real . Then is modular.
Note that the non-vanishing condition on is explicitly verifiable in theory (and in practice, see Theorem 5.1). There is a factorization
for some explicit meromorphic Artin -function . Our result actually only requires the non-vanishing of for . The non-vanishing of Artin -functions for is not implied by the GRH, and indeed Armitage found examples of number fields such that (see [Arm72]). Those constructions, however, arose from Artin -functions with real traces such that the global Artin root number was . Since all the representations of (indeed of ) are self-dual and definable over , the the Artin root number of any irreducible representation of these groups is automatically , and so is never forced to vanish at for sign reasons. Indeed, the non-vanishing of would follow if one assumed that (in addition to the GRH) that Artin -functions of irreducible representations have only simple zeros. The local condition at is not essential to the method, but the restriction on complex conjugation is completely essential. The reason we can prove anything non-trivial under these assumptions is because we can reduce to known cases of Artin for (projective) two-dimensional -extensions over quadratic extensions of . The assumption that complex conjugation is conjugate to ensures that these Artin representations are defined over (totally) real quadratic fields, and that the image of complex conjugation in has determinant . For such two dimensional representations, one can deduce modularity using results of Sasaki [Sas11a, Sas11b], which generalizes the approach of [BDSBT01].
I would like to thank Andrew Booker for establishing the non-vanishing of for for an explicit number field (see Theorem 5.3), demonstrating that one can effectively use the results of this paper to prove the Artin conjecture for particular -extensions. I would also like to thank Lassima Dembele for computing the Hilbert modular forms of weight and level one for . Finally, I would like to thank Kevin Buzzard, Peter Sarnak, and Dinakar Ramakrishnan for useful conversations.
2. Some Group Theory
Let be a finite subgroup, necessarily cyclic. Let denote the order of — it is divisible by . We define the group to be . There is a tautological map:
which realizes as a central extension of by the cyclic group of order . Let denote the kernel of the composite
The group is a central extension of by , and admits a faithful complex representation of dimension two. Any character of is determined by its restriction to .
Since the image of in is and the kernel is , it is clear that is a central extension. Central extensions of a group by a cyclic group of order are classified by . It is a result (essentially of Schur) that . Since is a perfect group, . Taking the cohomology of the Kummer exact sequence , we deduce that there is an isomorphism , and hence that both groups have order two if is even. It follows that is either the unique non-split central extension or . Since is not a subgroup of , it follows that is non-split. Any projective morphism admits a non-trivial central extension in of any any even degree — by uniqueness we identify these covers with which therefore admits a faithful representation . From this description, one may compute that the composite:
surjects onto the image of , and thus characters of are determined by their restriction to . ∎
Note that (respectively, ) admits an outer automorphism which is conjugation by an element of (respectively, ). We denote this automorphism by — this is not a particularly egregious abuse of notation since is compatible with the natural projection . We are interested in representations of , of , and of the group . The character table of is given as follows:
If is a representation of and is a representation of , say that if the action of on factors though the quotient , and the action of extends to an action of which is isomorphic to . On the other hand, if is a representation of and is a representation of , say that if the action of on extends to an action of which is isomorphic to .
We have the following:
The image of under lies in the scalar matrices, by Schur’s lemma. On the other hand, the involution fixes the centre of , and hence the restriction of to is the same as the restriction of to . The product of these restrictions it thus identified with , and thus the action of the centre on is trivial, and the action of factors through . On the other hand, the representation is preserved by the automorphism , and hence it lifts to a representation of , from which follows easily.
The action of the centre on is trivial, and thus it corresponds to a -dimensional representation of , which must be one of the two irreducible faithful representations of . There is a plethsym:
Viewing the left hand side as of an irreducible representation of , we may identify the non-trivial factor on the right hand side as the -dimensional irreducible representation of , which (from the character tables of and ) lifts to a representation of . We note, moreover, that this identification can equally be applied to . This establishes .
The claim follows directly from , as . ∎
On the other hand, we have the following:
There exists an irreducible four dimensional representation of such that
Let us first consider the projective representation:
The image depends only on the projective image of , and thus it factors through . Let us admit, for the moment, that this projective representation extends to a projective representation of . Then we obtain an identification of projective representations of :
Any two irreducible representations with isomorphic projective representations are twists of each other. Since any one dimensional character of is determined by its restriction to , and since and are the same restricted to , it follows that there must be an isomorphism:
It suffices, therefore, to establish that the fact about projective representations mentioned above. This is a fact which can be determined, for example, by an explicit computation with the Darstellungsgruppe of , the binary icosahedral group. Note that although the projective representation of factors through , it is not equivalent to either of the projective representations obtained from the linear representations of . ∎
We now describe, to some extent, the character of . We start by describing the projective representation associated to . That is, for an element , we give a matrix in lifting . In fact, we only do this for the odd permutations, since this is the only information we shall require. We use the same labeling of elements as in the character table of above. Here is a primitive th root of unity:
There is an isomorphism , and hence an isomorphism for some character . We have, moreover, that .
The representation preserves a non-degenerate generalized symplectic pairing:
There is an isomorphism
Since , there is certainly a non-degenerate pairing , it remains to determine whether this pairing is (generalized) symplectic or orthogonal. Yet the restriction of to is , which is irreducible and symplectic, and thus is symplectic. It follows that decomposes as plus some -dimensional representation of . Restricting to , we have the plethysm
Since the latter representation extends to , this shows that is either or . To distinguish between these two representations, let us compute on some conjugacy class in which maps to in . We must have
for some root of unity . It follows that has eigenvalues:
Note that since is (generalized) symplectic with similude character , the eigenvalues of are of the form . It follows that occurs to multiplicity at least two in the eigenvalues of , and thus . In particular, we deduce that
Yet, if has conjugacy class in , then whereas . ∎
Let be a lift of , and suppose that
Suppose instead that . Then has eigenvalues:
If , then the trace of is , whereas if then this trace is . Yet if is of type , then . ∎
3. Irreducible representations of dimension and
Let be an -extension satisfying the conditions of Theorem 1.1. Let denote the quadratic subfield of . By assumption, is real, and and both split in .
There exists an Galois extension with for some .
There is an exceptional isomorphism , giving rise to a representation
which factors through . The Galois cohomology group is trivial by a theorem of Tate (see [Ser77]). It follows that the projective representation lifts to a linear representation over with finite image. By a classification of subgroups of , the image must land in , proving the lemma. ∎
It follows that there exists a corresponding complex representation:
with projective image . By assumption, is odd at both real places of .
The representation is automorphic for .
This follows immediately from Theorem 1 of [Sas11b], given the assumed conditions on . ∎
Assume that is irreducible and of dimension or . Let denote the automorphic form for associated to by Lemma 3.2. There exists a conjugate representation given by applying the outer involution to , which corresponds (by another application of Lemma 3.2) to an automorphic form which we call .
Suppose that . The Asai transfer is automorphic and cuspidal for , as follows by Theorem D of [Ram02] and Theorem B of [PR11]. Yet, by Lemma 2.3 part , we deduce that corresponds (by Lemma 2.3) exactly to a twist of , from which the main result follows.
Suppose that . Then . We have already shown that is automorphic and corresponds to some cuspidal for . By a result of Kim [Kim03], the representation is automorphic for , and by [AR11], it is a simple matter to check that it is cuspidal. Hence corresponds (by Lemma 2.3) to , from which the main result follows. Alternatively, we may consider the automorphic form . The corresponding Galois representation has image , but is not isomorphic to its Galois conjugate (as has no irreducible representations of dimension three). Thus one may form the base change of to , which is cuspidal (by cyclic base change) and corresponds to the Galois representation .
4. Irreducible representations of dimension
Proving Langlands’ automorphy conjecture for or is somewhat harder, for reasons which we now explain. The symmetric fourth power of is automorphic and cuspidal by [Kim03] and [KS02] respectively. On the other hand, is invariant under the involution of , and thus, by multiplicity one [JS81b] and cyclic base change (Theorem 4.2 (p.202) of [AC89]), the corresponding twist arises from an automorphic form for . The Galois representation restricted to corresponds to this automorphic form. We would like to show that (or its quadratic twist) corresponds to . There is a well known problem, however, that since this descended form is not known a priori to admit a Galois representation, all we can deduce is that the collection of Satake parameters for and are the same as the collection of Frobenius eigenvalues of and . This implies that is holomorphic, but gives no information about . This problem is the main obstruction to proving the Artin conjecture for solvable representations.
The usual game in these situations is to play off several representations against each other and use functorialities known in small degrees. It turns out — in this instance — that it is profitable to work instead with the representation of the group in dimension four. Let denote one of the two dimensional representations of . Since is equal to its conjugate under (by Lemma 2.3), we deduce by multiplicity one that the same is true of . It follows by cyclic base change that there exists an automorphic form for such that . Conjecturally, is associated to the Galois representation:
Let denote the -tuple of the Satake parameters of . By abuse of notation, we write and for and respectively.
If , then consists of the eigenvalues of . If , then the union consists of the eigenvalues of together with the eigenvalues of .
This follows from the local compatibility between and . ∎
Since (by multiplicity one) we deduce (again by multiplicity one) that for some character such that is trivial. It follows that either or . Since , it should be the case that , although there does not seem to be any apparent way to prove this a priori. We shall, however, prove that in Lemma 4.6 below.
is of symplectic type, and is the corresponding similitude character. If is an unramified prime, then the Satake parameters of are of the form:
It suffices to show that is non-cuspidal, and then we can deduce the result from Theorem 1.1(ii) of [AR11]. Since is symplectic, we know that is non-cuspidal. Assume that is cuspidal. Then the base change of to is . In other words, becomes non-cuspidal after base change. By Theorem 4.2 (p.202) of [AC89], it follows that is the automorphic induction of a cuspidal form from . We deduce that , which is incompatible with the fact that the Satake parameters of do not all have multiplicity two. Thus is not cuspidal, and from the classification [AR11], we deduce that is symplectic (all other possibilities contradict the known structure of ). If has similitude character , then . It follows that . It is easy to check that is not induced from a quadratic subfield, and hence we must have . ∎
The second wedge decomposes as an isobaric sum of the similitude character together with another isobaric representation, and thus we may write character , and thus we may write
By considering Galois representations, we see that corresponds to , and conjecturally corresponds to . It also follows from this that is cuspidal for .
Let denote the -tuple of Satake parameters of . If , then the Satake parameters of are of the form
For the automorphic representations and , let
denote the sum of the Satake parameters. For the Galois representations and , let and denote the sum of the eigenvalues of and respectively.
Conjecturally, we have and . Note that and are real valued; the former because is a real representation, and the latter because consists of together with two pairs for a root of unity . We shall consider the following quantity:
For a prime , let
If then consists of the eigenvalues of , and consists of the eigenvalues of , and hence .
We now consider the possible values of as well as the structure of for with . Note that we do not know at this point whether or , so we shall have to take both possibilities into account.
Suppose that the conjugacy class of is in . Then from the (projective) character table of , we deduce that
By Lemma 2.6, we know that . Hence either if , or if . Using the known shape of , we deduce that one of the following possibilities holds (we only concern ourselves with identifying up to sign.)
Both here and in the two tables below, the first line of each table represents the (conjectural) reality.
Suppose that the conjugacy class of is in . Then from the (projective) character table of , we deduce that
By computing multiplicities in we deduce that or but we cannot pin down exactly — indeed, the group can (and does) contain distinct conjugacy classes with these same eigenvalues and with values of that differ (up to sign). It follows that the possibilities below are the same regardless whether or .
Suppose that the conjugacy class of is in . Then, as above, we may write
where . A key point in the computation below is that , and so . Here , and so if and otherwise. We deduce that the following possibilities may occur:
is meromorphic for , and is holomorphic in some neighbourhood of .
By By a theorem of Jacquet and Shalika ([JS81a]), the Rankin–Selberg -functions are meromorphic for with a simple pole each at . The same is true of the Artin -functions by Brauer’s theorem. ∎
There is an equality .
Assume otherwise. From the Euler product, we find that, as ,
Then, from the tables above, we compute that unless the projective image of is of type , in which case it is , or , in which case it is . By The Cebotarev density theorem, the class has density each respectively. Similarly, the class has Dirichlet density . Thus the RHS is asympotic to
This contradicts Lemma 4.5. ∎
Let denote the finite set of places for which is not unramified. For all primes outside a set of Dirichlet density zero, consists of the eigenvalues of . If lies in , then in has conjugacy class , and rather than .
Since , the result follows automatically for all conjugacy classes by our computation above except for the density claim concerning . Assume otherwise. Denote the set of such primes in these conjugacy classes with by . Then we compute that
Once more by Jacquet–Shalika, we deduce that the LHS is bounded, and hence the RHS also has order , and hence that has Dirichlet density zero. ∎
We now summarize what we have shown so far: Namely, in comparing the Artin representation with the automorphic form , we have that for a set of outside the set of density zero, the Satake parameters of agree with up to sign. In particular, outside the same set, the Satake parameters of agree with the eigenvalues of . This completes the proof of Theorem 1.1.
Unfortunately, we do not see an unconditional argument at this point for establishing an equivalence at all places. On the other hand, we know (by construction) precisely the Satake parameters of at the “troublesome” primes . The special form of these parameters will allow us to prove Theorem 1.2.
Let , where denotes the function
and , where the product is over the finite set of primes of bad reduction of and . Then the following holds:
There is an equality ,
extends to a meromorphic function on the complex plane.
is holomorphic and non-vanishing on the interval .
Either is finite, or has a pole on the real axis with .
We have that and , where the local factors agree for . For the exceptional , the corresponding polynomials are:
Comparing the two sides leads to the equality . We claim that the Gamma factors of both -functions involve only the standard factors and . This is true for by Artin, and is true for because of the identity:
Since and are both holomorphic and without zeros on , so is any ratio of products of such functions. The factors and for finite bad pri