Abstract
In this note we realize seven small simple groups as Galois groups over . The technique that we employ is the determination of the images of Galois representations attached to modular and automorphic forms, relying in two cases on recent results of Scholze on the existence of Galois representations attached to non-selfdual automorphic forms.
Keywords:
Galois representations; modular forms; inverse Galois theory; automorphic forms; finite simple groups MSC:
11F11; 11F12; 11F80; 12F12
1. Preliminaries
In his note [], David Zywina compiled a list of all simple groups up to a hundred million that are not yet known to be Galois groups over . The list contains only 14 groups. Most of them are classical groups, and we noticed that the technique of determining the images of the Galois representations attached to modular and automorphic forms, a technique that we employed several years ago in the first named author’s thesis (with the third named author as advisor) could be applied to prove that some of these groups are in fact Galois over . We succeed in doing so for seven of the simple groups in Zywina’s list. In this note we present the details of these computations.
Two-Dimensional Representations
We denote by the absolute Galois group . Given a prime ℓ, we will write for the mod ℓ cyclotomic character , sending each Frobenius element to for .
For any ring R, we denote by and the respective quotients of and by their subgroup of scalar matrices. The determinant yields a short exact sequence
so that an element of is in if and only if its determinant is a square.
Let and be positive integers, and let be a primitive Dirichlet character. We consider newforms of weight k, level N, and nebentypus . If the q-expansion of f is , the extension of generated by all is a number field, called the field of coefficients of f. Let be its ring of integers.
For any prime ℓ, Deligne [] constructs a continuous representation associated with f,
which is unramified at all primes . Moreover, for any prime in above ℓ, we have a representation
where is the -adic completion of , such that for all ,
We can compose with the reduction modulo to obtain a mod representation . Furthermore, we may projectivize by quotienting out scalar matrices, yielding
Our goal will be to determine the image of for specific forms f and for certain primes .
In our discussion, we shall need to look at the ramification of at ℓ, that is, its restriction to an inertia group , defined uniquely up to conjugacy. This is described by Theorem 2.5 in [].
Theorem 1.
Fix a prime for which . Let Λ be a prime ideal of dividing ℓ. Suppose . After conjugating by a matrix in , we have
2. Realization of Groups with
Let be an odd positive integer, a quadratic Dirichlet character, N a positive integer. We focus on newforms without CM having any nontrivial inner twists besides . Ribet [] proves that for every p not dividing the level N,
where c denotes complex conjugation. This implies that . We denote by the field of coefficients of f. We shall also consider the subfield of generated by .
We are going to use Theorem 3.1 from []. Let be the ℓ-adic representation associated with f, where and is the ring of integers of . Let
where and R is the ring of integers of . Considering as a character of , we can consider its kernel H, and K the corresponding fixed field, which is a quadratic extension of in our case. If we set , Theorem 3.1 of Ribet [] is as follows:
Theorem 2.
For all ℓ, we have the inclusion , which is an equality for almost every prime ℓ.
For each , there is some element such that . We can choose the different independently of ℓ and only depending on the coset . The full image of the representation is then given by Theorem 4.1 in [].
Theorem 3.
The image of is contained in the subgroup of generated by and the finite set of matrices
with . Moreover, the inclusion is an equality for all but finitely many primes ℓ.
Since is quadratic and H has index 2, we can choose for a prime p such that and , so that generates and is in .
Let be a prime in lying over ℓ, and let be its inertia degree. We also let and . We look at one of the induced representations associated with ,
Recall that we are assuming f has odd weight k. Considering that for any , the determinant of is (a square in ), and that is defined over by Theorem 2, we conclude that
We now have to impose conditions so that we also have By Theorem 3, the image is generated by and the class of a single matrix of the form in (1), which modulo scalars takes the form
The matrix is now in by the choice of . For it to be in , we only need
to be a square in . We assume from now on that is odd and .
Theorem 4.
If ℓ and λ satisfy one of the following conditions:
- 1.
- and λ is split in ;
- 2.
- and λ is inert in , and where p is a prime with ;
then we have
Proof.
Case 1:. In this case is a square in , and we need to be a square in . That is, it is enough that is in . We recall that is a generator of , and thus we see that if is a split prime in .
Case 2:. In this case is not a square in , and we need . This is satisfied as long as is an inert prime not dividing the conductor of the order of generated by . □
2.1. Discarding Possible Images
We have already proven how to achieve in cases where is odd using modular forms of odd weight. However, the image might be smaller in some cases which we now list.
Lemma 1.
From Dickson’s work [], the maximal subgroups of with are:
- Borel subgroups (i.e., conjugate to the subgroup of upper triangular matrices), corresponding to the case where is reducible;
- Dihedral subgroups;
- , when ;
- , when ;
- , when , .
- and , with .
The dihedral case corresponds to a subgroup which is the normalizer of a Cartan subgroup. Given a Cartan subgroup C and its normalizer N, we have that , and for all .
To show the image of some representation is equal to , we discard that the image is contained in one of the maximal subgroups as follows.
- 1.
- For the reducible case, we will usually be able to choose , so that . This means is the trivial character, and all our residual representations have diagonal entries equal to one when restricted to inertia by Theorem 1, as long as . Hence the characters , in the diagonal of are unramified outside N. Whenever N is prime, and because the prime-to-ℓ part of the conductor of divides N, one of these characters must be trivial and the other one has to be the nebentypus. Therefore, we have proved the following:Lemma 2.This yields a contradiction once we find a prime p with and . Alternatively, this gives a representation whose trace is defined over , while we may have examples of traces not in as long as .Let f be a newform of prime level N, weight k, quadratic nebentypus ψ which is the single inner twist, and a prime ℓ for which Theorem 4 says . If and , then .
- 2.
- After proving non-reducibility, we continue with the dihedral case (i.e., normalizer of Cartan). We can assume that the image is not inside a Cartan subgroup C because we have already dealt with the reducible case. To discard the image being in its normalizer N, one takes the nontrivial character
- 3.
- The groups , , , and are discarded by finding some element of large order. We can actually avoid considering the last three when is a nontrivial odd extension of because of Dickson’s congruences.
With these steps, we are able to show the following.
Theorem 5.
The groups , , and are Galois groups over .
Remark 1.
At the same period of time that this project was completed, the groups and have been independently realized as Galois groups over by similar methods by D. Zywina, cf. [].
2.2.
We consider a newform f in the orbit denoted as Newform orbit 31.5.b.b in [], of level , weight , and nebentypus the quadratic character associated to the field . The coefficient field is the degree-6 field with defining polynomial . Let be a root of this polynomial.
Since the level is prime, we see that f has no CM, as and . The element generates the field , which is a cubic extension of . In particular, this implies that is the only inner twist of f.
Let . It is inert in , so that . In turn, splits in as the product of two prime ideals, we fix one of them. We have and , so that . Since and splits completely in , Theorem 4 implies that
We now start discarding possible small images. Assume first that is reducible. We have and . Since , we may apply Lemma 2 to show that . However, this means the trace of would be zero for primes p such that , and we have . Hence, is irreducible.
To show the image is not contained in the normalizer of a Cartan subgroup, we only need to consider the quadratic primitive Dirichlet characters of conductor dividing , which are those associated to the quadratic fields
The first two have value at 3, while . The third character gives at 2, and we also have . These facts are incompatible with being in the normalizer of a Cartan subgroup.
The other maximal subgroup of , according to the congruences in Lemma 1 and point (3) in the discussion following Lemma 2, is . Let be a Frobenius element for 3 in . Modulo conjugation, we have
This matrix has order 124 in , and actually , so the element has order 31 in . Hence cannot be inside . We conclude that the projective image is the whole , which is a Galois group over .
2.3.
We consider a newform f in the orbit denoted as Newform orbit 43.5.b.b in [] of level , weight , and nebentypus the quadratic character associated to the field . The field is defined by the polynomial , it has degree 12 over . We let be a root of this polynomial. The q-expansion of f begins with
We have and , therefore, f has no CM. The coefficient generates a field with , which confirms that is the only inner twist of f.
We let . It splits as the product of two primes and in . Furthermore, remains inert in , we write . We check that and , , . Because , and , Theorem 4 implies that
Let us discard possible small images. If is reducible, and because its determinant is , it has to be conjugate to
where or 1, and , are characters ramified at most at . If both and were ramified at N, which is prime, then would divide the conductor of , which is not the case. Hence, one of them (say ) must be unramified at N and trivial, and the other one must be ramified at N and quadratic, that is, . Hence we find that and the trace of is defined over . However, , so has to be irreducible.
Now we look at the case when the image is inside the normalizer of a Cartan subgroup. We need to consider the quadratic primitive Dirichlet characters with conductor dividing , which are the ones associated to the fields
In the first two cases , but . In the third case, we have , and it is easy to check that . Hence, the image is not contained in a normalizer of Cartan subgroup.
By the congruences of Lemma 1 and the subsequent discussion, we still need to discard the image of being contained in and . To that effect, we consider the image of , which up to conjugation is
This matrix has order 13 in , so the image cannot be in such smaller subgroups. Therefore, .
2.4.
Let f be a newform in the orbit denoted as Newform orbit 31.7.b.c in [], of level , weight , and nebentypus the quadratic character associated to the field . The degree-12 field of coefficients is given by the polynomial . As usual we let be a root of this polynomial.
The form f does not have CM, since and . The field is generated over by and , so that is the unique inner twist of f.
The rational prime splits in as the product of three primes , and , with inertia degrees and 3, respectively. The prime is inert in , so that satisfies and . We have , in fact, generates . Since , Theorem 4 gives the inclusion .
In this example, we have again and . Since , if the representation were irreducible, by Lemma 2 we would have , but this cannot happen since and , so is irreducible. To rule out the normalizer of Cartan case, we look at the primitive Dirichlet characters with conductor dividing , namely the ones corresponding to the fields
We need to find some prime p with and . For the first two characters this is satisfied by , while for the third we may look at . Hence is not contained in a normalizer of a Cartan subgroup.
By Lemma 1, it remains to check that the projective image is not contained in or . To that effect, we note that is conjugate in to the matrix
This matrix has order in , and even in . Therefore, is not contained in such a smaller subgroup, and the image is all of .
2.5.
Let us consider a newform f of level , weight , and quadratic nebentypus associated with the field , in the orbit denoted as Newform orbit 67.3.b.b in []. Its coefficient field is given by the degree-10 polynomial , let be one of its roots in . The q-expansion of f begins with
We see f has neither CM nor inner twists besides , since the field is generated over by , has degree 2 over , and , while . The prime is inert in , while is inert in . Setting , this means and . We have , hence Theorem 4 says that .
We have and . Furthermore, , so we can apply Theorem 1 and Lemma 2 to show is irreducible. Indeed, if were reducible, we would have as usual , but and .
Next, we look at the possibility of being inside the normalizer of a Cartan subgroup of . Because the representation is unramified outside , we just need to find some p with and for any of the primitive quadratic Dirichlet characters associated with the fields
For the first two characters, this is the case for , for the third, we may use .
If we look at Lemma 1 and the discussion after, we need to check that the projective image is not contained in some maximal subgroup of , namely and . For instance, we can look at the image of , which up to conjugation is
This matrix has order both in and in . It follows that the projective image of is not contained in any smaller subgroup, and it is the full , as desired.
3.
Theorem 6.
The group is a Galois group over .
Proof.
We consider a newform f of level , weight 2, and trivial nebentypus in the orbit denoted as Newform orbit 67.3.b.b in []. We know f has no inner twists nor CM, since the form is Steinberg at the primes dividing the level. The coefficient field is the maximal totally real subfield of the cyclotomic field containing all 20th roots of unity. We have , and is generated by , whose irreducible polynomial is
This polynomial is also irreducible mod 3, so 3 is inert in and we may consider the representation
The determinant of is the mod 3 cyclotomic character . Hence, the projective image of is contained in . We will determine its image using the classification by Dickson in Lemma 1.
If was reducible we would have , since the level is squarefree (and thus the conductor of is squarefree with determinant ). Therefore, will have trace in , but has trace , which is a generator of . Hence, is irreducible.
We next consider the case where the projective image of is contained in the normalizer N of a Cartan subgroup C. As in Section 2.1, we only need to look at primitive quadratic Dirichlet characters with conductor dividing . What is more, 2 and 113 are primes where is Steinberg, so that for ,
This means cannot be ramified at 2 or 113, since it is a nontrivial quadratic character and has odd order. Therefore is the character associated with the quadratic field . However, we then have , while . We have thus discarded the dihedral case.
Lastly, we will see the projective image of is not contained in . The field of definition of the projective representation is that generated by the different (here is the nebentypus, which is trivial in this case). We have already shown that generates , therefore, the image cannot be contained in . We have no further maximal subgroups of because of Lemma 1 and the fact that is an extension of of degree larger than two.
Therefore, the image is maximal. □
4. and as Galois Groups over
The images of modular and geometric three-dimensional Galois representation have been studied in our previous paper []. As a consequence of the recent results of P. Scholze [], we know the existence of Galois representations associated with the mod p cohomology of the locally symmetric spaces for over F a totally real or CM field. Moreover, we have for characteristic 0 cohomology classes the existence of p-adic Galois representations by the recent result of Harris–Lan–Taylor–Thorne [] (also proved by Scholze). We state the result that we will use to obtain the Galois realization over of the group (cf. [], V.4.2; V.4.6).
Theorem 7.
Let F be an imaginary quadratic field. Let S be the pullback from finite set of primes of , which contains p and all places at which is ramified. Let π be a cuspidal automorphic representation of such that is regular algebraic, and such that is unramified at all finite places . Then there exists a unique continuous semisimple representation
such that for all finite places , the Satake parameters of agree with the eigenvalues of .
Lemma 3.
Let be the eigenform for the action of the Hecke algebra computed in [] Table 2 such that . The eigenform f is cuspidal.
Proof.
If f is not cuspidal, there is a unique decomposition
where is an automorphic form on and g is an automorphic form on . We remark that is algebraic, as a consequence it is a Hecke character of type and the finite part only ramifies at two and 11. According with the eigenvalues computed in [] we can determine the attached Satake polynomials which are of the form
In particular, if f is not cuspidal they will have as a factor the Satake character attached to , that is , where is a root of the unit with order coprime to 3, since the conductor of the character is and . Let , the polynomial is irreducible since we compute that it is irreducible modulo 5. Let be a root of it, the degree of over is therefore equal to 6. If then , so must be a root of unity of order 3 or 6, which is a contradiction.
By Theorem 7 we can consider the 5-adic Galois representation attached to f
Let the reduced Galois representation modulo 5. Because 5 is inert in , we have
As we have observed in [], the form of the characteristic polynomial of the Frobenius, implies that , where . So we have that the image is unitary. Moreover, since the determinant of is (cf. []), we have that
Theorem 8.
Proof.
We know that . From the classification of the maximal subgroups of in our case , we know that the maximal proper subgroups are of type reducible (two cases over ) or of type S, that is, or (cf. Liebeck’s Ph.D. thesis). The polynomial is irreducible modulo 5, as a consequence, the cases are not possible. On the other hand, modulo 5 is reducible and splits into a linear factor and a quadratic factor. The roots of the quadratic factor have order divisible by 13. That means that in the image of , there is an element M of order divisible by 13. Since 13 is relatively prime with , the order of is also divisible by 13. This exclude that the image is of the type S in the classification. □
As a consequence, we have that the group occurs as a Galois group over .
In the case of the group , we obtain that it is a Galois group over since conjecture 1’ of [] has been proved as a consequence of P. Scholze results modulo p (cf. [], cor. V.4.3). That means that to a Hecke eigenclass in the mod p cohomology of the congruence subgroup of we can attach a three-dimensional mod p semisimple Galois representation of . We consider a Hecke eigenclass over the field of level 167 for which several eigenvalues and the image of the mod 7 corresponding Galois representation have been computed in [], p. 222, and we conclude that thanks to Scholze’s result the realization of this group, already obtained as a result of these computations in [], now holds unconditionally.
Theorem 9
(Ash-McConnell; Scholze). The group is a Galois group over .
Author Contributions
Conceptualization, L.D., E.F. and N.V.; Investigation, L.D., E.F. and N.V.; Methodology, L.D., E.F. and N.V. All authors have read and agreed to the published version of the manuscript.
Funding
The first author is partially supported by the PID2019-107297GB-I00 grant of the MICINN (Spain). The second author is partially supported by the Spanish Ministry of Universities (FPU20/05059).
Conflicts of Interest
The authors declare no conflict of interest.
References
- Zywina, D. Inverse Galois Problem for Small Simple Groups. 2013 Preprint. Available online: http://pi.math.cornell.edu/~zywina/papers/smallGalois.pdf (accessed on 5 April 2022).
- Deligne, P. Formes Modulaires et Représentations l-adiques; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 1971; Volume 175. [Google Scholar]
- Edixhoven, B. The weight in Serre’s conjectures on modular forms. Invent. Math. 1992, 109, 563–594. [Google Scholar] [CrossRef]
- Ribet, K. Galois Representations Attached to Eigenforms with Nebentypus; Modular Functions of one Variable V; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 1977. [Google Scholar]
- Ribet, K. On l-adic representations attached to modular forms II. Glasg. Math. J. 1985, 27, 185–194. [Google Scholar] [CrossRef] [Green Version]
- Dickson, L.E. Linear Groups with an Exposition of the Galois Field Theory; Dover Publications: Mignola, NY, USA, 1958. [Google Scholar]
- Zywina, D. Modular forms and some cases of the inverse Galois problem. arXiv 2015, arXiv:1508.07916. [Google Scholar]
- The LMFDB Collaboration. The L-Functions and Modular Forms Database. Available online: http://www.lmfdb.org (accessed on 5 April 2022).
- Dieulefait, L.; Vila, N. On the images of modular and geometric three-dimensional Galois representations. Am. J. Math. 2004, 126, 335–361. [Google Scholar] [CrossRef] [Green Version]
- Scholze, P. On torsion in the cohomology of locally symmetric varieties. Ann. Math. 2015, 182, 945–1066. [Google Scholar] [CrossRef] [Green Version]
- Harris, M.; Lan, K.-W.; Taylor, R.; Thorne, J. On the rigid cohomology of certain Shimura varieties. Res. Math. Sci. 2016, 3, 37. [Google Scholar] [CrossRef] [Green Version]
- van Geemen, B.; van der Kallen, W.; Top, J.; Verberkmoes, A. Hecke Eigenforms in the Cohomology of Congruences Subgroups of SL(3,). Exp. Math. 1997, 6, 163–174. [Google Scholar] [CrossRef]
- Ash, A.; McConnell, M. Experimental indications of three-dimensional Galois representations from the cohomology of SL(3,). Exp. Math. 1992, 1, 209–223. [Google Scholar] [CrossRef]
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