# The simplest even Artin representations of degree 2 and the corresponding Maaß forms

What are the simplest numerical examples of even dihedral (resp. tetrahedral, resp. octahedral, resp. icosahedral) representations

$\rho:\mathrm{Gal}(\bar{\mathbf{Q}}|\mathbf{Q})\to\mathrm{GL}_2(\mathbf{C})$

and their associated Maaß forms $f_\rho\$ ? The word simplest can be taken to mean that the conductor of $\rho$ is small, or a small prime.

[Serre 1977] and [Buzzard 2012] provide many simple examples of odd Artin representations $\rho$ of degree $2$ and the associated cuspidal modular forms of weight $1$. For example, the splitting field of $T^3-T-1\$ gives rise to an odd dihedral representation of conductor $23$ whose associated weight-$1$ modular form is $$q\prod_{n>0}(1-q^n)(1-q^{23n}),$$ as discussed by Emerton in MO11747. For the simplest examples when the image of $\rho$ in $\mathrm{PGL}_2(\mathbf{C})$ is isomorphic to $\mathfrak{A}_4$, $\mathfrak{S}_4$ or $\mathfrak{A}_5$, see [Buzzard 2012].

I'm looking for something similar for even representations. I'm aware of [Vignéras 1985], so my real question is whether there has been further progress in constructing even Artin representations $\rho$ of degree $2$, or the Maaß forms which are supposed to correspond bijectively to such $\rho$ ?

[Buzzard 2012] Computing weight one modular forms over $\mathbf{C}$ and $\overline{\mathbf{F}}_p$. arXiv:1205.5077.

[Serre 1977] Modular forms of weight one and Galois representations. Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 193–268. Academic Press, London.

[Vignéras 1985] Représentations galoisiennes paires. Glasgow Math. J. 27, 223–237.

Addendum. (2012/06/14) Came across the short write-up Explicit Maass Forms by Kevin Buzzard which contains some nice examples.

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I have not read your question sufficiently carefully enough in the beginning. So I rather put my answer now only as a comment: "Here is something related in Sarnak's article pg.444-445: ams.org/journals/bull/2003-40-04/S0273-0979-03-00991-1 In short: If you are willing to assume that the Artin L-function is entire, the associated Maass wave form is given explicitly there. This is due to Andrew Booker: annals.math.princeton.edu/wp-content/uploads/…. –  Marc Palm May 29 '12 at 11:52
Hecke characters of real quadratic fields also give rise to even galois representations, and in turn maass forms. This is spelled out at the end of chapter 1 of Bump's Automorphic Forms and Representations. –  Dror Speiser May 29 '12 at 12:39
Came across the list of non-dihedral even representations of squarefree conductor $<857$ on p.156 of Booker (Andrew) and Strömbergsson (Andreas), Numerical computations with the trace formula and the Selberg eigenvalue conjecture. J. Reine Angew. Math. 607 (2007), 113–161. (degruyter.com/view/j/crll.2007.2007.issue-607/crelle.2007.047/…). –  Chandan Singh Dalawat May 29 '12 at 13:05
They do more (cf. Theorem 5); they have a complete list of all tetrahderal and octahedral representations of squarefree conductor $<3000$ on p.158. If the Artin conjecture is true for even icosahedral representations, then there are only two such representations of squarefree conductor $<3000$; they have conductors $1951$ and $2141$. –  Chandan Singh Dalawat May 29 '12 at 13:20
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## 1 Answer

I'll leave the dihedral and octahedral case to others, but for the tetrahedral ($A_4$) and icosahedral ($A_5$) case, I can give some answer.

For the tetrahedral case, the smallest conductor is 163. See my question: Does anyone want a pretty Maass form?

I have some (not very well documented) code to compute Artin L-function coefficients for this even tetrahedral Galois representation and thus the (provably, by Langlands) Maass form. I've posted this code on my webpage http://people.ucsc.edu/~weissman/

For the icosahedral case, a totally real $A_5$ extension of the rationals is given by the splitting field of $x^5 + 5 x^4 - 7 x^3 - 11 x^2 + 10 x + 3$. In my 1999 undergraduate senior thesis, at http://people.ucsc.edu/~weissman/MWSenThesis.pdf, I found some mild numerical evidence that the associated degree-3 L-function is entire. I can't find it written there, but I'm guessing it's the first, or among the first, $A_5$ extensions of $Q$. I would have chosen something of minimal conductor, to minimize the compute-time. I think I found this by looking in tables from J. Buhler's thesis.

Note that I didn't lift the projective representation $Gal \rightarrow PGL_2(C)$ to an honest 2-dimensional representation. This is a bit subtle, and I wasn't capable of that work at the time. Using the 3-dimensional representation (since $A_5$ has a faithful 3-dim representation) avoids this issue, capturing the adjoint square lift of the putative Maass form. I'd guess that lifting the projective representation would be possible now, if someone wanted to do the work.

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Somehow the file MWSenThesis.pdf seems to end prematurely on p.28 ! –  Chandan Singh Dalawat May 29 '12 at 16:26
Nope - I think that's just where it ends. Basically, I numerically checked the functional equation of the degree 3 Artin L-function. The bibliography is on p.16, and afterwards is an appendix. I'll check the print copy in my office to make sure it's all there, but I think that's it. –  Marty May 29 '12 at 16:34
Thanks for the clarification. –  Chandan Singh Dalawat May 29 '12 at 16:42
By the way, what is the totally real $\mathfrak{A}_4$-extension of $\mathbf{Q}$ cut out by the Artin representation corresponding to your "pretty Maaß form" in mathoverflow.net/questions/22908/… ? –  Chandan Singh Dalawat May 30 '12 at 3:38
It's the splitting field over $Q$ of the polynomial $x^4-x^3-7*x^2+2*x+9$. See the PDF printout of the SAGE code on my webpage for more. –  Marty May 30 '12 at 3:43
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