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Let $p=2$ or 3, and let $k$ be an algebraically closed field of char. $p.$ Let $E$ be the supersingular elliptic curve over $k$ (with $j=0$). Let $G$ be the automorphism group of $E,$ which has order 12 (resp. 24) when $p=3$ (resp. 2). Then the $\ell$-adic cohomology $H^1(E,\mathbb Q_{\ell})$ is a 2-dimensional representation of $G.$ Is it irreducible?

Since we know the group structure of $G$ (cf. Silverman's AEC, Appendix A, exercise..), does anyone have a reference for its character table?

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up vote 6 down vote accepted

If you have a supersingular curve in general, the $\ell$-adic cohomology is a faithful representation of the endomorphism ring, which is a quaternion order that splits over $\mathbb{Q}_\ell$. In the case of characteristic 2 and 3, the automorphism groups are non-abelian (in particular, isomorphic to $SL_2(\mathbb{F}_3)$ and $\mathbb{Z}/3\mathbb{Z} \rtimes \mathbb{Z}/4\mathbb{Z}$, respectively). For non-abelian groups, any faithful two dimensional representation (away from characteristic dividing the order) is automatically irreducible. You can find character tables of the groups in GAP, using commands like:

g := SpecialLinearGroup(2,3);
tbl := CharacterTable(g);
Display(tbl);

In SAGE, you can add "gap" to the beginnning of things:

g = gap.SpecialLinearGroup(2,3)
tbl = gap.CharacterTable(g)
gap.Display(tbl)

There might be a better way to do this.

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When $p = 2$, the group $G$ is the group of units in the Hurwitz order in the Hamilton quaternions $\mathbb H$ over $\mathbb Q$, i.e. the group $\{\pm 1,\pm i,\pm j,\pm k,(\pm 1 \pm i \pm j\pm k)/2\}$. It is a central extension of $A_4$ by a group of order $2$.

The $\ell$-adic cohomology is given by the standard two-dimensional representation of $M_2(\mathbb Q_{\ell}) := \mathbb H \otimes_{\mathbb Q} \mathbb Q_{\ell}$. This is an irrep. of $G$.

I can't write down the $p = 3$ case off the top of my head like this, but it will be similar, and the representation on $H^1(\mathbb Q_{\ell})$ should again be irreducible.

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