Abstract
These are the notes of a five-lecture course presented at the Computations with Modular Forms summer school, aimed at graduate students in number theory and related areas. Sections 1–4 give a sketch of the theory of reductive algebraic groups over Q, and of Gross’s purely algebraic definition of automorphic forms in the special case when G(R) is compact. Sections 5–9 describe how these automorphic forms can be explicitly computed, concentrating on the case of definite unitary groups; and Sects. 10 and 11 describe how to relate the results of these computations to Galois representations, and present some examples where the corresponding Galois representations can be identified, giving illustrations of various instances of Langlands’ functoriality conjectures.
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Notes
- 1.
Here “representation” is in the sense of algebraic groups: just a morphism of algebraic groups from G to GL n for some n.
- 2.
One has to be a little careful in defining this topology. One can equip GL n (A) with the subspace topology that comes from regarding it as an open subset of Mat n×n (A), where Mat n×n (A)≅A n has the product topology; but this is not the right topology, as inversion is not continuous (exercise!). Much better is to regard GL n (A) as a closed subset of Mat n×n (A)×A≅A n+1, given by {(m,x):det(m)x=1}. We then get a topology on G(A) for every linear group G by embedding it in GL n for some n.
- 3.
Note that to define G(Z p ) we need to choose an embedding into GL n , as above; but changing our choice of embedding will only affect finitely many primes, so it introduces no ambiguity in the restricted product.
- 4.
(Note added in press) To this list one could certainly add several more recent works, such as those of Chenevier-Lannes and Chenevier-Renard, which focus on the case of orthogonal groups attached to even unimodular lattices, and that of Dummigan on unitary groups of rank 4.
- 5.
Note that I didn’t write “basis” here, since it may very well happen that \(\mathcal{L}\) is not free as an \(\mathcal{O}_{E}\)-module if the class number of E is >1.
- 6.
So active, in fact, that substantial advances have been made in the time it has taken for these notes to be written up for publication; Dieulefait has recently given a proof of the automorphy of \(\operatorname{Sym}^{m}\) of a modular form for m=5, and Clozel and Thorne have proved the cases m=6 and m=8.
- 7.
Informally, an endoscopic subgroup is “the Levi factor of a parabolic subgroup that isn’t there”. Notice that definite groups cannot have parabolic subgroups, since their split rank is 0.
- 8.
Actually triples, but the third parameter is a twist by a power of the determinant and so doesn’t give you anything new.
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Loeffler, D. (2014). Computing with Algebraic Automorphic Forms. In: Böckle, G., Wiese, G. (eds) Computations with Modular Forms. Contributions in Mathematical and Computational Sciences, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-03847-6_2
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