Taking place at Université Pierre et Marie Curie, 8th of June, 2017.

David Kohel (Université d'Aix-Marseille) :

## Recognizing G_2

The *character method*, developed by Yih-Dar Shieh in his thesis, recognizes
a Sato-Tate from an associated Frobenius distribution. Previous methods
used moments of prescribed characters --- coefficients of a characteristic
polynomial of Frobenius. They correspond to symmetric product characters,
which decompose into direct sums of high multiplicity. As a result, the
moment sequences converge poorly to large integers.

The *character method* replaces the moments with a precomputed list of irreducible characters,
and from the orthogonality relations of characters implies that a Sato-Tate
group G is recognized by inner products yielding 0 or 1 (for which the
minimal precision to recognize one bit suffices).

We make explicit the character theory method for the exceptional Lie group G_2, and demonstrate
its effectiveness with certain character sums associated to families of curves
known to give rise to G_2 as its Sato-Tate group.

This is joint work with Yih-DarShieh.

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