Colloquium on the Geometry of Groups and Numbers

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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