Recognizing the Sato-Tate group UG_2

I2M, Aix-Marseille Université

Date(s) : 21/10/2021   iCal
11 h 00 min - 12 h 00 min

The character method, developed by Yih-Dar Shieh in his thesis, recognizes a Sato-Tate from an associated Frobenius distribution. Previous algorithms used moments of coefficients of a characteristic polynomial of Frobenius. The higher moments are degrees of the tensor product characters, which are direct sums with high multiplicities, hence the moment sequences converge (slowly, with sufficient precision) to large integers. The character method replaces the moments with a precomputed list of irreducible characters. From the orthogonality relations of characters, a Sato-Tate group G is recognized by inner products yielding 0 or 1 (for which only one bit of precision is required to determine its value).

The exceptional Lie group G_2, or rather its unitary subgroup UG_2, embeds as a subgroup of SO(7). We describe the character theory of orthogonal groups SO(2n+1), specialize the character theory method to UG_2, and to demonstrate its effectiveness with certain character sums associated to abelian factors of families of Jacboaisn known to give rise to the Sato-Tate group UG_2.


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