Notes on G2, determinants, and equidistribution

Nicholas M. Katz

doi:10.1016/j.ffa.2003.11.002

Notes on G₂, determinants, and equidistribution

Nicholas M. Katz

Mathematics

Research output: Contribution to journal › Review article › peer-review

14 Scopus citations

Abstract

We determine the exact shape of the G₂ equidistribution law for the one parameter family of exponential sums over F_p ^×, ∑ x mod p, x≠0 χ₂(x) exp(2πi(x⁷+tx)/p). Here χ₂(x) denotes the quadratic character (x/p), t in F_p, is the parameter, and p is any prime other than 2 or 7. This answers a question raised in Keating et al. (J. Phys. A Math. Gen. 36 (2003) 2943, footnote 3) and in Serre (pers. commun., March 7, 2002). We also analyze the analogous families when 7 is replaced by any odd integer n≥3.

Original language	English (US)
Pages (from-to)	221-269
Number of pages	49
Journal	Finite Fields and their Applications
Volume	10
Issue number	2
DOIs	https://doi.org/10.1016/j.ffa.2003.11.002
State	Published - Apr 2004

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Algebra and Number Theory
General Engineering
Applied Mathematics

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10.1016/j.ffa.2003.11.002

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abstract = "We determine the exact shape of the G2 equidistribution law for the one parameter family of exponential sums over Fp ×, ∑ x mod p, x≠0 χ2(x) exp(2πi(x7+tx)/p). Here χ2(x) denotes the quadratic character (x/p), t in Fp, is the parameter, and p is any prime other than 2 or 7. This answers a question raised in Keating et al. (J. Phys. A Math. Gen. 36 (2003) 2943, footnote 3) and in Serre (pers. commun., March 7, 2002). We also analyze the analogous families when 7 is replaced by any odd integer n≥3.",

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AB - We determine the exact shape of the G2 equidistribution law for the one parameter family of exponential sums over Fp ×, ∑ x mod p, x≠0 χ2(x) exp(2πi(x7+tx)/p). Here χ2(x) denotes the quadratic character (x/p), t in Fp, is the parameter, and p is any prime other than 2 or 7. This answers a question raised in Keating et al. (J. Phys. A Math. Gen. 36 (2003) 2943, footnote 3) and in Serre (pers. commun., March 7, 2002). We also analyze the analogous families when 7 is replaced by any odd integer n≥3.

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Notes on G₂, determinants, and equidistribution

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