Trace functions over finite fields and their applications. (English) Zbl 1408.11080
Zannier, Umberto (ed.), Colloquium De Giorgi 2013 and 2014. Lectures from the colloquium, Scuola Normale Superiore, Pisa, Italy, 2013 and 2014. Colloquia 5. Pisa: Edizioni della Normale, 7-35 (2015).
From the text: We survey our recent works concerning applications to analytic number theory of trace functions of \(\ell\)-adic sheaves over finite fields. …
The outline of the remainder of this survey is the following. In the next section, we define rigorously trace functions; there appears then a crucial definition for analytic applications, that of the conductor of a trace function, which measures its complexity, in such a way that uniformity with respect to \(p\) may be considered. These definitions are illustrated in Section 3 with many examples. We next discuss the crucial, extremely deep and extremely powerful quasi-orthogonality property, which follows from the Riemann Hypothesis over finite fields, and how we use it in [Geom. Funct. Anal. 25, No. 2, 580–657 (2015; Zbl 1344.11036)] and [Duke Math. J. 163, 1683–1736 (2014; Zbl 1318.11103)]. The last section discusses another, very concrete, application to the distribution of certain arithmetic functions in arithmetic progressions to large moduli, following [Mathematika 61, No. 1, 121–144 (2015; Zbl 1317.11080)].
We do not discuss some other papers, contenting ourselves with the following short indications:
(1) in [Math. Proc. Camb. Philos. Soc. 155, 277–295 (2013; Zbl 1307.11018)], we show that trace functions are “Gowers-uniform to all order”, unless they have a very special shape, providing in particular the first explicit examples of functions on \(\mathbb Z/p\mathbb Z\) with Gowers norms as small as those of “random” functions;
(2) in [Math. Res. Lett. 20, No. 2, 305–323 (2013; Zbl 1294.11101)], we use ideas of spherical codes to (roughly) bound from above the number of trace functions modulo \(p\) with bounded conductor;
(3) in [The sliding sum method for short exponential sums, preprint (2013), arxiv:1307.0135], we introduce a new method to estimate short exponential sums modulo primes, of length very close to \(\sqrt p\), and obtain improvements for trace functions of the classical Polyá-Vinogradov bound.
For the entire collection see [Zbl 1310.11005].
The outline of the remainder of this survey is the following. In the next section, we define rigorously trace functions; there appears then a crucial definition for analytic applications, that of the conductor of a trace function, which measures its complexity, in such a way that uniformity with respect to \(p\) may be considered. These definitions are illustrated in Section 3 with many examples. We next discuss the crucial, extremely deep and extremely powerful quasi-orthogonality property, which follows from the Riemann Hypothesis over finite fields, and how we use it in [Geom. Funct. Anal. 25, No. 2, 580–657 (2015; Zbl 1344.11036)] and [Duke Math. J. 163, 1683–1736 (2014; Zbl 1318.11103)]. The last section discusses another, very concrete, application to the distribution of certain arithmetic functions in arithmetic progressions to large moduli, following [Mathematika 61, No. 1, 121–144 (2015; Zbl 1317.11080)].
We do not discuss some other papers, contenting ourselves with the following short indications:
(1) in [Math. Proc. Camb. Philos. Soc. 155, 277–295 (2013; Zbl 1307.11018)], we show that trace functions are “Gowers-uniform to all order”, unless they have a very special shape, providing in particular the first explicit examples of functions on \(\mathbb Z/p\mathbb Z\) with Gowers norms as small as those of “random” functions;
(2) in [Math. Res. Lett. 20, No. 2, 305–323 (2013; Zbl 1294.11101)], we use ideas of spherical codes to (roughly) bound from above the number of trace functions modulo \(p\) with bounded conductor;
(3) in [The sliding sum method for short exponential sums, preprint (2013), arxiv:1307.0135], we introduce a new method to estimate short exponential sums modulo primes, of length very close to \(\sqrt p\), and obtain improvements for trace functions of the classical Polyá-Vinogradov bound.
For the entire collection see [Zbl 1310.11005].
MSC:
| 11L05 | Gauss and Kloosterman sums; generalizations |
| 11N25 | Distribution of integers with specified multiplicative constraints |
| 11N37 | Asymptotic results on arithmetic functions |
| 11T23 | Exponential sums |
| 11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |
