A general stratification theorem for exponential sums, and applications (Q2754272)

We are concerned with the size of the general exponential sum NEWLINE\[NEWLINES(V;f,g;p,\psi,\chi):= \sum_{x\in V(\mathbb F_p)} \chi(g(x)) \psi(f(x_1,\dots, x_n)+ (h_1x_1+\cdots+ h_nx_n)),NEWLINE\]NEWLINE where \(p\) is a prime number, \(V\) a locally closed subscheme of \(\mathbb A_\mathbb Z^n\) with \(\dim(V_\mathbb C)\leq d\), \(f\) a polynomial of \(\mathbb Z[X_1,\dots, X_n]\), \(\psi\) a nontrivial additive character of \(\mathbb F_p\), \(\chi\) a multiplicative character of \(\mathbb{F}_p^\times\), and \(g\) an invertible function on \(V\). Following the ideas contained in [\textit{N. Katz} and \textit{G. Laumon}, Publ. Math., Inst. Hautes Étud. Sci. 62, 145--202 (1985; Zbl 0603.14015)], we improve the results of that paper by proving that there exists a constant \(C\), depending only on \((n,d,V,f)\), and closed subschemes of \(\mathbb A_\mathbb Z^n\), called \(X_j\) \((j=1,\dots, n)\) of relative dimension \(\leq n-j\), depending only on \((n,d,V,f)\), \(\mathbb A_\mathbb Z^n\supset X_1\supset\cdots\supset X_n\), such that we have NEWLINE\[NEWLINES|(V;f,g;p,\psi,\chi)|\leq Cp^{\frac d2+ \frac{j-1}{2}},NEWLINE\]NEWLINE for \(h\in \mathbb A_{\mathbb F_p}^n-X_j (\mathbb F_p)\). This is Theorem 1. For \(f\equiv 0\) and \(j=1\), we recover the result of Katz and Laumon. NEWLINENEWLINENEWLINEIn Theorem 2, we are concerned with the particular case where \(f\equiv 0\), \(g\equiv 1\), \(V\) is smooth and irreducible, and we improve by 1 the upper bound of the dimensions of the \(X_j\), under the assumption that the \(A\)-number attached to the situation is at least one. For this notion of \(A\)-number, see \textit{N. Katz} [Perversity and exponential sums, Algebraic Number Theory -- in honor of K. Iwasawa, Adv. Stud. Pure Math. 17, 209--259 (1989; Zbl 0755.14008)]. We give new sufficient conditions for an \(A\)-number to be \(\geq 1\), by comparison with \(B\)-numbers, and establishing links between these two numbers, for hypersurfaces defined by \(F(x)= \alpha\), where \(F\) is a homogeneous polynomial. We give very simple criteria for the nonvanishing of these numbers (congruence considerations, for example). NEWLINENEWLINENEWLINEWe then give applications of the above results, coming from algebraic geometry, to analytic number theory. The first one is a consequence of Theorem 1 and shows the equidistribution of values of very general polynomial vectors modulo \(p\), with arguments in small boxes, in the following sense. Suppose \(n\geq 1\), \(r\geq 1\) are integers, \(P_1(X),\dots, P_r(X)\) are \(r\) polynomials in \(\mathbb Z[X_1,\dots, X_n]\) such that the total degree of any linear combination \(a_1P_1+\cdots+ a_rP_r\) (with coefficients \((a_1,\dots, a_r)\in \mathbb Z^r- \{0\}\)) is at least two. Let \(\varphi(x)\) be a real function of \(x\), tending to \(+\infty\) with \(x\). Then, for \(p\to\infty\), the set of vectors of \(\mathbb R^r\) NEWLINE\[NEWLINE\Biggl\{ \biggl( \frac{P_1(x_1,\dots, x_n)}{p},\dots, \frac{P_r(x_1,\dots, x_n)}{p} \biggr);\;0\leq x_1,\dots, x_n\leq \sqrt{p}\cdot \log p\cdot \varphi(p)\Biggr\},NEWLINE\]NEWLINE is equidistributed modulo 1. NEWLINENEWLINENEWLINEThe second application concerns the equidistribution of points of general varieties modulo \(p\) in small boxes and extends the results of \textit{E. Fouvry} [Isr. J. Math. 120, 81--96 (2000; Zbl 1010.11045)]. The last one improves on [\textit{E. Fouvry}, Bull. Soc. Math. Fr. 127, 95--113 (1999; Zbl 0993.11046)] and deals with the divisibility properties of the class number \(h(n)\) of quadratic fields. We prove the existence of an absolute \(c_0> 0\) and an absolute \(x_0\) such that, for \(x> x_0\), we have the lower bound NEWLINE\[NEWLINE\sharp\{p\leq x;\;p\equiv 1\bmod 4,\;p+4\text{ squarefree, }3\nmid h(p+4)\}\geq c_0 \frac{x}{\log x}.NEWLINE\]NEWLINE In other words, we prove that for a positive proportion of primes \(p\), \(p+4\) is a fundamental discriminant such that the ideal class group of the real quadratic field \(\mathbb Q(\sqrt{p+4})\) has no element of order 3. In the above mentioned paper of E. Fouvry, an inequality of that type was proved, but with the prime variable \(p\) replaced by the integer variable \(n\equiv 1\) modulo 4 having one prime factor, or two distinct prime factors one of which is congruent to 3 modulo 4. (Note that for such an \(n\), the number \(h(n)\) is odd.) We also use sieve techniques and the Davenport-Heilbronn theory, but the improvement is due to the fact that we can prove that the \(A\)-number attached to the hypersurface defined by \(\Delta_3(x)= \alpha\) \((\alpha\neq 0)\) is nonzero. (Here \(\Delta_3\) is the discriminant of a binary cubic form.) Theorem 2 can then be applied.