Gauss sums, Kloosterman sums, and monodromy groups.

*(English)*Zbl 0675.14004
Annals of Mathematics Studies, 116. Princeton, NJ: Princeton University Press. viii, 246 p. $ 52.50 (1988).

The connection between exponential sums and the theory of L-functions of varieties over finite fields was emphasized by Hasse, Davenport, Weil and Deligne.

The book under review provides a sheaf-theoretic approach to the study of Gauss sums and Kloosterman sums by combining a comprehensive treatment of the main representation-theoretic ideas involved, together with their applications. This point of view helps the reader to get a better understanding of some of Deligne’s deep achievements in this context.

The easiest examples of Kloosterman sums are those given by the complex numbers \[ Kl(p,n,a)=\sum_{x_ 1...x_ n=a;\quad x_ 1,...,x_ n mod p}\exp (\frac{2\pi i}{p}(x_ 1+...+x_ n)), \] where \(n\geq 2\) and p is a prime. In general, Kloosterman sums occur naturally as the inverse Fourier transforms of monomials in Gauss sums. They depend on an additive character \(\psi: ({\mathbb{F}}_ q,+)\to E^*,\) E being a finite extension of \({\mathbb{Q}}\), on n multiplicative characters \(\chi_ i: {\mathbb{F}}^*q\to E^*\) and on n strictly positive integers \(b_ i.\)

Let \(E_{\lambda}\) be the \(\lambda\)-adic completion of E, \({\mathfrak O}_{\lambda}\) its ring of integers, for an \(\ell\)-adic place \(\lambda\) of E (\(\ell \neq p)\). By a theorem of Deligne, there exists a lisse sheaf of free \({\mathfrak O}_{\lambda}\)-modules of finite rank on \({\mathbb{G}}_ m\otimes {\mathbb{F}}_ q\), denoted by \(Kl=Kl_{\lambda}(\psi;\chi_ 1,...,\chi_ n;b_ 1,...,b_ n)\), whose local Frobenius traces are equal, up to sign, to the Kloosterman sums in question. The Kloosterman sheaves Kl are pure of weight \(n-1,\) tame at zero and totally wild at infinity with Swan conductor equal to one. The proof of this existence theorem is given in chapter 4 for \(n=1\) and in chapter 5 for general n by successively convolving those sheaves constructed in the case \(n=1.\)

Chapters 6 and 7 deal with the study of the local monodromy at zero and at infinity of a convolution of sheaves. For the special case of Kloosterman sheaves, it is shown in theorem 7.3.2 that the characters occurring in the local monodromy at zero of Kl are precisely the characters \(\chi\) of \({\mathbb{F}}^*_ q\) such that \(\chi^{b_ i}=\chi_ i\) for some value of \(i=1,...,n\); each of these characters occur in a single Jordan block. At infinity, it is proved in chapter 10, that the character of \(Kl_{\lambda}\), as representation of the wild ramification group \(p_{\infty}\), is independent of both \(\lambda\) (this is clear) and of the particular choice of the multiplicative characters \(\chi_ 1,...,\chi_ n.\)

The global monodromy of the sheaves \(Kl_ n(\psi)=Kl(\psi;1,...,^{n}1;1,...,^{n}1)\otimes_{E_{\lambda}}E_{ \lambda}((n-1)/2)\) is calculated in chapter 11. If \(G_{geom}\) denotes the Zariski closure in Gl(n) of the image of \(\pi_ 1({\mathbb{G}}_ m\otimes {\bar {\mathbb{F}}}_ q,\bar x)\) under the monodromy representation \(\rho: \pi_ 1({\mathbb{G}}_ m\otimes {\mathbb{F}}_ q,\bar x)\to Gl(n,E_{\lambda}),\) theorem 11.1 tells us that \(G_{geom}\) is equal to Sp(n), Sl(n), SO(n) or \(G_ 2\) according to n even; pn odd; \(p=2\), n odd, \(n\neq 7\); or \(p=2\) and \(n=7\), respectively. This result leads to a rather concrete “equidistribution theorem” as we are going to see.

For \(n=2\) the Kloosterman sum \(Kl(p,a)=Kl(p,2,a)\) is a real number and as a consequence of the “Riemann hypothesis” for curves over finite fields, there exists a unique angle \(\theta(p,a)\in [0,\Pi]\) for which \(- Kl(p,a)=2\sqrt{p} \cos \theta (p,a)\). The fact that the geometric monodromy group of the rank-two sheaf \(Kl_{\psi}(2)\) is a Zariski-dense subgroup of Sl(2) yields, via Weil II [cf. P. Deligne, Publ. Math., Inst. Hautes Étud. Sci. 52, 137-252 (1980; Zbl 0456.14014)], that as \(p\to \infty\), the \(p-1\) angles \(\{\theta (p,a)\}_{a\in {\mathbb{F}}^*_ p}\) become equidistributed in \([0,\Pi].\)- More general Kloosterman sums of n variables are attached to the sheaves \(Kl_{\psi}(n)\). Their “angles” \(\theta ({\mathbb{F}}_ q,\psi,a)\) must now be understood as conjugacy classes in a compact maximal subgroup K of \(G_{geom}\) of the Frobenius elements \(\rho_{\lambda}(F_ a)^{ss}\). Deligne’s equidistribution theorem for pure sheaves \({\mathcal F}\) of weight zero, the proof of which is explained in chapter 3, can be applied and it yields the desired explicit equidistribution property for the “angles” in K.

Further related questions are also considered. For example, it is proved in chapter 8 that the category of lisse, \(\lambda\)-adic sheaves on \({\mathbb{G}}_ m\otimes {\mathbb{F}}_ q\) which are pure, tame at zero and completely ramified at infinity is stable under convolution. Moreover, those sheaves with Swan conductor at infinity equal to one are, up to twist, a sheaf of Kloosterman type.

The book is read with pleasure. The inclusion of motivating questions and of fundamental background (with proofs!) in the first chapters and all over the book is something so unusual in these levels of knowledge that the reviewer thinks it deserves to be stressed.

The book under review provides a sheaf-theoretic approach to the study of Gauss sums and Kloosterman sums by combining a comprehensive treatment of the main representation-theoretic ideas involved, together with their applications. This point of view helps the reader to get a better understanding of some of Deligne’s deep achievements in this context.

The easiest examples of Kloosterman sums are those given by the complex numbers \[ Kl(p,n,a)=\sum_{x_ 1...x_ n=a;\quad x_ 1,...,x_ n mod p}\exp (\frac{2\pi i}{p}(x_ 1+...+x_ n)), \] where \(n\geq 2\) and p is a prime. In general, Kloosterman sums occur naturally as the inverse Fourier transforms of monomials in Gauss sums. They depend on an additive character \(\psi: ({\mathbb{F}}_ q,+)\to E^*,\) E being a finite extension of \({\mathbb{Q}}\), on n multiplicative characters \(\chi_ i: {\mathbb{F}}^*q\to E^*\) and on n strictly positive integers \(b_ i.\)

Let \(E_{\lambda}\) be the \(\lambda\)-adic completion of E, \({\mathfrak O}_{\lambda}\) its ring of integers, for an \(\ell\)-adic place \(\lambda\) of E (\(\ell \neq p)\). By a theorem of Deligne, there exists a lisse sheaf of free \({\mathfrak O}_{\lambda}\)-modules of finite rank on \({\mathbb{G}}_ m\otimes {\mathbb{F}}_ q\), denoted by \(Kl=Kl_{\lambda}(\psi;\chi_ 1,...,\chi_ n;b_ 1,...,b_ n)\), whose local Frobenius traces are equal, up to sign, to the Kloosterman sums in question. The Kloosterman sheaves Kl are pure of weight \(n-1,\) tame at zero and totally wild at infinity with Swan conductor equal to one. The proof of this existence theorem is given in chapter 4 for \(n=1\) and in chapter 5 for general n by successively convolving those sheaves constructed in the case \(n=1.\)

Chapters 6 and 7 deal with the study of the local monodromy at zero and at infinity of a convolution of sheaves. For the special case of Kloosterman sheaves, it is shown in theorem 7.3.2 that the characters occurring in the local monodromy at zero of Kl are precisely the characters \(\chi\) of \({\mathbb{F}}^*_ q\) such that \(\chi^{b_ i}=\chi_ i\) for some value of \(i=1,...,n\); each of these characters occur in a single Jordan block. At infinity, it is proved in chapter 10, that the character of \(Kl_{\lambda}\), as representation of the wild ramification group \(p_{\infty}\), is independent of both \(\lambda\) (this is clear) and of the particular choice of the multiplicative characters \(\chi_ 1,...,\chi_ n.\)

The global monodromy of the sheaves \(Kl_ n(\psi)=Kl(\psi;1,...,^{n}1;1,...,^{n}1)\otimes_{E_{\lambda}}E_{ \lambda}((n-1)/2)\) is calculated in chapter 11. If \(G_{geom}\) denotes the Zariski closure in Gl(n) of the image of \(\pi_ 1({\mathbb{G}}_ m\otimes {\bar {\mathbb{F}}}_ q,\bar x)\) under the monodromy representation \(\rho: \pi_ 1({\mathbb{G}}_ m\otimes {\mathbb{F}}_ q,\bar x)\to Gl(n,E_{\lambda}),\) theorem 11.1 tells us that \(G_{geom}\) is equal to Sp(n), Sl(n), SO(n) or \(G_ 2\) according to n even; pn odd; \(p=2\), n odd, \(n\neq 7\); or \(p=2\) and \(n=7\), respectively. This result leads to a rather concrete “equidistribution theorem” as we are going to see.

For \(n=2\) the Kloosterman sum \(Kl(p,a)=Kl(p,2,a)\) is a real number and as a consequence of the “Riemann hypothesis” for curves over finite fields, there exists a unique angle \(\theta(p,a)\in [0,\Pi]\) for which \(- Kl(p,a)=2\sqrt{p} \cos \theta (p,a)\). The fact that the geometric monodromy group of the rank-two sheaf \(Kl_{\psi}(2)\) is a Zariski-dense subgroup of Sl(2) yields, via Weil II [cf. P. Deligne, Publ. Math., Inst. Hautes Étud. Sci. 52, 137-252 (1980; Zbl 0456.14014)], that as \(p\to \infty\), the \(p-1\) angles \(\{\theta (p,a)\}_{a\in {\mathbb{F}}^*_ p}\) become equidistributed in \([0,\Pi].\)- More general Kloosterman sums of n variables are attached to the sheaves \(Kl_{\psi}(n)\). Their “angles” \(\theta ({\mathbb{F}}_ q,\psi,a)\) must now be understood as conjugacy classes in a compact maximal subgroup K of \(G_{geom}\) of the Frobenius elements \(\rho_{\lambda}(F_ a)^{ss}\). Deligne’s equidistribution theorem for pure sheaves \({\mathcal F}\) of weight zero, the proof of which is explained in chapter 3, can be applied and it yields the desired explicit equidistribution property for the “angles” in K.

Further related questions are also considered. For example, it is proved in chapter 8 that the category of lisse, \(\lambda\)-adic sheaves on \({\mathbb{G}}_ m\otimes {\mathbb{F}}_ q\) which are pure, tame at zero and completely ramified at infinity is stable under convolution. Moreover, those sheaves with Swan conductor at infinity equal to one are, up to twist, a sheaf of Kloosterman type.

The book is read with pleasure. The inclusion of motivating questions and of fundamental background (with proofs!) in the first chapters and all over the book is something so unusual in these levels of knowledge that the reviewer thinks it deserves to be stressed.

Reviewer: P.Bayer

##### MSC:

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14G25 | Global ground fields in algebraic geometry |

11T23 | Exponential sums |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |