##
**Exponential sums and differential equations.**
*(English)*
Zbl 0731.14008

Annals of Mathematics Studies, 124. Princeton, NJ: Princeton University Press. ix, 430 p. $ 22.50/pbk; $ 65.00/hbk (1990).

This book, as often the production of the author, is a model of elegance and clarity. It is written in the most precise and sophisticated modern style, but its contents are so classically concrete, that even an unexperienced reader has a chance of understanding the main points of the “high-technology” which is used. Actually this book represents a wonderful occasion for quickly gaining some familiarity with \({\mathcal D}\)-modules, derived categories, differential algebra in the tannakian category context, \(\ell\)-adic cohomology, but also with classical representation theory, waisting no time in abstract technicism. Of course, continuous reference to N. Bourbaki’s Lie algebra chapters, P. Deligne’s Weil papers and Z. Mebkhout’s book on Grothendieck’s six operations for \({\mathcal D}\)-modules might be necessary.

The main point of the book is in the interplay between the two notions of “one-parameter families of exponential sums over finite fields” and “classical differential equations with polynomial coefficients”. In the previous families of sums, the characteristic \(p\) of the finite field k is variable (so, these sums are really parametrized by an arithmetic surface): one then fixes a prime \(\ell\) and the discussion holds for sufficiently large p. The book starts with a chapter on classical representation theory. This is the basis for all the subsequent explicit calculations of Lie groups that appear in the sequel. - Then comes a discussion of differential equations with polynomial coefficients over open curves. Among the topics discussed are: The extension of differential modules defined on an open set of a curve to the curve itself (in particular the notion of “middle extension”), Fourier transforms, slopes at infinity, convolution products over \({\mathbb{A}}^ 1\) or \({\mathbb{G}}_ m\) and differential Galois groups.

The third part of the book develops the analogous concepts for \(\ell\)- adic perverse sheaves: here the slopes at infinity are replaced by the breaks of \(\ell\)-adic representations of inertia groups of the points at infinity and differential Galois groups by geometric monodromy groups. In these second and third parts, the results of the first one are applied to give an explicit calculation of the differential Galois group and of the geometric monodromy group of, respectively, certain differential operators and certain \(\ell\)-adic smooth sheaves. In suitably corresponding situations these turn out to be equal. The explicit calculations rely heavily on results of representation theory, given in chapter 1, some of which classical and some new (the author ascribes many of them to Ofer Gabber, whose name appears at several steps along this book).

In the last part of the book, a general comparison theorem in this direction is proven, which compares, for an object which lives on \(\mathbb{A}^1\) over \(\mathbb{Z}\), the differential Galois group of the \(\mathcal D\)-module Fourier transform of its \(\mathbb{C}\)-fibre to the geometric monodromy group of the \(\ell\)-adic Fourier transform of its \(\mathbb{F}_p\)-fibres, for variable primes p. Most of the book is dedicated to the following “hypergeometric case”, appearing both in the \({\mathcal D}\)-module and the \(\ell\)-adic (i.e. “exponential sums”) contexts. (We will only consider here the irregular singular case.) One starts with two natural numbers \(n>m\) and rational numbers \((\alpha_1,\ldots,\alpha_n,\beta_1,\ldots,\beta_m)\) with \(\alpha_i-\beta_j\not\in\mathbb{Z}\). For large \(p\) and a finite field \(k\) of characteristic \(p\), these may be considered as multiplicative characters of \(k\); let \(\psi\) be a nontrivial additive character of \(k\). For any finite extension \(E\) of \(k\) and \(t\in E^{\times}\) let \(V(n,m;t)\subset \mathbb{G}_m^{n+m}\) denote the hypersurface defined by the equation \(\prod^n_{i=1}x_i =t(\prod^m_{j=1}y_j)\).

On the classical side, let \(D\) denote the derivation \(xd/dx\) of \(\mathbb{C}[x,1/x]/\mathbb{C}\). The exponential sums in this case are of the form: \[ \mathrm{Hyp}(\psi,\alpha\text{'s},\beta\text{'s})(E,t)= \]

\[ \sum_{V(n,m;t)(E)}\psi \circ \operatorname{Tr}_{E/k}\Bigl(\sum_i x_i - \sum_j y_j\Bigr)\prod^n_{i=1}\alpha_i\circ \mathrm{Norm}_{E/k}(x_i) \prod^m_{j=1}\beta_j^{-1}\circ \mathrm{Norm}_{E/k}(y_j). \] The corresponding differential operator (or differential module on \(\mathbb{G}_m)\), is \[ \mathrm{Hyp}(\alpha\text{'s},\beta\text{'s})=\prod^n_{i=1}(D- \alpha_i) -x\prod^m_{j=1}(D-\beta_j). \] Now the sum \(\mathrm{Hyp}(\psi,\alpha\text{'s},\beta\text{'s})(E,t)\) is obtained as trace function of Frobenius on an \(\ell\)-adic “hypergeometric complex”: \(\mathrm{Hyp}(!,\psi;\alpha\text{'s};\beta\text{'s})\), a perverse object of \(D_c^b(\mathbb{G}_m,\overline{\mathbb{Q}}_\ell)\), which in fact in general comes from a lisse \(\ell\)-adic sheaf \(\mathcal H_p=\mathcal H(!,\psi;\alpha\text{'s};\beta\text{'s})\) on \(\mathbb{G}_m\). In particular, \([n- m]^* \mathcal H_p\) has a geometric monodromy group \(G_{\text{geom},p}\) which somehow controls \(\mathrm{Hyp}(\psi,\alpha\text{'s},\beta\text{'s})(E,t)\). As we said before, this construction may be performed for variable \(p\) and the point is that the groups \(G_{\text{geom},p}\) tend, in a suitable sense, to be independent of \(p\), for \(p\gg 0\). They equal an important classical invariant of \(\mathrm{Hyp}(\alpha\text{'s},\beta\text{'s})\), (the derived group of the connected component of the identity in) its differential Galois group \(G_{\text{gal}}\). (Analogous results hold for the regular singular case, corresponding to \(m=n.)\)

The author has formulated a general conjecture on this phenomenon; he has also explained the relevance of this conjecture to the equidistribution of exponential sums varying on an arithmetic surface. The conjecture is here proven to hold for the previous hypergeometric examples, by explicit independent calculation of all the groups involved and, in greater generality, via a theoretical, non-computational method.

The main theme and tool of this work is the perfect analogy between \(\ell\)-adic and classical or \(\mathcal D\)-module coefficients for the cohomology of algebraic varieties. Somehow \(p\)-adic (rigid) cohomology for varieties of characteristic \(p\) remains between the lines, as well as the corresponding monodromy theory: in this case \(\mathcal D\)-module coefficients are the only available and analogous results would be in the style of rigid GAGA. We should however recall, that a typical feature of Dwork’s theory is the independence of p of the deformation equations that control families of twisted exponential sums over fields of characteristic \(p\). This fact, also due to the author (in his PhD thesis), established the link between the deformation equations of Dwork and those of Picard-Fuchs and is in some sense the crucial point also in the present discussion.

We are fascinated from the problematics raised by this book and plan to study it more thoroughly in the next future.

The main point of the book is in the interplay between the two notions of “one-parameter families of exponential sums over finite fields” and “classical differential equations with polynomial coefficients”. In the previous families of sums, the characteristic \(p\) of the finite field k is variable (so, these sums are really parametrized by an arithmetic surface): one then fixes a prime \(\ell\) and the discussion holds for sufficiently large p. The book starts with a chapter on classical representation theory. This is the basis for all the subsequent explicit calculations of Lie groups that appear in the sequel. - Then comes a discussion of differential equations with polynomial coefficients over open curves. Among the topics discussed are: The extension of differential modules defined on an open set of a curve to the curve itself (in particular the notion of “middle extension”), Fourier transforms, slopes at infinity, convolution products over \({\mathbb{A}}^ 1\) or \({\mathbb{G}}_ m\) and differential Galois groups.

The third part of the book develops the analogous concepts for \(\ell\)- adic perverse sheaves: here the slopes at infinity are replaced by the breaks of \(\ell\)-adic representations of inertia groups of the points at infinity and differential Galois groups by geometric monodromy groups. In these second and third parts, the results of the first one are applied to give an explicit calculation of the differential Galois group and of the geometric monodromy group of, respectively, certain differential operators and certain \(\ell\)-adic smooth sheaves. In suitably corresponding situations these turn out to be equal. The explicit calculations rely heavily on results of representation theory, given in chapter 1, some of which classical and some new (the author ascribes many of them to Ofer Gabber, whose name appears at several steps along this book).

In the last part of the book, a general comparison theorem in this direction is proven, which compares, for an object which lives on \(\mathbb{A}^1\) over \(\mathbb{Z}\), the differential Galois group of the \(\mathcal D\)-module Fourier transform of its \(\mathbb{C}\)-fibre to the geometric monodromy group of the \(\ell\)-adic Fourier transform of its \(\mathbb{F}_p\)-fibres, for variable primes p. Most of the book is dedicated to the following “hypergeometric case”, appearing both in the \({\mathcal D}\)-module and the \(\ell\)-adic (i.e. “exponential sums”) contexts. (We will only consider here the irregular singular case.) One starts with two natural numbers \(n>m\) and rational numbers \((\alpha_1,\ldots,\alpha_n,\beta_1,\ldots,\beta_m)\) with \(\alpha_i-\beta_j\not\in\mathbb{Z}\). For large \(p\) and a finite field \(k\) of characteristic \(p\), these may be considered as multiplicative characters of \(k\); let \(\psi\) be a nontrivial additive character of \(k\). For any finite extension \(E\) of \(k\) and \(t\in E^{\times}\) let \(V(n,m;t)\subset \mathbb{G}_m^{n+m}\) denote the hypersurface defined by the equation \(\prod^n_{i=1}x_i =t(\prod^m_{j=1}y_j)\).

On the classical side, let \(D\) denote the derivation \(xd/dx\) of \(\mathbb{C}[x,1/x]/\mathbb{C}\). The exponential sums in this case are of the form: \[ \mathrm{Hyp}(\psi,\alpha\text{'s},\beta\text{'s})(E,t)= \]

\[ \sum_{V(n,m;t)(E)}\psi \circ \operatorname{Tr}_{E/k}\Bigl(\sum_i x_i - \sum_j y_j\Bigr)\prod^n_{i=1}\alpha_i\circ \mathrm{Norm}_{E/k}(x_i) \prod^m_{j=1}\beta_j^{-1}\circ \mathrm{Norm}_{E/k}(y_j). \] The corresponding differential operator (or differential module on \(\mathbb{G}_m)\), is \[ \mathrm{Hyp}(\alpha\text{'s},\beta\text{'s})=\prod^n_{i=1}(D- \alpha_i) -x\prod^m_{j=1}(D-\beta_j). \] Now the sum \(\mathrm{Hyp}(\psi,\alpha\text{'s},\beta\text{'s})(E,t)\) is obtained as trace function of Frobenius on an \(\ell\)-adic “hypergeometric complex”: \(\mathrm{Hyp}(!,\psi;\alpha\text{'s};\beta\text{'s})\), a perverse object of \(D_c^b(\mathbb{G}_m,\overline{\mathbb{Q}}_\ell)\), which in fact in general comes from a lisse \(\ell\)-adic sheaf \(\mathcal H_p=\mathcal H(!,\psi;\alpha\text{'s};\beta\text{'s})\) on \(\mathbb{G}_m\). In particular, \([n- m]^* \mathcal H_p\) has a geometric monodromy group \(G_{\text{geom},p}\) which somehow controls \(\mathrm{Hyp}(\psi,\alpha\text{'s},\beta\text{'s})(E,t)\). As we said before, this construction may be performed for variable \(p\) and the point is that the groups \(G_{\text{geom},p}\) tend, in a suitable sense, to be independent of \(p\), for \(p\gg 0\). They equal an important classical invariant of \(\mathrm{Hyp}(\alpha\text{'s},\beta\text{'s})\), (the derived group of the connected component of the identity in) its differential Galois group \(G_{\text{gal}}\). (Analogous results hold for the regular singular case, corresponding to \(m=n.)\)

The author has formulated a general conjecture on this phenomenon; he has also explained the relevance of this conjecture to the equidistribution of exponential sums varying on an arithmetic surface. The conjecture is here proven to hold for the previous hypergeometric examples, by explicit independent calculation of all the groups involved and, in greater generality, via a theoretical, non-computational method.

The main theme and tool of this work is the perfect analogy between \(\ell\)-adic and classical or \(\mathcal D\)-module coefficients for the cohomology of algebraic varieties. Somehow \(p\)-adic (rigid) cohomology for varieties of characteristic \(p\) remains between the lines, as well as the corresponding monodromy theory: in this case \(\mathcal D\)-module coefficients are the only available and analogous results would be in the style of rigid GAGA. We should however recall, that a typical feature of Dwork’s theory is the independence of p of the deformation equations that control families of twisted exponential sums over fields of characteristic \(p\). This fact, also due to the author (in his PhD thesis), established the link between the deformation equations of Dwork and those of Picard-Fuchs and is in some sense the crucial point also in the present discussion.

We are fascinated from the problematics raised by this book and plan to study it more thoroughly in the next future.

Reviewer: Francesco Baldassarri (Padova)

### MSC:

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

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

12-02 | Research exposition (monographs, survey articles) pertaining to field theory |

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

11T23 | Exponential sums |

12H25 | \(p\)-adic differential equations |

14G15 | Finite ground fields in algebraic geometry |

12H20 | Abstract differential equations |