1. Introduction

Let ${A}$ be an abelian variety over an algebraically closed field ${k}$, of dimension ${g}$. One of the basic tools in analyzing the properties of ${A}$ is the study of line bundles on ${A}$. It’s a little non-intuitive to me why this is the case, so I’m going to try to motivate the topic.

Given ${A}$, we are interested in questions of the following form: What is the structure of the ${n}$-torsion points ${A[n]}$? To compute ${|A[n]|}$, we are reduced to computing the degree of multiplication by ${n}$, $\displaystyle n_A: A \rightarrow A$

(which is a morphism of varieties). In fact, we will show that ${n_A}$ is a finite flat morphism, and determine the degree of ${n_A}$, which is thus the cardinality of the fiber over ${0}$. The determination will be done by analyzing how ${n_A^*}$ acts on line bundles. For a symmetric line bundle ${\mathcal{L}}$ over ${A}$, one can prove the crucial formula $\displaystyle n_A^* \mathcal{L} \simeq \mathcal{L}^{n^2},$

and comparing the Hilbert polynomials of ${\mathcal{L}}$ and ${n_A^* \mathcal{L}}$, one can get as a result $\displaystyle \deg n_A = n^{2g} .$

Another way of phrasing this deduction is the following. By the Hirzebruch-Riemann-Roch formula and the parallelizability of an abelian variety, we have $\displaystyle \chi(\mathcal{L}) = \frac{c_1(\mathcal{L})^g}{g!}$

for any line bundle ${\mathcal{L} \in \mathrm{Pic}(A)}$. Consequently, in view of the asserted formula ${n_A^* \mathcal{L} \simeq \mathcal{L}^{n^2}}$, we find for any line bundle ${\mathcal{L}}$: $\displaystyle \chi(n_A^* \mathcal{L}) = n^{2g} \chi(\mathcal{L}).$

Choosing ${\mathcal{L}}$ to be a high power of a very ample line bundle, we will have ${\chi(\mathcal{L}) \neq 0}$. Now we can appeal to the following result:

Theorem 1 Let ${X}$ be a proper scheme over a field. Let ${G}$ be a finite group scheme, and let ${\pi: P \rightarrow X}$ be a ${G}$-torsor. Then if ${\mathcal{F}}$ is a coherent sheaf on ${X}$, we have $\displaystyle \chi(\pi^* \mathcal{F}) = (\deg \pi) \chi(\mathcal{F}).$

It follows from this result that ${\deg n_A = n^{2g}}$. For ${n}$ prime to the characteristic, the morphism ${n_A}$ can be seen to be separable, and it follows as a result there are ${n^{2g}}$ points of ${n}$-torsion on ${A}$. (more…)

The topic of this post is a curious functor, discovered by Deligne, on the category of sheaves over the affine line, which is a “sheafification” of the Fourier transform for functions.

Recall that the classical Fourier transform is an almost-involution of the Hilbert space ${L^2(\mathbb{R})}$. We shall now discuss the Fourier-Deligne transform, which is an almost-involution of the bounded derived category of ${l}$-adic sheaves, ${\mathbf{D}^b_c(\mathbb{A}^1_{\kappa}, \overline{\mathbb{Q}_l})}$. The Fourier transform is defined by multiplying a function with a character (which depends on a parameter) and integrating. Analogously, the Fourier-Deligne transform will twist an element of ${\mathbf{D}^b_c(\mathbb{A}^1_{\kappa}, \overline{\mathbb{Q}_l})}$ by a character depending on a parameter, and then take the cohomology.

More precisely, consider the following: let $G$ be a LCA group, $G^*$ its dual. We have a canonical character on $\phi: G \times G^* \to \mathbb{C}^*$ given by evaluation. To construct the Fourier transform $L^2(G) \to L^2(G^*)$, we start with a function $f: G \to \mathbb{C}$. We pull back to $G \times G^*$, multiply by the evaluation character $\phi$ defined above, and integrate along fibers to give a function on $G^*$.

Everything we’ve done here has a sheaf-theoretic analog, however: pulling back a function corresponds to the functorial pull-back of sheaves, multiplication by a character corresponds to tensoring with a suitable line bundle, and integration along fibers corresponds to the lower shriek push-forward. Much of the classical formalism goes over to the sheaf-theoretic case. One can prove an “inversion formula” analogous to the Fourier inversion formula (with a Tate twist).

Why should we care? Well, Laumon interpreted the Fourier transform as a suitable “deformation” of the cohomology of a suitable sheaf on the affine line, and used it to give a simplified proof of the main results of Weil II, without using scary things like vanishing cycles and Picard-Lefschetz theory. The Fourier transform also behaves very well with respect to perverse sheaves: it is an auto-equivalence of the category of perverse sheaves, because of the careful way in which it is calibrated. Its careful use can be used to simplify some of the arguments in BBD that also rely on other scary things.