Vanishing theorems for constructible sheaves on abelian varieties over finite fields

Abstract

Let $\kappa$ be a field, finitely generated over its prime field, and let k denote an algebraically closed field containing $\kappa$. For a perverse $\overline{\mathbb {Q}}_\ell$-adic sheaf $K_0$ on an abelian variety $X_0$ over $\kappa$, let K and X denote the base field extensions of $K_0$ and $X_0$ to k. Then, the aim of this note is to show that the Euler–Poincare characteristic of the perverse sheaf K on X is a non-negative integer, i.e. $\chi (X,K)=\sum _\nu (-1)^\nu \dim _{\overline{\mathbb {Q}}_\ell }(H^\nu (X,K))\ge 0$. This generalizes the result of Franecki and Kapranov [9] for fields of characteristic zero. Furthermore we show that $\chi (X,K)=0$ implies K to be translation invariant. This result allows to considerably simplify the proof of the generic vanishing theorems for constructible sheaves on complex abelian varieties of [11]. Furthermore it extends these vanishing theorems to constructible sheaves on abelian varieties over finite fields.

Appendix

Let $A_n=\Lambda \{\!\{t_1,\ldots ,t_n\}\!\}$ denote the ring of power series in the variables $t_1,\ldots ,t_n$ with coefficients in some finite extension field $E_\lambda$ of $\mathbb {Q}_\ell$ that are convergent in $\Lambda$ with some positive radius r of convergence with respect to the nonarchimedean norm on $\Lambda$, with r and $E_\lambda$ depending on the power series.

Lemma 7

The Weierstraß preparation theorem holds for $A_n$. The ring $A_n$ is a regular noetherian local ring of Krull dimension n, hence $A_n$ is a normal factorial domain.

Proof

Any substitution $t_i \mapsto \lambda _i \cdot t_i$ for $\lambda _i\in \Lambda ^*$ defines an automorphism of $A_n$. For the proof of the Weierstraß preparation theorem suppose given $G\in A_n$ and a $t_1$-regular $F\in A_n$ such that $F(t_1,0,\ldots ,0) =c\cdot t_1^a$ plus terms of higher order. We have to show the existence of $U\in A_n$ and $R_0,\ldots ,R_{a-1}\in A_{n-1}=\Lambda \{\!\{ t_2,\ldots ,t_n\}\!\}=:\Lambda \{\!\{ T'\}\!\}$ such that $G(T) = U(T)\cdot F(T) + \sum _{i=0}^{a-1} R_i(T') \cdot t_1^i$. We may assume that F is not a unit in $A_n$ and that $G(0,\ldots ,0)$ has absolute value $\le 1$. Then, by a suitable substitution $T \mapsto \lambda \cdot T$, we can assume $F,G \in {\mathfrak {o}}_\lambda [[t_1,\ldots ,t_n]] \subset A_n$ for some subring ${\mathfrak {o}}_\lambda$ of $\Lambda$ that is finite over $\mathbb {Z}_\ell$ with maximal ideal $m_\lambda$. Replacing F by $c^{-1-a}F(ct_1,c^{a+1}t_2,\ldots ,c^{a+1}t_n)$ for some nonzero $c\in {\mathfrak {o}}_\lambda$, we may assume $F(t_1,0,\ldots , 0)=t_1^a + \sum _{i>a} c_i(T')\cdot t_1^i$ for $c_i(T') \in A_{n-1}$ and $c^{-1-a}F(ct_1,c^{a+1}t_2,\ldots , c^{a+1}t_n)\in {\mathfrak {o}}_\lambda [[t_1,\ldots ,t_n]]$. Again, by replacing F(T) with $b^{-a}F(bt_1,b^at_2,\ldots ,b^at_n)$ for some $b\in m_\lambda$, we can assume $F(T) \equiv t_1^a$ modulo $(m_\lambda ,t_2,\ldots ,t_n)$. Since $G(bct_1,c(bc)^at_2,\ldots , c(bc)^at_n)$ is in ${\mathfrak {o}}_\lambda [[t_1,\ldots ,t_n]]$ and $b^{a}c^{1+a}$ is a unit in $A_n$, the proof of the preparation theorem is reduced to [10, prop. A.2.1(i)], i.e. the assertion that the Weierstraß preparation theorem holds for ${\mathfrak {o}}_\lambda [[t_1,\ldots ,t_n]]$. Since, up to a linear coordinate change, any nontrivial ideal I of $A_n$ contains a $t_1$-regular element $F(T)=t_1^a + c_1(T')t_1^{a-1} + \cdots + c_a(T')$ with coefficients $c_\nu \in \Lambda \{\!\{t_2,\ldots ,t_n\}\!\} \cong A_{n-1}$, by the Weierstraß preparation theorem then F and $I'= I \cap \{ A_{n-1} t_1^{a-1} + \cdots + A_{n-1}\}$ generate I. To show that $A_n$ is noetherian we can assume $A_{n-1}$ to be noetherian by induction, so $I'$ is a finitely generated $A_{n-1}$-module and its generators together with F generate I as an $A_n$-module. This proves that $A_n$ is noetherian. It is easy to see that for any powers series $F(T)\in A_n$ with $F(0) \ne 0$ the formal power series 1 / F(T) again has positive radius of convergency. Hence $A_n$ is a local ring with maximal ideal $m=(t_1,\ldots ,t_n)$. Since $\hat{A}_n$ is isomorphic to the regular ring $\hat{R}_n$ of formal powers series over $\Lambda$, $\hat{A}_n$ is a regular local ring. $\square$

The regular noetherian ring $R_n = \Lambda \otimes _{\mathbb {Z}_\ell } {\mathbb {Z}}_\ell [[t_1,\ldots ,t_{n}]]$ is a subring of the ring $A_n$. The completions $\hat{A}_n$ resp. $\hat{R}_n$ of $A_n$ resp. $R_n$ with respect to $m=(t_1,\ldots , t_n)$ coincide with the formal power series ring $\Lambda [[t_1,\ldots ,t_n]]$. Any power series $P(t_1,\ldots , t_n)\in R_n$ with $P(0,\ldots ,,0)\ne 0$ is a unit in $A_n$. Hence the localisation $\tilde{R}_n=(R_n)_{m}$ of $R_n$ with respect to its maximal ideal $m=(t_1,\ldots , t_n)$ is a local noetherian subring of $A_n$. Both local rings $\tilde{R}_n$ and $A_n$ have the same completion (for their maximal ideals), namely the formal power series ring $\hat{R}_n$. The ideals I in $m \subset R_n$ correspond one-to-one to ideals in $\tilde{R}_n$. Since Noetherian local rings are Zariski rings [21, p. 264], any ideal I of a noetherian local ring R can be recovered from its completion $\hat{I}$ by intersection $I= \hat{I} \cap R$, and furthermore $\hat{I} = I \cdot \hat{R}$ holds. See [21, VIII, §$2$, thm. 5, cor. 2] resp. [21, VIII, §$4$, thm. 8], and for the definition of Zariski rings [21, VIII §4]. For $I\subset R_n$ this applies for both $\tilde{R}_n \cdot I\subset \tilde{R}_n$ and $A_n\cdot I \subset A_n$, hence I can be recovered from $\tilde{R}_n \cdot I$, which can be recovered from $\hat{R}_n \cdot I$ or $A_n\cdot I$. In particular, for any ideal $I\subset R_n$ contained in $(t_1,\ldots ,t_n)$ the ideal $\tilde{I} = I \cdot A_n = \hat{I} \cap A_n$ generated by I in $A_n$ is maximal resp. zero if and only if $\tilde{I}$ is maximal resp. zero in $A_n$. Similarly, two ideals $J,J'$ in $A_n$ are equal iff $J\cdot \hat{R}_n$ and $J'\cdot \hat{R}_n$ are equal.

Lemma 8

A normal noetherian domain R is factorial if and only if every prime ideal I of height $ht(I)=1$ is a principal ideal.

Proof

[Bourbaki 7.3, no. 2, thm. 1] or [15, thm. 3.2,5.3, p. 6 cor.]. $\square$

Lemma 9

Let p(X) be a $x_1$-regular homogenous polynomial $x_1^a + \sum _{\nu < a} c_\nu (X') \cdot x_1^\nu$ of degree $a>0$ in $A_n=\Lambda \{\!\{ x_1,x_2,\ldots ,x_n\}\!\}$ with coefficients $c_\nu (X')$ in $A_{n-1}=\Lambda \{\!\{x_2,\ldots ,x_n\}\!\}$ for $i=0,\ldots ,a-1$ and with $c_0(X')$ in $(x_2,\ldots ,x_n)$. Then $A_n$ is a finite ring extension of its subring $\Lambda \{\!\{ p(X),x_2,\ldots ,x_n\}\!\}$.

Proof

Any g(X) in $\Lambda \{\!\{ X\}\!\}$ can be written in the form $g(X)=u(X) \cdot p(X) + \sum _{i=0}^{a-1} r_i(X') \cdot x_1^i$ by the Weierstraß preparation theorem (Lemma 7). If we apply this iteratively for u(X) instead of g(X) and continue, we obtain formal power series $f_i(y_1,\ldots ,y_n)\in \Lambda [[y_1,\ldots ,y_n]]$ so that $g(X) = \sum _{i=0}^{a-1} f_i(p(X),x_2,\ldots ,x_n)\cdot x_1^i$ holds in $\Lambda [[x_1,\ldots ,x_n]]$. To prove $\Lambda \{\!\{ X\}\!\} = x_1^{a-1} \cdot \Lambda \{\!\{ p,X'\}\!\} + x_1^{a-2}\Lambda \{\!\{ p,X'\}\!\} + \cdots + \Lambda \{\!\{ p,X'\}\!\}$ it suffices to show $f_i\in \Lambda \{\!\{ Y \}\!\}$. If g(X) and p(X) both have integral coefficients in ${\mathfrak {o}}_\lambda$ with $p(X)\equiv x_1^a$ modulo $(m_\lambda ,x_2,\ldots ,x_n)$, then all $f_i(Y)$ have integral coefficients by the Weierstraß preparation theorem for the ring ${\mathfrak {o}}_\lambda [[x_1,\ldots ,x_n]]$. The general case can be easily reduced to this by the method used in the proof of Lemma 7. We may assume $g(0)=0$ and replace g(X) by $g(c \cdot x_1,c^a \cdot x_2,\ldots ,c^a \cdot x_n)$ and p(X) by $\tilde{p}(X)=c^{-a} \cdot p(c \cdot x_1, c^a \cdot x_2,\ldots , c^a x_n)$ to show $f_i(y_1,\ldots ,y_n)= c^{-i} \cdot g_i(c^{-a}\cdot y_1, y_2,\ldots , y_n)$ for certain $g_i\in {\mathfrak {o}}_\lambda [[x_1,\ldots ,x_n]]$. Hence all $f_i(y_1,\ldots , y_n)$ are locally convergent. $\square$