## 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.

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## Appendix

### 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. \(\square \)

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Weissauer, R. Vanishing theorems for constructible sheaves on abelian varieties over finite fields.
*Math. Ann.* **365**, 559–578 (2016). https://doi.org/10.1007/s00208-015-1307-8

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DOI: https://doi.org/10.1007/s00208-015-1307-8