Skip to main content
Log in

Vanishing theorems for constructible sheaves on abelian varieties over finite fields

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Abbes, A., Saito, T.: Ramification and cleanliness. Tohoku Math J Centennial Issue 63(4), 775–853 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison Wesley (1969)

  3. Beilinson, A.A., Bernstein, J., Deligne, P.: Faiscaux pervers. In: Analyse et topologie sur les espaces singuliers (I), asterisque 100, SMF (1982)

  4. Ekedahl, T.: On the adic formalism. In: The Grothendieck Festschrift, vol. II, Progr. Math., vol. 87, pp. 197–218. Birkhäuer, Boston (1990)

  5. Deligne, P.: La conjecture de Weil, II. Inst. Hautes Etudes Sci. Publ. Math. 52, 137–252 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  6. Deligne, P.: Finitude de l’extensions de \(\mathbb{Q}\) engendree par les traces de Frobenius, en characterisque finie. Moscow Math. J. 12 (2012)

  7. Drinfeld, V.: On a conjecture of Deligne. Moscow Math. J. 12 (2012)

  8. Drinfeld, V.: On a conjecture of Kashiwara. Math. Res. Lett. 8, 713–728 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Franecki, J., Kapranov, M.: The Gauss map and a noncompact Riemann–Roch formula for constructible sheaves on semiabelian varieties. Duke Math. J. 104(1), 171–180 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gabber, O., Loeser, F.: Faiscaux pervers \(\ell \)-adiques sur un tore. Duke Math. J. 83(3), 501–606 (1996)

    Article  MathSciNet  Google Scholar 

  11. Krämer, T., Weissauer, R.: Vanishing Theorems for constructible sheaves on abelian varieties. J. Algebr. Geom. 24, 531–568 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Krämer, T., Weissauer, R.: On the Tannaka group attached to the Theta divisor of a generic principally polarized abelian variety. Math. Z. arXiv:1309.3754 (2015, to appear)

  13. Lafforgue, L.: Chtoucas de Drinfeld et correspondence de Langlands. Invent. Math. 147(1), 1–241 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Laumon, G.: Letter to Gabber and Loeser (22/12/91)

  15. Samuel, P.: Lectures on Unique Factorization Domains. Tata Institute, Bombay (1964)

    MATH  Google Scholar 

  16. Serre, J.P.: Algebraic Groups and Class Fields. Springer, Berlin (1988)

  17. Weissauer, R.: On the rigidity of BN-sheaves. arXiv:1204.1929

  18. Weissauer, R.: Why certain Tannaka groups attached to abelian varieties are almost connected.arXiv:1207.4039

  19. Weissauer, R.: Degenerate perverse sheaves on abelian varieties. arXiv:1204.2247

  20. Weissauer, R.: A remark on the rigidity of BN-sheaves. arXiv:1111.6095 (2011)

  21. Zariski, O., Samuel, P.: Commutative Algebra, vol. II. Springer, Berlin (1960)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rainer Weissauer.

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. \(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

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

Navigation