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1. Problems

Here are the problems discussed during the problem session.


    1. Add remark

      Problem 1.1.

      [Y. Zahrin] Let A be an abelian variety over a number field K. Does there exist a finite extension K'/K such that for a set of primes v of K' of density 1, A/\F_v is ordinary?
          It is known that if \dim A \leq 2. The result is also true for some other varieties, such as K3 surfaces or ordinary cubic fourfolds. Look at Ogus, LN 900.


        • Add remark

          Problem 1.2.

          [D. Ulmer] Let X/\mathbb C be a surface of general type. Then Lang’s conjecture implies that there exists a constant N, depending only on X, such that if C \subseteq X is rational, then K_X \cdot C \leq N.

          What happens if X is an elliptic surface? Is there a bound on K_X \cdot C as C varies through rational curves?
              Ulmer conjectures that there is a bound. However, this result is false over finite fields.


            • Add remark

              Problem 1.3.

              [J. Achter] Let K be a number field, and let A/K be an abelian variety. Do there exist infinitely many places v of K such that the reduction A/\F_v is not ordinary? More strongly, do there exist infinitely many places v of K such that the reduction A/\F_v is supersingular?
                  This result was proven by Elkies for \dim A = 1 over number fields K with a real place. Poonen says that the supersingular part is probably false, since the supersingular locus in the moduli space of abelian varieties has codimension >1.


                • Add remark

                  Problem 1.4.

                  [V. Vologodsky] (a) Let K be an algebraically closed field, X/K an abelian variety. The Voevodsky motive F is given as a sheaf of groups F(S) = \textrm{Mor}_K(S,X). Is F geometric (i.e. in the triangulated subcategory generated by smooth projective varieties)?

                  (b) (This is a stronger variant of (a)): Is every principally polarized abelian variety a direct summand of a Jacobian integrally (as an abelian variety)?


                    • Add remark

                      Problem 1.5.

                      [R. Pries] Let p be a prime number. For a fixed dimension g and a fixed p-rank f, what is the smallest field of definition of a simple abelian variety over \overline{\F_p} with dimension g and p-rank f?


                        • Add remark

                          Problem 1.6.

                          [C. Voisin] Is every abelian variety the quotient of \textrm{Jac}(C), where C is a smooth plane curve?
                              Ciliberto: Yes, if every surface S of general type is birational to S' \subseteq \mathbb P^3, where S' has only isolated singularities.


                            • Add remark

                              Problem 1.7.

                              [C. Voisin] Let X be a principally polarized abelian variety of dimension \geq 4. Is \theta^2 \in CH^2(X)/\textrm{tors} divisible by 2?
                                  Voisin thinks the answer may be "no", by using the method of Esnault-Bloch.


                                • Add remark

                                  Problem 1.8.

                                  [H. Esnault] Let A be an abelian variety over a finite field K, and u: A \to A a fixed automorphism. Then u induces u^*: H^*(\overline{A}, \Q_l) \to H^*(\overline{A}, \Q_l). Let \alpha_i be its eigenvalues, and let |\alpha_i| denote the archimedean size of \alpha_i. Take the subring in H^*(\overline{A}, \Q_l) generated by a polarization and stabilized by u. Is the maximum archimedean size of the eigenvalues of the subring equal to the maximum archimedean size of the eigenvalues of H^*(\overline{A}, \Q_l)?
                                      OK if \dim A = 2, or if K = \C and A is a smooth projective variety.

                                      Cite this as: AimPL: Cohomological methods in abelian varieties, available at http://aimpl.org/cohomabelian.