Recent zbMATH articles in MSC 11G10https://zbmath.org/atom/cc/11G102022-07-25T18:03:43.254055ZWerkzeugGalois symbol maps for abelian varieties over a \(p\)-adic fieldhttps://zbmath.org/1487.111052022-07-25T18:03:43.254055Z"Hiranouchi, Toshiro"https://zbmath.org/authors/?q=ai:hiranouchi.toshiroLet \(k\) be a finite extension of \(\mathbb{Q}_p\) with resident field \(\mathbb{F}\). Let \(X\) be a projective smooth geometrically connected curve over \(k\). The reciprocity maps \(k(x)^{\times} \to \pi_1^{\mathrm{ab}}(x)\) of the local class field theory of \(k(x)\) for some closed point \(x\) of \(X\) induce the reciprocity map \(\rho\) from \(SK_1(X):= \mathrm{Coker}(\delta: K_2(k(X)) \to \oplus_{x \in X_0}k(x)^{\times})\) to the abelian fundamental group \(\pi_1^{\mathrm{ab}}(X)\) of \(X\). Let \(V(x) = \mathrm{Ker}(\delta: SK_1(X) \to k^{\times})\). Then there is a direct sum decomposition \(V(X) = V(X)_{\mathrm{div}} \oplus V(X)_{\mathrm{fin}}\) where \(V(X)_{\mathrm{div}}\) is the maximal divisible subgroup of \(V(X)\) and \(V(X)_{\mathrm{fin}}\) is a finite group. The main goal of this paper is to study the structure of \(V(X)_{\mathrm{fin}}\).
Let \(\mathscr{X}\) be the regular model of \(X\) over \(O_k\) with \(\mathscr{X} \otimes_{O_k} k \simeq X\). The Jacobian variety \(\mathrm{Jac}(\mathscr{X})\) has generic fiber \(J = \mathrm{Jac}(X)\) and special fiber \(\overline{J} = \mathrm{Jac}(\mathscr{X} \otimes_{O_k} \mathbb{F})\). The following is the main result of this paper.
Theorem: Suppose that \(J[p] \subset J(k)\), \(\overline{J}\) has ordinary reduction, and \(k(\mu_{p^{N+1}})/k\) is a nontrivial totally ramified extension where \(\mu_{p^{N+1}}\) is the group of \(p^{N+1}\)-th roots of unity, and \(N = \mathrm{max}\{n \mid J[p^n] \subset J(k)\}\). Then we have \(V(X)_{\mathrm{fin}} \simeq (\mathbb{Z}/p^N)^{\oplus g} \oplus \overline{J}(\mathbb{F})\) where \(g = \mathrm{dim}(J)\).
Reviewer: Xiao Xiao (Utica)Mass formula and Oort's conjecture for supersingular abelian threefoldshttps://zbmath.org/1487.140622022-07-25T18:03:43.254055Z"Karemaker, Valentijn"https://zbmath.org/authors/?q=ai:karemaker.valentijn"Yobuko, Fuetaro"https://zbmath.org/authors/?q=ai:yobuko.fuetaro"Yu, Chia-Fu"https://zbmath.org/authors/?q=ai:yu.chia-fuThe paper deals with mass stratification and conjecture by \textit{S. J. Edixhoven} et al. [Bull. Sci. Math. 125, No. 1, 1--22 (2001; Zbl 1009.11002)] on the automorphism groups of generic (supersingular) abelian threefolds for supersingular locus of the Siegel modular variety of degree 3. The authors of the paper under review give the number of strata and obtaine the explicit mass formula for each stratum. The classification of possible automorphism groups on each strata of \(\alpha\)-number one is also given. These give the Oort conjecture for investigated polarized abelian threefolds in the case \(p > 2.\) For supersingular abelian surfaces for any odd prime \(p\) the Oort conjecture is proved by \textit{T. Ibukiyama} [J. Math. Soc. Japan 72, No. 4, 1161--1180 (2020; Zbl 1471.14091)]. Let \((X, \lambda)\) be the \(p\)-power degree polarized abelian variety over an algebraically closed field \(k\) of characteristic \(p\) and \(x = (X_0, \lambda_0)\) be polarized supersingular abelian variety of \(p\)-power degree over \(k\). An abelian variety defined over \(k\) is supersingular if it is isogenous to a product of supersingular elliptic curves and is superspecial if it is isomorphic to a product of supersingular elliptic curves. The authors of the paper under review first define for any integer \(d \ge 1\) the (coarse) moduli space \({\mathcal A}_{g,d}\) over \(\overline{\mathbb F}_p\) of \(g\)-dimensional abelian varieties \((X, \lambda)\) with polarization degree \(\deg \lambda = d^2\) and for any \(m \ge 1\) the supersingular locus \({\mathcal G}_{g,p^m}\) of supersingular abelian varieties in \({\mathcal A}_{g,p^m}\). For abelian variety \((X,\lambda)\) let \(X^\bot\) be its dual and respectively let \(G\) and \(G^\bot\) their \(p\)-divicible groups. A polarization \(\lambda\) is an isogeny which is symmetric \((\lambda: X \to X^\bot)^\bot = \lambda\) with the identification \(X = X^{\bot \bot}\) by \textit{D. Mumford} [Abelian varieties. London: Oxford University Press (1970; Zbl 0223.14022)] and by \textit{T. Oda} and \textit{F. Oort} [in: Proc. int. Symp. on algebraic geometry, Kyoto. 595--621 (1977; Zbl 0402.14016)]. A quasi-polarization \(\lambda: G \to G^\bot\) of a \(p\)-divisible group \(G\) is a symmetric isogeny of \(p\)-divisible groups such that \((\lambda: G \to G^\bot)^\bot = \lambda\) by \textit{F. Oort} [Ann. Math. (2) 152, No. 1, 183--206 (2000; Zbl 0991.14016)]. Then the authors define the set \(\Lambda_x\) of isomorphim classes of \(p\)-power polarization degree polarized abelian varieties \((X, \lambda)\) over \(k\). The set \(\Lambda_x\) consists of those polarized abelian varieties whose assosiated quasi-polarized \(p\)-divisible groups satisfy \((X, \lambda)[p^\infty] = (X_0, \lambda_0)[p^\infty] .\) The set \(\Lambda_x\) is finite by \textit{C.-F. Yu} [J. Aust. Math. Soc. 78, No. 3, 373--392 (2005; Zbl 1137.11323)] The mass of \(\Lambda_x\) is defined by \(\mathrm{Mass}(\Lambda_x) = \sum_{(X, \lambda) \in \Lambda_x} \frac{1}{\#\mathrm{Aut}(X,\lambda)}.\) Sections 4 and 5 are concerned with computing the mass for principally polarized supersingular abelian threefolds respectivly for \(\alpha\)-number \(\ge 2\) (Theorem 4.3) and for \(\alpha\)-number 1 (Theorem 5.21). Section 6 deals with ``the automorphism groups of principally polarised abelian threefolds \((X, \lambda)\) over an algebraically closed field \(k \supseteq {\mathbb F}_p\) with \(\alpha(X) = 1\).'' The main result of the section is Theorem 6.4. The section includes also the discussion of arithmetic properties of definite quaternion algebras over rational numbers and the superspecial case. The section is ended with some open problems. The case of a set-theoretic intersection of the Fermat curve and a curve \(\Delta\) defined by authors in Section 5 is treated in the Appendix.
Reviewer: Nikolaj M. Glazunov (Kyïv)Points of small height on semiabelian varietieshttps://zbmath.org/1487.140632022-07-25T18:03:43.254055Z"Kühne, Lars"https://zbmath.org/authors/?q=ai:kuhne.lars.1Let \(K\) be a number field, and \(X\) a projective variety over \(K\). For a place \(\nu\) of \(K\), denote by \(\mathbb{C}_{\nu} := \widehat{\overline{K}_{\nu}}\) and \(X_{\mathbb{C}_{\nu}}^{\operatorname{an}}\) the corresponding analytic space. A ``generic'' sequence of closed points \(\{x_i\}\) of \(X\) is a sequence such that no infinite subsequence is contained in a proper closed subvariety of \(X\).
Let \(\overline{L}\) be a line bundle on \(X\) endowed with a set of metrics; this defines a height function on closed points of \(X\), which we denote by \(h_{\overline{L}}\), which we assume for now to be positive. A ``small'' sequence of closed points \(\{x_i\}\) of \(X\) is a sequence such that \(h_{\overline{L}}(x_i)\to 0\).
Then the equidistribution conjecture of small points can be formulated in the following way.
Conjecture 1 (Equidistribution conjecture): Let \(\{x_i\}\) be a generic small sequence of closed points of \(X\). Then, for every place \(\nu\) of \(K\), the measures
\[
\frac{1}{\#O_{\nu}(x_i)}\sum_{y \in O_{\nu}(x_i)}\delta_y \; \text{ converge weakly to } \frac{1}{\operatorname{deg}_L(X)}c_1(\overline{L}_{\nu})^{\wedge \operatorname{dim}(X)},
\]
where \(O_{\nu}(x_i) = \left(x_i \otimes_K\mathbb{C}_{\nu}\right)^{\operatorname{an}}\) is the analytic \(0\)-cycle of \(X_{\mathbb{C}_{\nu}}^{\operatorname{an}}\) associated to \(x_i\), \(\delta_y\) is the Dirac measure supported at \(y\), and \(c_1\left(\overline{L}_{\nu}\right)^{\wedge \operatorname{dim}(X)}\) is a measure associated to \(\overline{L}_{\nu}\).
When \(X\) is an algebraic group and \(L\) is a line bundle on \(X\), there is a canonical way of associating a metric to \(L\) (hence defining a height function). For instance, these height functions uniquely identify torsion points of \(X\left(\overline{K}\right)\) with closed points of zero height. In this context, one can state the following conjecture.
Conjecture 2 (Bogomolov conjecture): Let \(Y\) be a geometrically irreducible algebraic subvariety of \(X\), which is not an irreducible component of an algebraic subgroup of \(X\). Then there exists an \(\varepsilon >0\) such that the set
\[
\left\{y \in Y\left(\overline{K}\right) \; | \; h_{\overline{L}}(y) \leq \varepsilon \right\}
\]
is not Zariski dense in \(X\).
The equidistribution conjecture was proven for abelian varieties by \textit{L. Szpiro} et al. [Invent. Math. 127, No. 2, 337--347 (1997; Zbl 0991.11035)] where they initiated a method relaying on the arithmetic Hilbert--Samuel theorem. A more general equidistribution theorem was then proven by Yuan who extends their principle (see [\textit{X. Yuan}, Invent. Math. 173, No. 3, 603--649 (2008; Zbl 1146.14016)]). These techniques are however not useful in the case of semiabelian varieties, since they relay on generic sequences of points whose height converges towards the height of the ambient variety, which in the case of semiabelian varieties is negative unless it is almost split. On the other hand, the Bogomolov conjecture was settled for abelian varieties and algebraic tori by \textit{S. Zhang} [J. Amer. Math. Soc. 8, No. 1, 187--221 (1995; Zbl 0861.14018); Ann. of Math. (2) 147, No. 1, 159--165 (1998; Zbl 0991.11034)]. Regarding semiabelian varieties, both conjectures were only known whenever the abelian variety is ``almost split'' due to the work of \textit{A. Chambert-Loir} [Ann. Sci. École Norm. Sup. (4) 33, No. 6, 789--821 (2000; Zbl 1018.11034)], where his main obstruction is the negativity of the hieght of semiabelian varieties in the non-split case. The Bogomolov conjecture was proven by \textit{S. David} and \textit{P. Philippon} [C. R. Acad. Sci. Paris Sér. I Math 331, 387--592 (2000; Zbl 0972.11059)] using a different approach.
In the present article, the author proves both statements in the case of general semiabelian varieties using an asymptotoc adaption of the techniques initiated by Szpiro, Ullmo and Zhang in [loc. cit.] avoiding the main obstructions occurring in the work of Chambert-Loir.
Reviewer: Ana María Botero (Regensburg)Mirror symmetry of Calabi-Yau manifolds fibered by \((1,8)\)-polarized abelian surfaceshttps://zbmath.org/1487.140862022-07-25T18:03:43.254055Z"Hosono, Shinobu"https://zbmath.org/authors/?q=ai:hosono.shinobu"Takagi, Hiromichi"https://zbmath.org/authors/?q=ai:takagi.hiromichiThe paper under review studies mirror symmetry of a family of Calabi-Yau threefolds fibered by \((1,8)\)-polarized abelian surfaces, with vanishing Euler characteristic. Such a Calabi-Yau threefold has \(h^{1,1}=h^{2,1}=2\), and is denoted by \(V_{8,w}^1\). The Heisenberg group \({\mathcal{H}}_8=<\sigma,\tau>\) acts on \(V_{8,w}^1\) as \(\mathbb{Z}_8\times\mathbb{Z}_8\). The quotients \(V_{8,w}^1/\mathbb{Z}_8\times \mathbb{Z}_8\) and \(V_{8,w}^1/\mathbb{Z}_8\) with a subgroup \(\mathbb{Z}_8=<\tau>\subset\mathbb{Z}_8\times\mathbb{Z}_8\) have been studied in the article of [\textit{M. Gross} and \textit{S. Pavanelli}, Proc. Am. Math. Soc. 136, No. 1, 1--9 (2008; Zbl 1127.14036)]. In that article it was conjectured that mirror of the quotient \(V_{8,w}^1/\mathbb{Z}_8\) is given by \(V_{8,w}^1/\mathbb{Z}_8\times\mathbb{Z}_8\). In fact, mirror symmetry was first obtained by Pavenelli stydying locally near a special boundary point by computing Gromov-Witten invariants.
A purpose of this paper is to extend a local mirror symmetry to a global one.
Conjecture 1=Proposition 7.9: Mirror of the Calabi-Yau threefold \(V_{8,w}^1\) is given by \(V_{8,w}^1/\mathbb{Z}_8\) with a subgroup \(\mathbb{Z}_8=<\tau>\subset\mathbb{Z}_8\times\mathbb{Z}_8=<\sigma,\tau>\). Then the mirror of the quotient \(V_{8,w}^1/\mathbb{Z}_8\) is given by \(V_{8,w}^1/\mathbb{Z}_8\times\mathbb{Z}_8\).
This is done by constructing families \(\mathcal{V}_{\mathbb{Z}_8}^1\to\mathbb{P}_{\Delta}\) and \(\mathcal{V}_{\mathbb{Z}_8}\times\mathbb{Z}_8^1\to\mathbb{P}_{\Delta}\) for \(V_{8,w}^1/\mathbb{Z}_8\) and \(V_{8,w}^1/\mathbb{Z}_8\times\mathbb{Z}_8\), respectively, over a toric variety \(\mathbb{P}_{\Delta}\). The Picard-Fuchs differential equations of these families are determined and it is shown that they are identical. Also It is shown that there are degeneration points \(A, B, C\) and \(A^{\prime}, B^{\prime}, C^{\prime}\) on a suitable resolution of \(\mathbb{P}_{\Delta}\), where the mirror correspondences are observed: \(A\leftrightarrow V_{S,w}^1,\,\, B\leftrightarrow V_{S,w}^1/\mathbb{Z}_8\times\mathbb{Z}_8,\,\, C\leftrightarrow V_{S,w}^1/\mathbb{Z}_8\) and \(A^{\prime}, B^{\prime}, C^{\prime}\) corresponding to birational models of each. Mirror correspondences are confirmed by computing Gromov-Witten invariants of stable maps for some genus \(g\), e.g., \(g=0, 1, 2\), near the boundary points.
From the calculations of Gromov-Witten invariants, the generating functions of these invariants are shown to have quasi-modular properties. For the genus zero Gromov-Witten invariants is the following observation.
Conjecture 2=Observation 5.9: The generating functions \(Z_{0,n}^A(q)\) have the following forms: \[Z_{0,n}^A(q)=P_{0,n}^A(E_2,E_4,E_6)(64/\bar{\eta}(q)^8)^7\] where \(\bar{\eta}(q):=\prod_{n\geq 1} (1-q^n)\) and \(P_{0,n}^A\) are quasi-modular forms of weight \(4(n-1)\) expressed in terms of \(E_2(q), E_4(q)\) and \(E_6(q)\) with \(P_{0,1}^A=1\). Similarly, \(Z_{0,n}^B(q)=\frac{1}{64} Z_{0,n}^A(q^8),\, Z_{0,n}^C{q}=\frac{1}{8}Z_{0,n}^A(q^2)\).
This is established numerically computing the BPS numbers.
For higher genus Gromov-Witten invariants, and for the generating functions \(Z_{g,n}^M(q)\) for \(M=A,B,C\), conjectures are formulated.
Conjecture 1.3: The generating functions \(Z_{g,n}^M(q)\) (\(M=A,B\)) are expressed in terms of quasi-modular forms as \[Z_{g,n}^A=P_{g,n}^A(E_2, S, T, U)(64/\bar{\eta}(q)^8)^n,\, Z_{g,n}^B=P_{g,n}^B(E_2,S,T,U)(1/\bar{\eta}(q^8)^8)^n\] where \(P_{g,n}^A\) and \(P_{g,n}^B\) are polynomials of degree \(2(g+n-1)\) of Eisenstein series \(E_2\) and \(S:=\theta(q)^4,\, T:=\theta(q^2)^4,\, U:=\theta(q)^2\theta(q^2)^2\).
For \(M=C\), there is also a conjectural formula for \(Z_{g,n}^C(q)\).
For \(n=1\), the conjectural formulas take simpler forms. The cojecture 1.3 was confirmed for \(n=g=1\).
Reviewer: Noriko Yui (Kingston)