Recent zbMATH articles in MSC 14Khttps://zbmath.org/atom/cc/14K2021-03-30T15:24:00+00:00WerkzeugPeriodic and quasi-periodic solutions of Toda lattice via hyperelliptic \(\sigma\) functions.https://zbmath.org/1455.140662021-03-30T15:24:00+00:00"Matsutani, Shigeki"https://zbmath.org/authors/?q=ai:matsutani.shigekiSummary: In this report, I summarize results in the paper \textit{Y. Kodama} et al. [Ann. Inst. Fourier 63, No. 2, 655--688 (2013; Zbl 1279.14044)] to pose a problem to give an explicit relation between periodic and quasi-periodic solutions of Toda lattice. For a hyperelliptic curve \(X_g\) of genus \(g\), we have a quasi-periodic solution of Toda lattice in terms of the hyperelliptic \(\sigma\) function and its addition theorem. Using the division polynomial of \(X_g\), we find \(2N\)-division points in its Jacobi variety and then have \(N\)-periodic solution of Toda-lattice. It is well-known that the \(N\)-periodic solution is associated with a hyperellptic curve \(\widehat{X}_{g,N-1}\) of genus \(N-1\) rather than \(g\). However it is not clear how \(X_g\) and \(\widehat{X}_{g,N-1}\) are connected geometrically, though the problem is very simple and natural. In this report, after I give a review of the recent development of \(\sigma\) function theory of higher genus and show the summary of our previous work, I give some comments on the problem.Recursion relations on the power series expansion of the universal Weierstrass sigma function.https://zbmath.org/1455.330112021-03-30T15:24:00+00:00"Eilbeck, J. Chris"https://zbmath.org/authors/?q=ai:eilbeck.john-chris"Ônishi, Yoshihiro"https://zbmath.org/authors/?q=ai:onishi.yoshihiroSummary: The main aim of this paper is an exposition of the theory of Buchstaber and Leykin on the heat equations for the multivariate sigma functions. We treat only the elliptic curve case, but keeping the most general elliptic curve equation, which may be useful for number theoretic applications.Torsion group schemes as iterative differential Galois groups.https://zbmath.org/1455.120062021-03-30T15:24:00+00:00"Maurischat, Andreas"https://zbmath.org/authors/?q=ai:maurischat.andreasThe paper is devoted to the development of Galois theory for extensions of fields of nonzero characteristic. For such fields, one can construct (see [\textit{A. Maurischat}, Trans. Am. Math. Soc. 362, No. 10, 5411--5453 (2010; Zbl 1250.13009)]) a Galois theory similar to Kolchin's theory for extensions of differential fields, if one replaces ordinary derivations by the so-called higher derivations. Since the fields of constants arising in this case are usually not algebraically closed, finite group schemes are used instead of algebraic groups. The paper contains the main concepts and basic results of this theory as an introduction. Its ``main part is to find computable criteria when higher derivations are iterative derivations, and furthermore when an iterative derivation on the function field of an abelian variety is compatible with the addition map''. For the inverse problem of Galois theory, it is shown ``that torsion group schemes of abelian varieties in positive characteristic occur as iterative differential Galois groups of extensions of iterative differential fields''.
Reviewer: Mykola Grygorenko (Kyïv)Some specialization theorems for families of abelian varieties.https://zbmath.org/1455.140872021-03-30T15:24:00+00:00"Zannier, Umberto"https://zbmath.org/authors/?q=ai:zannier.umberto-mSummary: Consider an algebraic family \(\pi:\mathcal{A}\to B\) of abelian varieties, defined over \(\overline{\mathbb{Q}}\). We shall be concerned with properties of the generic fiber of \(\mathcal{A}\) which are preserved on restricting to some (or `many') suitable special fibers. We shall focus on instances like torsion for values of a section, endomorphism rings, existence of generic and special isogenies, illustrating some known results and some applications. Another, more recent, issue which we shall briefly discuss concerns the existence of abelian varieties over \(\overline{\mathbb{Q}}\) not isogenous to a Jacobian. We shall conclude with a few comments on other specialization issues.On the arithmetic of \(K3\) surfaces with complex multiplication and its applications.https://zbmath.org/1455.140762021-03-30T15:24:00+00:00"Ito, Kazuhiro"https://zbmath.org/authors/?q=ai:ito.kazuhiroSummary: This survey article is an outline of author's talk at the RIMS Workshop Algebraic Number Theory and Related Topics (2017). We study arithmetic properties of \(K3\) surfaces with complex multiplication (CM) generalizing the results of Shimada for \(K3\) surfaces with Picard number 20. Then, following Taelman's strategy and using Matsumoto's good reduction criterion for \(K3\) surfaces with CM, we construct \(K3\) surfaces over finite fields with given \(L\)-function, up to finite extensions of the base fields. We also prove the Tate conjecture for self-products of \(K3\) surfaces over finite fields by CM lifts and the Hodge conjecture for self-products of \(K3\) surfaces with CM proved by Mukai and Buskin.Logarithmic abelian varieties. VI: Local moduli and GAGF.https://zbmath.org/1455.140852021-03-30T15:24:00+00:00"Kajiwara, Takeshi"https://zbmath.org/authors/?q=ai:kajiwara.takeshi"Kato, Kazuya"https://zbmath.org/authors/?q=ai:kato.kazuya"Nakayama, Chikara"https://zbmath.org/authors/?q=ai:nakayama.chikaraSummary: This is Part VI of our series of papers on log abelian varieties. In this part, we study local moduli and GAGF of log abelian varieties.
For Part I--V see [the authors, Nagoya Math. J. 189, 63--138 (2008; Zbl 1169.14031); J. Math. Sci., Tokyo 15, No. 1, 69--193 (2008; Zbl 1156.14038); Nagoya Math. J. 210, 59--81 (2013; Zbl 1280.14008); Nagoya Math. J. 219, 9--63 (2015; Zbl 1329.14090); ibid. 64, 21--82 (2018; Zbl 1420.14102)].Extending finite-subgroup schemes of semistable abelian varieties via log-abelian varieties.https://zbmath.org/1455.140862021-03-30T15:24:00+00:00"Zhao, Heer"https://zbmath.org/authors/?q=ai:zhao.heerSummary: We show -- for a semistable abelian variety \(A_K\) over a complete discrete valuation field \(K\) -- that every finite-subgroup scheme of \[A_K\] extends to a log finite-flat group scheme over the valuation ring of \(K\) endowed with the canonical log structure. To achieve this, we first give a positive answer to a question of Nakayama, namely whether every weak log-abelian variety over an \textit{fs (fine and saturated)} log scheme with its underlying scheme locally noetherian is a sheaf for the Kummer-flat topology. We also give several equivalent conditions defining isogenies of log-abelian varieties.