Recent zbMATH articles in MSC 11Ghttps://zbmath.org/atom/cc/11G2022-07-25T18:03:43.254055ZWerkzeugWhat positive integers \(n\) can be presented in the form \(n=(x+y+z)(1/x+1/y+1/z)\)?https://zbmath.org/1487.110322022-07-25T18:03:43.254055Z"Nguyen Xuan Tho"https://zbmath.org/authors/?q=ai:nguyen-xuan-tho.Summary: This paper shows that the equation in the title does not have positive integer solutions when \(n\) is divisible by 4. This gives a partial answer to a question by Melvyn Knight, see \textit{A. Bremner} et al. [Math. Comput. 61, No. 203, 117--130 (1993; Zbl 0808.11022)]. The proof is a mixture of elementary \(p\)-adic analysis and elliptic curve theory.Elliptic curves with all quartic twists of the same root numberhttps://zbmath.org/1487.110552022-07-25T18:03:43.254055Z"Byeon, Dongho"https://zbmath.org/authors/?q=ai:byeon.dongho"Han, Gyeoul"https://zbmath.org/authors/?q=ai:han.gyeoulFor a number field \(K\) and an elliptic curve, \(E/K\), defined over \(K\), we can associate an \(L\)-function, \(L(E/K,s)\), known as its Hasse-Weil \(L\)-function. This function is defined for \(\mathrm{Re}(s)>3/2\) and the Hasse-Weil conjecture asserts that it has an analytic continuation to the entire complex plane. The Hasse-Weil conjecture also states that this \(L\)-function satisfies a functional equation under \(s \longleftrightarrow 2-s\) with a factor of \(\pm 1\) on one side. This factor of \(\pm 1\) is called the (global) root number of the curve, \(w(E/K)\). The functional equation implies that \(w(E/K)=(-1)^{\mathrm{ord}_{s=1} L(E/K,s)}\), so, under the Birch Swinnerton-Dyer conjecture, the root number tells us the parity of the rank of Mordell-Weil group, \(E(K)\).
In this paper, the authors complete an investigation of when the root number stays constant for all twists of an elliptic curve.
There are four types of twists: quadratic, cubic, quartic and sextic. All but the third of these (the quartic twists) have been considered in previous work. See [\textit{T. Dokchitser} and \textit{V. Dokchitser}, Acta Arith. 137, No. 2, 193--197 (2009; Zbl 1275.11097)] and
[\textit{D. Byeon} and \textit{N. Kim}, J. Number Theory 136, 22--27 (2014; Zbl 1284.11089)].
Here the authors address the remaining case, that of the quartic twists. Only elliptic curves with \(j\)-invariant \(1728\) defined by the equation \(y^{2}=y^{2}+ax\) can have quartic twists and these twists are given by \(E_{D}/K: y^{2}=x^{3}+aDx\) where \(D \in K^{\times}/ \left( K^{\times} \right)^{4}\).
Their Theorem~1.1 states that the root numbers of all quartic twists are constant if and only if \(\sqrt{-1} \in K\). Furthermore, the root number is always \(+1\) if \(\sqrt{-1} \in K\). If \(\sqrt{-1} \not\in K\), then there are infinitely many \(D\) such that \(w \left( E_{D}/K \right)=+1\) and infinitely many \(D\) such that \(w \left( E_{D}/K \right)=-1\).
\textit{K. Česnavičius} [J. Reine Angew. Math. 719, 45--73 (2016; Zbl 1388.11032)] already proved that if \(\sqrt{-1} \in K\), then any
elliptic curve with \(j\)-invariant \(1728\) over \(K\) has root number \(+1\). Here the
authors use the structure of the Galois group \(\mathrm{Gal} \left( L/K_{v} \right)\),
where \(K_{v}\) is a local field with respect to a place \(v|2\) and \(L=K_{v}(E[3])\),
to establish the behaviour when \(\sqrt{-1} \not\in K\).
Reviewer: Paul Voutier (London)Non-divisible point on a two-parameter family of elliptic curveshttps://zbmath.org/1487.110562022-07-25T18:03:43.254055Z"Petit, Valentin"https://zbmath.org/authors/?q=ai:petit.valentinIn this paper, the author studies the family of elliptic curves \(\mathcal{E}_n(t)\) over \(\mathbb{Q}\) given by the equation \(y^2 = x^3+ tx^2 - n^2(t+3n^2)x+n^6\). The main result is that if \(n\) is a positive integer and \(t\geq \max(100n^2,n^4\) or \(t\leq \min(-100n^2,-2n^4)\), and \(t^2+2n^2t+9n^4\) is squarefree, then the \((0,n^3)\) is not dividible on \(\mathcal{E}_n(t)\), i.e. there does not exist a point \(P\in \mathcal{E}_n(t)\) and an integer \(\ell \geq 2\) such that \(\ell P=(0,n^3)\).
Reviewer: Andrej Dujella (Zagreb)Relating the Tate-Shafarevich group of an elliptic curve with the class grouphttps://zbmath.org/1487.110572022-07-25T18:03:43.254055Z"Prasad, Dipendra"https://zbmath.org/authors/?q=ai:prasad.dipendra"Shekhar, Sudhanshu"https://zbmath.org/authors/?q=ai:shekhar.sudhanshuLet \(p\) be some prime, let \(E\) be an elliptic curve over \(\mathbb{Q}\), and let \(K = \mathbb{Q}(E[p])\) denote the \(p\)-division field of \(E\). In this manuscript, the authors formulate a precise relationship between the Tate-Shafarevich group of \(E\) with a quotient of the class group of \(K\) on which \(G = \text{Gal}(K) \cong \text{GL}_2(\mathbb{Z}/p\mathbb{Z})\) operates by the standard representation over \(\mathbb{Z}/p\mathbb{Z}\).
The work begins by studying various Galois cohomology groups associated to elliptic curves. The main technical input of this work is Lemma 2.6, which proves:~for \(E/\mathbb{Q} \) with either good or multiplicative reduction at a prime \(p\geq 3\) and \(\ell\) any prime of \(\mathbb{Q}\) and \(w\mid \ell\) a prime of \(K\) over \(\ell\), if the \(\ell\)-Tamagawa number \(c_{\ell}(E)\) is co-prime to \(p\) for all primes \(\ell\neq p\), then \(H^1(I_{w},E[p])^{\Gamma_{w}}\) is non-zero precisely when \(\ell \neq p\) is a prime in \(\mathfrak{T}\), which consists of primes such that either \(E\) has split multiplicative reduction at \(\ell\) and \(E(\mathbb{Q}_{\ell})[p]\) has rank 1, or \(E\) has nonsplit multiplicative reduction at \(\ell\) and \(E(\mathbb{Q}_{\ell}^{\text{ur}})[p]\) has rank 1, where \(I_{w}\) is the inertia subgroup of \(G\) at \(w\), \(D_w\) is the decomposition subgroup of \(G\) at \(w\), and \(\Gamma_w = D_w/I_w\).
With this lemma and other results concerning the \(p\)-rank of the Selmer group of \(E\) under restriction maps, the authors prove their main result, which is Theorem 4.2. Their theorem states that for \(E\) such that \(E\) has either good or multiplicative reduction at \(p\), \(E(\mathbb{Q}_p)[p] = 0\), \(c_{\ell}(E)\) is a \(p\)-adic unit for every finite prime \(\ell\neq p\), \(E[p]\) is an irreducible \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\)-representation, then
\[
\text{rank}_{\mathbb{F}_p} (\text{Sel}_p(E/\mathbb{Q})) - 1 \leq \text{rank}_{\mathbb{F}_p} \text{Hom}_G(\text{Cl}_K/p\text{Cl}_K,E[p])\leq \text{rank}_{\mathbb{F}_p} (\text{Sel}_p(E/\mathbb{Q})) + \#\mathfrak{T},
\]
where \(\mathfrak{T}\) is the set of primes from Lemma 2.6.
To conclude, the authors give an example to illustrate their man theorem and a discussion on how to construct unramified Galois extensions of \(K\) using ideas of Ribet and congruences of cusp forms with Eisenstein series.
Reviewer: Jackson Morrow (Berkeley)The greatest common valuation of \(\phi_n\) and \(\psi_n^2\) at points on elliptic curveshttps://zbmath.org/1487.110582022-07-25T18:03:43.254055Z"Voutier, Paul"https://zbmath.org/authors/?q=ai:voutier.paul-m"Yabuta, Minoru"https://zbmath.org/authors/?q=ai:yabuta.minoruLet \(K\) be a finite extension of the \(p\)-adic field \(\mbox{Q}_p\), \(\pi\) a uniformiser for \(K\), and \(v\) a normalized valuation of \(K\). Let \(E\) be an elliptic curve defined over \(K\) by a minimal Weierstrass model, and \(\{\psi_n(X, Y )\}_{n\geq 1}\) the sequence of associated division polynomials. Also, consider the sequence of polynomials, \(\{\phi_n(X, Y )\}_{n\geq 1}\), such that \[x([n]P) = \frac{\phi_n(x(P), y(P))}{\psi_n^2(x(P), y(P))}, \] for any point, \(P = (x(P), y(P)) \in E(K)\). Let \(\lambda_v : E(K)\setminus \{O\} \rightarrow R\) be the local height function for \( E\) at \(v\). In this paper, for every point \(P\in E(K)\) of infinite order and \(n\geq 1\), the quantity \(k_{n,v}(P) = \min\{v(\phi_n(P)),v(\psi^2(P))\}\) is computed. More precisely, if \(P\) modulo \(\pi\) is non-singular, then \[k_{n,v}(P) = \min\{0, n^2v(x(P))\},\] and if \(P\) modulo \(\pi\) is singular, then \[ k_{n,v}(P) = -2\frac{[K : \mbox{Q}_p]}{ \log |k|} \lambda_v(P) n^2 + \epsilon_{v,n}(P),\] where \(\epsilon_{v,n}(P)\) is explicitly given. Note that this result improves significantly previous estimates.
Reviewer: Dimitros Poulakis (Thessaloniki)Non-vanishing theorems for central \(L\)-values of some elliptic curves with complex multiplicationhttps://zbmath.org/1487.110592022-07-25T18:03:43.254055Z"Coates, John"https://zbmath.org/authors/?q=ai:coates.john-h"Li, Yongxiong"https://zbmath.org/authors/?q=ai:li.yongxiongThe aim of this paper is to extend non-vanishing theorems of Rohrlich for the complex \(L\)-seires of certain quadratic twists of Gross curves. This is obtained by exploiting \(\mathbb{Z}_2\)-Iwasawa theory for the restriction of scalars to quadratic imaginary fields of the Gross curve.
To fix the notation, let \(K=\mathbb{Q}(\sqrt{-q})\) where \(q\equiv 7\pmod{8}\) is a prime number. Let \(H=K(j(\mathcal{O}_K))\) be the Hilbert class field of \(K\), where \(\mathcal{O}_K\) is the ring of integers of \(K\) and \(j\) is the classical modular function. It is known thanks to a result by Gross that there exists a unique elliptic curve \(A\) defined over \(\mathbb{Q}(j(\mathcal{O}_K)\) with complex multiplication by \(\mathcal{O}_K\), minimal discriminant \((-q^3)\), which is a \(\mathbb{Q}\)-curve.
The prime \(2\) splits in \(K\) as \((2)=\mathfrak{p}\bar{\mathfrak{p}}\), with \(\mathfrak{p}\neq\bar{\mathfrak{p}}\). For each integer \(n\geq 1\), let \(A_{\mathfrak{p}^n}\) be the Galois module of \(\mathfrak{p}^n\)-division points of \(A(\bar{\mathbb Q})\), and define \(\mathfrak{F}_\infty=H(A_{\mathfrak{p}^\infty})\) where \(A_{\mathfrak{p}^\infty}= \cup_{n\geq 1}A_{\mathfrak{p}^n}\). Define \(\mathcal{G}=\mathrm{Gal}(\mathfrak{F}_\infty/H)\). Then \(\mathcal{G}\) is isomorphic to \(\mathbb{Z}_2^\times\) via the character \(\rho_\mathfrak{p}\) which gives the action of \(\mathcal{G}\) on \(A_{\mathfrak{p}^\infty}\); this allows to exploit Iwasawa techniques for this particular \(\mathbb{Z}_2\)-extension.
Suppose now that \(q\equiv 7\pmod{16}\). The main results are the following:
\begin{itemize}
\item For all \(\chi:\mathcal{G}\rightarrow\bar{\mathbb{Q}}^\times\) of finite order, \(L(A/H,\chi,1)\neq 0\).
\item For all finite extensions \(F/H\) contained in \(\mathfrak{F}_\infty\), \(A(F)\) is finite and \(\mathfrak{p}\)-primary part \({\text{Ш}}_\mathfrak{p}(A/F)\) of the Tate-Shafarevich group \({\text{Ш}}(A/F)\) of \(A/F\) is finite.
\end{itemize}
These results are generalised to two families of twists of \(A\) by quadratic characters. One of the ideas involved in the proof is to look at the restriction of scalars from \(H\) to \(K\) of \(A\), and exploit the \(\mathbb{Z}_2\)-Iwasawa theory for the extension \(K_\mathfrak{p}\).
Reviewer: Matteo Longo (Padova)Supersingular twisted Edwards curves over prime fields. II: Supersingular twisted Edwards curves with the \(j\)-invariant equal to \(66^3\)https://zbmath.org/1487.110602022-07-25T18:03:43.254055Z"Bessalov, A. V."https://zbmath.org/authors/?q=ai:bessalov.a-v"Kovalchuk, L. V."https://zbmath.org/authors/?q=ai:kovalchuk.l-vSummary: Theorems on the conditions for the existence of supersingular Edwards curves over a prime field with the \(j\)-invariant equal to \(66^3\) and with other values of \(j\)-invariants are formulated and proved. A generalization of the results obtained earlier is presented, which uses isomorphisms of curves in Legendre and Edwards forms.
For Part I, see [the authors, Cybern. Syst. Anal. 55, No. 3, 347--353 (2019; Zbl 1443.11116); translation from Kibern. Sist. Anal. 2019, No. 3, 3--10 (2019)].Comments on efficient batch verification test for digital signatures based on elliptic curveshttps://zbmath.org/1487.110612022-07-25T18:03:43.254055Z"Hakuta, Keisuke"https://zbmath.org/authors/?q=ai:hakuta.keisuke"Ochiai, Hiroyuki"https://zbmath.org/authors/?q=ai:ochiai.hiroyuki"Takagi, Tsuyoshi"https://zbmath.org/authors/?q=ai:takagi.tsuyoshiSummary: Batch verification for digital signature scheme is a method to verify multiple signatures simultaneously. The complex exponent test (CE test for short) proposed by Cheon and Lee is one of the most efficient batch verification tests for several digital signature schemes on certain types of elliptic curves (including Koblitz curves). The security of the CE test relies essentially on the cardinality of a subset of a residue ring of the endomorphism ring of an elliptic curve over an ideal. They have evaluated the cardinality of the above subset, and have illustrated the effectiveness of the CE test by using the evaluation. The aim of this paper is to point out that their evaluation contains a flaw. The flaw is generally related to two roots of a quadratic equation which is used in their argument. We mend the flaw of their evaluation. Our correct evaluation shows that the CE test can achieve the same security as the underlying signature scheme on Koblitz curves. As a result, the CE test is a secure batch verification when the underlying signature scheme uses Koblitz curves.Functional transcendence for the unipotent Albanese maphttps://zbmath.org/1487.110622022-07-25T18:03:43.254055Z"Hast, Daniel Rayor"https://zbmath.org/authors/?q=ai:hast.daniel-rayorGiven a curve \(\mathcal X\) defined over the ring of integers of a \(p\)-adic field \(K\), a basepoint \(b \in \mathcal X(\mathcal O_{K})\), and a positive integer \(n\), the (\(n\)th) unipotent Albanese map \(j_{n}\) is a rigid \(K\)-analytic map from \(X(\mathcal O_{K})\) to a certain unipotent quotient \(\Pi^{\mathrm{dR}}_{n}/F^0\) of the de Rham fundamental group of \(\mathcal X\). This map was originally constructed by \textit{M. Kim} [Publ. Res. Inst. Math. Sci. 45, No. 1, 89--133 (2009; Zbl 1165.14020)], who also proved that the image of \(j_{n}\) is Zariski-dense. This observation plays a crucial role in the \(p\)-adic `nonabelian Chabauty-Kim method' for computing/bounding the set of rational points on a curve defined over \(\mathbb Q\). Specifically, if \(Z\subseteq \Pi^{\mathrm{dR}}_{n}/F^0\) is an algebraic subvariety of positive codimension, \(j_{n}^{-1}(Z) \subseteq X(\mathcal O_{K})\) has lower dimension than \(X_{K}\). In particular, if \(X_{K}\) is a curve, then \(j_{n}^{-1}(Z)\) is finite. In the Chabuaty-Kim method, one takes \(Z\) to be a proper subvariety containing the image of the global points on the curve.
Hast's main theorem is a functional transcendence result refining Kim's observation that the image of the unipotent Albanese map \(j_{n}\) is Zariski dense. This theorem says that the dimension of any irreducible analytic component \(W\) of the intersection of the graph of \(j_{n}\) with an algebraic subvariety of \(X_{K} \times \Pi^{\mathrm{dR}}_{n}/F^0\) is at most the expected dimension for such an intersection, unless the projection of \(W\) to \(X_{K}\) is contained in a `weakly special subvariety' (in the sense of [\textit{B. Klingler}, ``Hodge loci and atypical intersections: conjectures'', Preprint, \url{arXiv:1711.09387}]).
As applications of the main theorem, Hast generalizes the abelian Chabauty's method (the \(n = 1\) case) and the nonabelian Chabauty-Kim method to study rational points on higher dimensional varieties \(V\) and (by replacing a curve \(X\) over a number field \(F\) with the restriction of scalars \(\mathrm{Res}^{F}_{\mathbb Q}\)) the \(F\)-rational points on curves. For instance, under a rank versus dimension inequality for the Albanese variety \(A\) of \(V\), Hast proves that the image of the rational points on \(V\) under the Albanese map is contained in a finite union of cosets of proper abelian subvarieties of \(A\). Moreover, Hast extends the Chabauty-Kim method to prove finiteness of \(F\)-rational points on curves in essentially all cases where the method previously recovered finiteness of \(\mathbb Q\)-rational points on curves. This includes: integral points on \(\mathbb P^1\) minus at least 3 points, CM elliptic curves minus the origin, curves with a dominant map to a curve with CM Jacobian, and (assuming either the Fontaine-Mazur conjecture or the Bloch-Kato conjecture) all hyperbolic curves.
To prove the main result, Hast first uses the Ax-Schanuel conjecture for variations of mixed Hodge structure (now a theorem of \textit{K. C. T. Chiu} [``Ax-Schanuel for variations of mixed Hodge structures'', Preprint, \url{arXiv:2101.10968}] and of \textit{Z. Gao} and \textit{B. Klingler} [``The Ax-Schanuel conjecture for variations of mixed Hodge structures'', \url{arXiv:2101.10938}]) to deduce functional transcendence properties for the complex analogue of the unipotent Albanese map. Hast then converts the complex functional transcendence results to \(p\)-adic functional transcendence results by a formal argument comparing the power series involved in the complex and \(p\)-adic constructions. For the applications to Chabauty's method and the Chabauty-Kim method, Hast proves a number of inequalities relating dimensions of various cohomology groups and quotients of the pro-unipotent fundamental group of the curve. Many of the applications to the Chabauty-Kim method over number fields were discovered independently and simultaneously by \textit{N. Dogra} [``Unlikely intersections and the Chabauty-Kim method over number fields'', Preprint, \url{arXiv:1903.05032}] by very different methods. Dogra's work uses only \(p\)-adic techniques, as opposed to the complex analytic techniques used in Hast's work.
Reviewer: Nicholas Triantafillou (Athens)On a torsion analogue of the weight-monodromy conjecturehttps://zbmath.org/1487.110632022-07-25T18:03:43.254055Z"Ito, Kazuhiro"https://zbmath.org/authors/?q=ai:ito.kazuhiroSummary: We formulate and study a torsion analogue of the weight-monodromy conjecture for a proper smooth scheme over a non-archimedean local field. We prove it for proper smooth schemes over equal characteristic non-archimedean local fields, abelian varieties, surfaces, varieties uniformized by Drinfeld upper half spaces, and set-theoretic complete intersections in projective smooth toric varieties. In the equal characteristic case, our methods rely on an ultraproduct variant of Weil II established by Cadoret.Hyperelliptic curves and newform coefficientshttps://zbmath.org/1487.110642022-07-25T18:03:43.254055Z"Dembner, Spencer"https://zbmath.org/authors/?q=ai:dembner.spencer"Jain, Vanshika"https://zbmath.org/authors/?q=ai:jain.vanshikaConsider \(f\) a normalized newform of even weigh \(2k\) and level \(N\) with \(q\)-expansion \(f(z)=q+\sum_{n\geq 2} a_f(n)q^n\), such that \(a_f(n)\) are integers. The paper under review study which integers \(\alpha\) could appear at \(a_f(m)\) for some natural \(m\geq 2\). Lehmer conjectured, still unproven, that the weight 12 modular form \(\Delta(z)=\sum_{n\geq 1}\tau(n)q^n\) always satisfy \(\tau(n)\neq0\). Recent work has focused on the variant problem of showing that \(\tau(n)\) never takes certain values, see for example the works of \textit{M. R. Murty} et al. [Bull. Soc. Math. Fr. 115, 391--395 (1987; Zbl 0635.10020)], \textit{J. S. Balakrishnan} et al. in [``Variants of Lehmer's conjecture for Ramanujan's tau-function'', J. Number Theory (to appear)] and \textit{J. S. Balakrishnan} et al. [``Variants of Lehmer's speculation for newforms'', Preprint, \url{arXiv:2005.10354}]. From the work of [loc. cit.], \(a_f(n)\neq\pm\ell^m\) for a fixed odd prime \(\ell\), it is enough to study if \(a_f(p^{d-1})\neq \pm\ell^{m}\), where \(d\) is one of odd primes dividing \(\ell(\ell^2-1)\). For \(d\geq 7\), solutions to \(a_f(p^{d-1})=\pm\ell^m\) give solutions of Thue equations, which are well-studied. For \(d=3\), from the work in [loc. cit.], \(a_f(p^2)=\pm\ell^m\) are related with solutions of the hyperelliptic curves \(Y^2=X^{2k-1}+\pm\ell^m\) (already known when \(\ell^n\leq 100\) [\textit{Y. Bugeaud} et al., Compos. Math. 142, No. 1, 31--62 (2006; Zbl 1128.11013); \textit{C. F. Barros}, On the Lebesgue-Nagel equation and related subjects. University of Warwick (Ph.D. thesis) (2010)]), and for \(d=5\) are related with solutions of \(Y^2=5 X^{2(2k-1)}+4(\pm\ell^m)\). Again, inspired from [\textit{J. S. Balakrishnan} et al., ``Variants of Lehmer's speculation for newforms'', Preprint, \url{arXiv:2005.10354}], these equations for \(d=5\) are studied by Fibonacci type sequences as linear combination of the Fibonacci numbers with the Lucas numbers. In particular, in this paper the authors prove that for all odd primes \(\ell<100\) and all naturals we have \(\tau(n)\neq\pm\ell,\pm 5^m\). For general newforms \(f\) of even weight and level \(N\), the paper under review, obtained results of the flavour of Theorem 1.4 in [\textit{J. S. Balakrishnan} et al., ``Variants of Lehmer's speculation for newforms'', Preprint, \url{arXiv:2005.10354}], by use of modular method and a theorem of \textit{K. A. Ribet} on level-lowering [Int. Mat. Res. Not. 1991, No. 2, 15--19 (1991; Zbl 0728.11029)]. In particular if \(p\nmid a_f(p)\), then \(a_f(p^2)\neq m^j\) for any \(j\geq 4\) dividing \(2k-1\) and any nonzero integer \(m\), in particular, if \(2k\geq 6\), then \(a_f(p^2)\neq m^{2k-1}\). The authors obtains also a statement on \(a_f(p^4)\) under Frey-Mazur conjecture.
Reviewer: Francesc Bars Cortina (Bellaterra)Compatibility of special value conjectures with the functional equation of zeta functionshttps://zbmath.org/1487.110652022-07-25T18:03:43.254055Z"Flach, Matthias"https://zbmath.org/authors/?q=ai:flach.matthias"Morin, Baptiste"https://zbmath.org/authors/?q=ai:morin.baptisteThis article is a continuation of their previous article [Doc. Math. 23, 1425--1560 (2018; Zbl 1404.14024)] where they formulated a conjecture describing the leading Taylor coefficient of the Zeta function \(\zeta(\mathcal{X},s)\) of a proper, regular arithmetic scheme \(\mathcal{X}\).
The aim of the paper under review is to prove their special value conjecture for \(\zeta(\mathcal{X},s)\) is compatible with the functional equation of \(\zeta(\mathcal{X},s)\). Their conjecture needed an inexplicit correction rational factor \(C(\mathcal{X},n)\). The reason why \(C(\mathcal{X},n)\) is inexplicit is that \(C(\mathcal{X},n)\) is defined by the \(p\)-adic Hodge theory so that it fits into the Tamagawa number conjecture.
The authors introduce the tractable rational factor \(C_{\infty}(\mathcal{X},n)\) defined by \[C_{\infty}(\mathcal{X},n):=\prod_{i\leq n-1\leq;j}(n-1-i)!^{(-1)^{i+j}\dim_{\mathbb{Q}}H^i(\mathcal{X}_{\mathbb{Q}},\Omega^i)}\] and they conjecture \(C(\mathcal{X},n)^{-1}=C_{\infty}(\mathcal{X},n)\). In this paper, they prove that their special value conjecture is compatible with the functional equation of \(\zeta(\mathcal{X},s)\) if \(C(\mathcal{X},n)^{-1}\) is replaced by \(C_{\infty}(\mathcal{X},n)\).
Reviewer: Kazuma Morita (Sapporo)Centralized variant of the Li criterion on functions fieldshttps://zbmath.org/1487.110822022-07-25T18:03:43.254055Z"Bllaca, Kajtaz H."https://zbmath.org/authors/?q=ai:bllaca.kajtaz-h"Mazhouda, Kamel"https://zbmath.org/authors/?q=ai:mazhouda.kamelLet \(K\) be a function field over a finite field \(\mathbb F_q\) and \(X\) be its smooth projective curve of genus \(g\) over \(\mathbb F_q\). The zeta function of \(K\) is expressed as \[ Z_K(T)=\frac{L(T)}{(1-T)(1-qT)}, \] where \[ L(T)=\prod_{j=1}^{2g}(1-\alpha_j T)\in\mathbb Z[T] \] is a polynomial of degree \(2g\). We order the indices \(j\in\{1,\ldots,g\}\) so that \(\alpha_{g+j}=\overline{\alpha_j}\) and put \(\theta_j\) as \(\alpha_j=q^{1/2}\exp(i\theta_j)\). The completed zeta function is defined by \[ \xi_K(s)=q^{gs}L(q^{-s}). \] The Voros-Li coefficients \(\lambda_{K,\omega}^V(n)\) are defined by \[ \log\xi_K\left(\frac12+\frac{\sqrt{\omega s}}{1-s}\right) -\log\xi_K(1/2)=\sum_{n=1}^\infty\frac{\lambda_{K,\omega}^V(n)}n s^n. \] Assume \(\xi_K(1/2)\ne0\). The first main theorem in this article asserts the equivalence of GRH for \(\xi_K\) and the inequalities \[ \lambda_{K,\omega}^V(n)\ge0\quad(\text{ for all } n\ge 1) \] for \(0<\omega<4\min_j(\theta_j/\log q)^2\). The authors also introduce another type of Li-type coefficients which they call the Sekatskii-Voros-Li coefficients \(\lambda_{K,l}^{SV}(n)\) defined by \[ \lambda_{K,l}^{SV}(n) =\frac1{(2n-1)!}\frac{d^{2n}}{ds^{2n}}[L_{2n,l}(s)\log\xi_K(s)]_{s=1/2}, \] where \[ L_{2n,l}(s) =4\sum_{k=0}^{n-1}(n-k)A_{k,l,n}(s-1/2)^{2k} \] and \[ A_{k,l,n}=l^{2k-2n}\sum_{m=k}^n\binom {2n}{2m}\binom m k. \] The second main theorem shows that the GRH for \(\xi_K\) is equivalent to the non-negativity \[ \lambda_{K,l}^{SV}(n)\ge0\quad(\text{ for all } n\ge 1) \] with \(l>\max_j\left(\frac{\log q}{\theta_j}\right)\).
As applications they deduce formulas and give observations for the case of \(n=1\) and \(n=2\) by using GRH proved to be true over \(K\).
Reviewer: Shin-ya Koyama (Yokohama)Rubin's conjecture on local units in the anticyclotomic tower at inert primeshttps://zbmath.org/1487.111002022-07-25T18:03:43.254055Z"Burungale, Ashay"https://zbmath.org/authors/?q=ai:burungale.ashay-a"Kobayashi, Shinichi"https://zbmath.org/authors/?q=ai:kobayashi.shinichi"Ota, Kazuto"https://zbmath.org/authors/?q=ai:ota.kazutoLet \(p \geq 5\) be a prime. The authors prove a conjecture of \textit{K. Rubin} [Invent. Math. 88, 405--422 (1987; Zbl 0623.14006)] on the structure of local units along the anticyclotomic \(\mathbb{Z}_p\)-extension of the unramified quadratic extension \(\Phi\) of \(\mathbb{Q}_p\). \par We briefly sketch Rubin's conjecture. Fix a Lubin-Tate formal group \(\mathcal{F}\) over the ring of integers \(\mathcal{O}\) in \(\Phi\) for the uniformizer \(\pi := -p\). Let \(\Phi_n\) be the field obtained from \(\Phi\) by adjoining all \(\pi^{n+1}\)-torsion points of \(\mathcal{F}\) and set \(\Phi_{\infty} := \bigcup_n \Phi_n\). Then there is a natural isomorphism \(\kappa:\mathrm{Gal}(\Phi_{\infty}/\Phi) \simeq \mathcal{O}^{\times}\). Moreover, one has a decomposition
\[
\mathrm{Gal}(\Phi_{\infty}/\Phi_0) \simeq G^+ \times G^-,
\]
where both \(G^{\pm}\) are isomorphic to \(\mathbb{Z}_p\) and \(\mathrm{Gal}(\Phi/\mathbb{Q}_p)\) acts upon \(G^+\) and \(G^-\) by \(+1\) and \(-1\), respectively. We may identify \(G^+\) and \(G^-\) with the Galois groups of the cyclotomic and the anticyclotomic \(\mathbb{Z}_p\)-extension of \(\Phi\), respectively. \par Let \(U_n\) be the group of principal units in \(\Phi_n\) and consider the inverse limit \(\varprojlim_n U_n \otimes_{\mathbb{Z}_p} \mathcal{O}\) with respect to the norm maps. We denote the part of this limit upon which \(\mathrm{Gal}(\Phi_0/\Phi)\) acts via the Teichmüller character by \(U_{\infty}\). Then \(U_{\infty}\) is free of rank \(2\) over the Iwasawa algebra \(\mathcal{O}[[\mathrm{Gal}(\Phi_{\infty}/\Phi_0)]]\). The main object of study is the quotient
\[
V^{\ast}_{\infty} := U_{\infty}^{\ast} / (\sigma-1),
\]
where \(U_{\infty}^{\ast}\) is \(U_{\infty}\) with the Galois action twisted by \(\kappa^{-1}\) and \(\sigma\) is a topological generator of \(G^+\). Then \(V^{\ast}_{\infty}\) is a free \(\Lambda := \mathcal{O}[[G^-]]\)-module of rank \(2\). Rubin defines two subspaces \(V^{\ast, \pm}_{\infty}\) of \(V^{\ast}_{\infty}\) and conjectures that one has a decomposition
\[
V^{\ast}_{\infty} = V^{\ast,+}_{\infty} \oplus V^{\ast,-}_{\infty}.
\]
He showed that both subspaces are free of rank \(1\) and that their intersection is trivial. \par The strategy of proof of Rubin's conjecture is now as follows. Consider the Coates-Wiles derivative
\[
\delta: V^{\ast}_{\infty} / V^{\ast,-}_{\infty} \rightarrow \mathcal{O}.
\]
The authors prove that (i) there is \(\xi \in V^{\ast,+}_{\infty}\) such that \(\delta(\xi) \in \mathcal{O}^{\times}\) and (ii) there is an isomorphism \(V^{\ast}_{\infty}/V^{\ast,-}_{\infty} \simeq \Lambda\). Under this identification, \(\xi\) is not contained in the maximal ideal of \(\Lambda\) (as \(\delta(\xi) \in p \mathcal{O}\) otherwise) and hence is a generator of the quotient \(V^{\ast}_{\infty} / V^{\ast,-}_{\infty}\). Rubin's conjecture follows. \par For (i) the authors construct a certain auxiliary imaginary quadratic field \(K\) and an elliptic curve \(E\) over its Hilbert class field with complex multiplication by \(\mathcal{O}_K\). By a criterion of Rubin it would be sufficient to construct such an \(E\) with good supersingular reduction at \(p\) whose central \(L\)-value is \(p\)-divisible. The authors generalize this approach to allow more general \(L\)-values (they consider twists by certain Hecke characters). The proof of (i) then crucially relies on work of \textit{T. Finis} [Ann. Math. (2) 163, No. 3, 767--807 (2006; Zbl 1111.11047)]. For (ii) the authors make use of the theory of quasi-canonical lifts of \textit{B.H. Gross} [Invent. Math. 84, 321--326 (1986; Zbl 0597.14044)] to construct an optimal system of local points of the formal group. This is the main difficulty of the present work. \par Finally, we note that the main result of this article makes the work of \textit{A. Agboola} and \textit{B. Howard} [Math. Res. Lett. 12, No. 5--6, 611--621 (2005; Zbl 1130.11058)] on a variant of the Iwasawa main conjecture unconditional.
Reviewer: Andreas Nickel (Essen)Galois 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)The density of fibres with a rational point for a fibration over hypersurfaces of low degreehttps://zbmath.org/1487.140582022-07-25T18:03:43.254055Z"Sofos, Efthymios"https://zbmath.org/authors/?q=ai:sofos.efthymios"Visse-Martindale, Erik"https://zbmath.org/authors/?q=ai:visse-martindale.erikSummary: We prove asymptotics for the proportion of fibres with a rational point in a conic bundle fibration. The base of the fibration is a general hypersurface of low degree.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)On \(\ell\)-adic Galois polylogarithms and triple \(\ell\)th power residue symbolshttps://zbmath.org/1487.140712022-07-25T18:03:43.254055Z"Shiraishi, Densuke"https://zbmath.org/authors/?q=ai:shiraishi.densukeFor \(\ell=2\) (resp. \(3\)), let \(\mu_\ell=\{\pm 1\}\) (resp. \(\{1,\zeta_3^{\pm 1}\}\)) denote the set of \(\ell\)-th roots of unity. When three primes \(\mathfrak{p}_i\) (\(i=1,2,3\)) of \(\mathbb{Q}(\mu_\ell)\) have pairwise trivial power residue symbols, \textit{M. Morishita} [Knots and primes. An introduction to arithmetic topology. Based on the Japanese original (Springer, 2009). Berlin: Springer (2012; Zbl 1267.57001)] and \textit{F. Amano} et al. [Res. Number Theory 4, No. 1, Paper No. 7, 29 p. (2018; Zbl 1444.11222)] introduced the triple \(\ell\)-th power residue symbol \([\mathfrak{p}_1,\mathfrak{p}_2,\mathfrak{p}_3]_\ell\in\mu_\ell\) as an analog of the Milnor invariant of three knots in the space that are linked as total but pairwise unlinked. Based on a preceding work by \textit{H. Hirano} and \textit{M. Morishita} [J. Number Theory 198, 211--238 (2019; Zbl 1456.11216)], the paper under review shows a formula that expresses \([\mathfrak{p}_1,\mathfrak{p}_2,\mathfrak{p}_3]_\ell\in\mu_\ell\) with a special value of the \(\ell\)-adic Galois dilogarithmic function studied by \textit{H. Nakamura} and \textit{Z. Wojtkowiak} [Proc. Symp. Pure Math. 70, 285--294 (2002; Zbl 1191.11022)] that enables one to interpret a reciprocity law of type \([\mathfrak{p}_1,\mathfrak{p}_2,\mathfrak{p}_3]_\ell [\mathfrak{p}_2,\mathfrak{p}_1,\mathfrak{p}_3]_\ell=1\) (due to Rédei, Amano-Mizusawa-Morishita) as a consequence of the \(\ell\)-adic dilogarithmic functional equation between \(Li_2(z)\) and \(Li_2(1-z)\) shown in [\textit{H. Nakamura} and \textit{Z. Wojtkowiak}, Lond. Math. Soc. Lect. Note Ser. 393, 258--310 (2012; Zbl 1271.11068)]. In the Appendix, computational examples of \([\mathfrak{p}_1,\mathfrak{p}_2,\mathfrak{p}_3]_3\) for various specific primes \(\mathfrak{p}_i\) (\(i=1,2,3\)) are presented.
Reviewer: Hiroaki Nakamura (Osaka)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)Essential dimension and pro-finite group schemeshttps://zbmath.org/1487.140962022-07-25T18:03:43.254055Z"Bresciani, Giulio"https://zbmath.org/authors/?q=ai:bresciani.giulioSummary: A. Vistoli observed that, if Grothendieck's section conjecture is true and \(X\) is a smooth hyperbolic curve over a field finitely generated over \(\mathbb{Q}\), then \(\underline{\pi}_1(X)\) should somehow have essential dimension 1. We prove that an infinite, pro-finite etale group scheme always has infinite essential dimension. We introduce a variant of essential dimension, the fce dimension, fced \(G\), of a pro-finite group scheme \(G\), which naturally coincides with ed \(G\) if \(G\) is finite, but has a better behaviour in the pro-finite case. Grothendieck's section conjecture implies fced \(\pi_1(X) = \operatorname{dim} X = 1\) for \(X\) as above. We prove that, if \(A\) is an abelian variety over a field finitely generated over \(\mathbb{Q}\), then fced \(\underline{\pi}_1(A) = \mathrm{fced} TA = \operatorname{dim} A\).Solution of the Euler top equations taking into account their time reversibilityhttps://zbmath.org/1487.370712022-07-25T18:03:43.254055Z"Abrarov, D. L."https://zbmath.org/authors/?q=ai:abrarov.d-lSummary: We show that, in the Euler case, the solution of the Euler-Poisson equations can be represented by the normalized exponential of the \(\zeta\)-function of an elliptic curve of a special form over the field \(\mathbb Q\) of rational numbers. This function yields a special form of the general solution of the Euler-Poisson equations in exponentials of \(L\)-functions of elliptic curves over \(\mathbb Q\), which we obtained earlier [Mekh. Tverd. Tela 37, 42--68 (2007; Zbl 1487.70023)]. We compare the solution obtained with the classical solution.Degree gaps for multipliers and the dynamical André-Oort conjecturehttps://zbmath.org/1487.371042022-07-25T18:03:43.254055Z"Ingram, Patrick"https://zbmath.org/authors/?q=ai:ingram.patrickGiven \(f(z)\in \mathbb{C}[z]\), we say that \(f\) is PCF (post-critically finite) if the orbit of every critical point is finite. \textit{M. Baker} and \textit{L. De Marco} [Forum Math. Pi 1, Paper No. e3, 35 p. (2013; Zbl 1320.37022)] proved that the family of cubic polynomials with a fixed point of fixed multiplier \(\lambda\) has infinitely many PCF polynomials if and only if \(\lambda=0\). This result was then generalized to cubic polynomials with a periodic point of fixed multiplier and to quadratic rational maps. Here the author studies the same problem for families of polynomials parametrized on a curve. He proves that, if \(f_t\) is a family of polynomials parametrized over a curve \(X\) with a marked periodic point \(P\) of multiplier \(\lambda_f(P)\in\mathbb{C}(X)\) and \(f_t\) is PCF for infinitely many \(t\in \mathbb{C}(X)\), then \(\lambda_f(P)=0\) identically on \(X\), assuming that \(\deg(\lambda_f(P))\) is strictly smaller than the critical height of \(f\). He also shows that, if \(\lambda_f(P)=0\) identically on \(X\), then \(f_t\) can be PCF only for finitely many \(t\in \mathbb{C}(X)\).
Reviewer: Matteo Verzobio (Pisa)