Recent zbMATH articles in MSC 11Rhttps://zbmath.org/atom/cc/11R2021-04-16T16:22:00+00:00WerkzeugGlobally analytic \(p\)-adic representations of the pro-\(p\) Iwahori subgroup of \(\mathrm{GL}(2)\) and base change. II: A Steinberg tensor product theorem.https://zbmath.org/1456.112102021-04-16T16:22:00+00:00"Clozel, Laurent"https://zbmath.org/authors/?q=ai:clozel.laurentThe current work is the second part of author's paper, the first part of which is [Bull. Iran. Math. Soc. 43, No. 4, 55--76 (2017; Zbl 1423.11187)]. The first part of this paper is devoted to the study of Iwasawa algebra of the pro-\(p\) Iwahori subgroup of GL\((2, L)\) for an unramified extension \(L\) of degree \(r\) of \(\mathbb{Q}_p\) and gave a presentation of it by generators and relations, imitating [Doc. Math. 16, 545--559 (2011; Zbl 1263.22011)]. A natural base change map then appears that, however, is well defined only for the globally analytic distributions on the groups, seen as rigid-analytic spaces. In Section 1 of [Bull. Iran. Math. Soc. 43, No. 4, 55--76 (2017; Zbl 1423.11187)], the author stated that this should be related to a construction of base change for representations of these groups, similar to \textit{R. Steinberg}'s tensor product theorem [Nagoya Math. J. 22, 33--56 (1963; Zbl 0271.20019)] for algebraic groups over finite fields.
In this paper under review, the author gives such a construction, and show that it is compatible with the (\(p\)-adic) Langlands correspondence in the case of the principal series for GL\((2)\). He exploits the base change map for globally analytic distributions constructed there, relating distributions on the pro-\(p\) Iwahori subgroup of GL\((2)\) over \(\mathbb{Q}_p\) and those on the pro-\(p\) Iwahori subgroup of GL\((2, L)\) where \(L\) is an unramified extension of \(\mathbb{Q}_p\). This is used to obtain a functor, the `Steinberg tensor product', relating globally analytic \(p\)-adic representations of these two groups. We are led to extend the theory, sketched by \textit{M. Emerton} [Locally analytic vectors in representations of locally \(p\)-adic analytic groups. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1430.22020)], of these globally analytic representations. In the last section, the author showed that this functor exhibits, for principal series, Langlands' base change (at least for the restrictions of these representations to the pro-\(p\) Iwahori subgroups.)
For the entire collection see [Zbl 1401.20003].
Reviewer: Wei Feng (Beijing)Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves.https://zbmath.org/1456.111012021-04-16T16:22:00+00:00"Bhargava, M."https://zbmath.org/authors/?q=ai:bhargava.manjul"Shankar, A."https://zbmath.org/authors/?q=ai:shankar.arul|shankar.ananth-n"Taniguchi, T."https://zbmath.org/authors/?q=ai:taniguchi.takashi.1"Thorne, F."https://zbmath.org/authors/?q=ai:thorne.frank"Tsimerman, J."https://zbmath.org/authors/?q=ai:tsimerman.jacob"Zhao, Y."https://zbmath.org/authors/?q=ai:zhao.yueqing|zhao.yude|zhao.yongan|zhao.yingmin|zhao.yanxing|zhao.yongling|zhao.yannan|zhao.yanliang|zhao.yongqiang|zhao.yuxiao|zhao.yanying|zhao.yaru|zhao.yunping|zhao.youjie|zhao.yane|zhao.yaowu|zhao.yongwang|zhao.yaqin|zhao.yuesheng|zhao.yiyuan|zhao.yibin|zhao.yixing|zhao.yu|zhao.yingshuai|zhao.yongxiang|zhao.yongchen|zhao.yafan|zhao.yuhua|zhao.yanxiang|zhao.yunpeng|zhao.yonglong|zhao.yuyuan|zhao.yongchun|zhao.yuqi|zhao.yuejing|zhao.yuanjun|zhao.yajuan|zhao.yanzhu|zhao.yuzhe|zhao.yongda|zhao.yiwen|zhao.yidong|zhao.yanqi|zhao.yanyong|zhao.yongcai|zhao.yuqing|zhao.yuzhou|zhao.yanyun|zhao.yuane|zhao.yongzhi|zhao.yaming|zhao.yuchun|zhao.yapu|zhao.yongkai|zhao.yingqi|zhao.yaobing|zhao.yihong|zhao.yihui|zhao.yanda|zhao.yougang|zhao.yize|zhao.yaomin|zhao.yuling|zhao.yan|zhao.yili|zhao.yunlong|zhao.yixuan|zhao.yanfang|zhao.ye|zhao.yanjia|zhao.yani|zhao.yanxia|zhao.yongxia|zhao.yunsong|zhao.yizhen|zhao.yanmeng|zhao.yuansong|zhao.yibing|zhao.yinlong|zhao.yijin|zhao.yicai|zhao.yuelong|zhao.yajun|zhao.yuli|zhao.yanxin|zhao.youhui|zhao.yingfeng|zhao.yanju|zhao.yiqiang|zhao.younan|zhao.yaqing|zhao.yongbo|zhao.yongkang|zhao.yuxin|zhao.yuzhuang|zhao.yuhuai|zhao.yonghui|zhao.yanli|zhao.yangyang|zhao.yufan|zhao.yingying|zhao.yifei|zhao.yancai|zhao.yilin|zhao.yulin|zhao.yuhan|zhao.yanjuan|zhao.yang|zhao.yaping|zhao.yongchao|zhao.youqun|zhao.yudong|zhao.yuanxiang|zhao.yinghai|zhao.yongzhe|zhao.yunbo|zhao.yuefei|zhao.yizhi|zhao.yuwen|zhao.yimin|zhao.yongxin|zhao.yuliang|zhao.yadong|zhao.yianhe|zhao.yigong|zhao.yong|zhao.yuzhang|zhao.yongqian|zhao.yueyuan|zhao.yishu|zhao.yaozong|zhao.yuying|zhao.yuqiu|zhao.ying|zhao.yuhai|zhao.yanchun|zhao.you|zhao.yongyi|zhao.yingliang|zhao.yonghua|zhao.yicheng|zhao.yuehua|zhao.yarlwen|zhao.yuxiang|zhao.yingxue|zhao.youxuan|zhao.yinglu|zhao.yijia|zhao.yumeng|zhao.yunmei|zhao.yongdong|zhao.yihan|zhao.yongping|zhao.yinchao|zhao.yongtao|zhao.yuzhong|zhao.yuntao|zhao.yitian|zhao.yangsheng|zhao.yuanlu|zhao.yingzi|zhao.yanyu|zhao.yanguang|zhao.yinghui|zhao.yunfan|zhao.yumin|zhao.yanan|zhao.yucan|zhao.yunhong|zhao.yongchang|zhao.yanwei|zhao.yanjie|zhao.yanhua|zhao.yanqing|zhao.yehua|zhao.yukun|zhao.yuandi|zhao.yingtao|zhao.yaowen|zhao.yiming|zhao.yiwu|zhao.yanping|zhao.yuanying|zhao.yunge|zhao.yongjie|zhao.yanbin|zhao.yanchang|zhao.yaohong|zhao.yajing|zhao.yunwei|zhao.yongshen|zhao.yuchen|zhao.yanwen|zhao.yanhong|zhao.yonghong|zhao.yingchao|zhao.yuemin|zhao.yanqin|zhao.yeye|zhao.yayun|zhao.yueyu|zhao.yueqiang|zhao.yifen|zhao.yushu|zhao.yuanhe|zhao.yuejen|zhao.yizheng|zhao.yadi|zhao.yunyuan|zhao.yuna|zhao.youjian|zhao.yile|zhao.yuxia|zhao.yanyan|zhao.yunzhuan|zhao.yuexu|zhao.yian|zhao.yipeng|zhao.ya|zhao.yibao|zhao.yongjuan|zhao.yanfen|zhao.yanlu|zhao.yingbo|zhao.yuwei|zhao.yingnan|zhao.yinglin|zhao.yunfeng|zhao.yexi|zhao.yunxin|zhao.yalun|zhao.yingxiu|zhao.yanfeng|zhao.yurong|zhao.yunlei|zhao.yuhang|zhao.yanhui|zhao.yisi|zhao.yuning|zhao.yuanyuan|zhao.yigeng|zhao.yubo|zhao.yongwei|zhao.youxing|zhao.yibo|zhao.yuqin|zhao.yuge|zhao.yecheng|zhao.yuyun|zhao.yongfang|zhao.yongcheng|zhao.yueling|zhao.yufei|zhao.yiyi|zhao.yanjun|zhao.yinglei|zhao.yuanshan|zhao.yuchao|zhao.yuanzhang|zhao.yanmin|zhao.yaling|zhao.yangzhang|zhao.yijun|zhao.yuandong|zhao.yijiu|zhao.yanbo|zhao.yao|zhao.yusheng|zhao.yingdong|zhao.yueying|zhao.yingxin|zhao.yafei|zhao.yun|zhao.yunsheng|zhao.yumei|zhao.yuefeng|zhao.yunjie|zhao.yi|zhao.yanlei|zhao.yanzheng|zhao.yongshun|zhao.yage|zhao.yating|zhao.yeqing|zhao.yi.1|zhao.yujie|zhao.yanfei|zhao.yuoli|zhao.yanling|zhao.yuzhen|zhao.yinchuan|zhao.yongye|zhao.yuhuan|zhao.yuping|zhao.yufu|zhao.yinshan|zhao.yaonan|zhao.yige|zhao.yufang|zhao.yujuan|zhao.yinan|zhao.yufeng|zhao.yunfei|zhao.yazhou|zhao.yuda|zhao.yanzhong|zhao.yaxi|zhao.youyi|zhao.yunbin|zhao.yinghong|zhao.yunhe|zhao.yagu|zhao.yueqin|zhao.yichun|zhao.yaqun|zhao.yushan|zhao.yaohua|zhao.yiming.1|zhao.yakun|zhao.yongchi|zhao.yanlong|zhao.yongrui|zhao.yue|zhao.yongliang|zhao.yinyu|zhao.yusong|zhao.yanfa|zhao.yinliang|zhao.yingjie|zhao.yueru|zhao.yuming|zhao.yuhong|zhao.yongjun|zhao.yiping|zhao.yongyao|zhao.yonggan|zhao.yupeng|zhao.yin|zhao.yanru|zhao.yifan|zhao.yanting|zhao.yaoqing|zhao.yuhui|zhao.yahong|zhao.yulong|zhao.yanlin|zhao.yanbing|zhao.yuting|zhao.yucheng|zhao.yongsheng|zhao.yihua|zhao.yichuan|zhao.yingcui|zhao.yingchun|zhao.yunwang|zhao.yonggang|zhao.yaoshun|zhao.yuee|zhao.yujia|zhao.yunliang|zhao.yuzhuo|zhao.yuejuan|zhao.youjun|zhao.yixin|zhao.yuan|zhao.yandong|zhao.yuqianGiven a number field \(K\) of degree \(n\) and any positive integer \(m\), it is conjectured that \(h_m(K) = O_{\varepsilon, n, m}(|\mathrm{Disc}(K)|^{\varepsilon})\), where \(h_m(K)\) denotes the size of the \(m\)-torsion subgroup of the class group of \(K\), and \(\mathrm{Disc}(K)\) is the discriminant of \(K\). There are several works concerning with finding upper bounds for \(h_m(K)\) in the literature, say [\textit{J. S. Ellenberg} and \textit{A. Venkatesh}, Int. Math. Res. Not. 2007, No. 1, Article ID rnm002, 18 p. (2007; Zbl 1130.11060); \textit{H. A. Helfgott} and \textit{A. Venkatesh}, J. Am. Math. Soc. 19, No. 3, 527--550 (2006; Zbl 1127.14029); \textit{E. Landau}, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1918, 79--97 (1918; JFM 46.0266.02); \textit{L. B. Pierce}, Forum Math. 18, No. 4, 677--698 (2006; Zbl 1138.11049)].
In the paper under review, the authors develop a uniform method to provide nontrivial bounds for the case \(m=2\) and all degrees \(n \geq 3\). Indeed, in Theorem 1.1, the authors prove that \[h_2(K) < O_\varepsilon (|\mathrm{Disc}(K)|)^{\frac{1}{2} - \delta_n + \epsilon}\] with \(\delta_n= .2215 \cdots,\) for \(n=3, 4\) and \(\delta_n = 1/2n\) for \(n\geq 5\).
Using their result in the case \(n=3\), in Theorem 1.2, they give upper bounds on the size of \(2\)-Selmer groups, ranks, and the number of integral points on the elliptic curves (in terms of their discriminants) improving the known ones in [\textit{A. Brumer} and \textit{K. Kramer}, Duke Math. J. 44, 715--743 (1977; Zbl 0376.14011); \textit{H. A. Helfgott} and \textit{A. Venkatesh}, J. Am. Math. Soc. 19, No. 3, 527--550 (2006; Zbl 1127.14029)]. Considering their results in the case \(n>4\), in Theorem 1.3, they obtain upper bounds on the size of the \(2\)-Selmer groups, ranks of the Jacobian of the hyperelliptic curves \(C: y^2=f(x)\) over \(\mathbb Q\) in terms of \(|\mathrm{Disc}(K)|\), where \(f\) is a separable polynomial of degree \(n\) and \(K\) is the étale \(\mathbb Q\)-algebra \({\mathbb Q}[x]/f(x)\), which improve the previous result in [\textit{A. Brumer} and \textit{K. Kramer}, Duke Math. J. 44, 715--743 (1977; Zbl 0376.14011)]. As an other consequence of the case \(n=3\) of Theorem 1.1, the authors obtain an upper bound for the number of isomorphism classes of quartic fields having Galois group \(A_4\) and discriminant less than a given \(X>0\). Their results improve those of the previous works [\textit{A. M. Baily}, J. Reine Angew. Math. 315, 190--210 (1980; Zbl 0421.12007); \textit{S. Wong}, Proc. Am. Math. Soc. 133, No. 10, 2873--2881 (2005; Zbl 1106.11041)]. The key tools for proving Theorem 1.1 are given in Theorems 1.5 and 1.6.
Finally, in Theorem 1.7, the authors prove an analogous of Theorems 1.5 and 1.6 to the case of the function fields to obtain a nontrivial upper bound on the \(2\)-torsion points in \(\mathrm{Pic}^0(C)(k)\), where \(C\) ia an algebraic curve of genus \(g\) on a finite field \(k\), in terms of \(g\) and the cardinal number of \(k\).
Reviewer: Sajad Salami (Rio de Janeiro)Universal quadratic forms and indecomposables over biquadratic fields.https://zbmath.org/1456.110362021-04-16T16:22:00+00:00"Čech, Martin"https://zbmath.org/authors/?q=ai:cech.martin"Lachman, Dominik"https://zbmath.org/authors/?q=ai:lachman.dominik"Svoboda, Josef"https://zbmath.org/authors/?q=ai:svoboda.josef"Tinková, Magdaléna"https://zbmath.org/authors/?q=ai:tinkova.magdalena"Zemková, Kristýna"https://zbmath.org/authors/?q=ai:zemkova.kristynaThe paper has two explicit results: A totally positive definite universal classic quadratic form over \(\mathbb{Q}(\sqrt{2}, \sqrt{3})\) (resp. \(\mathbb{Q}(\sqrt{6}, \sqrt{19})\)) must have at least 5 variables (resp. 6 variables). The basis for the proof is the study of additively indecomposable algebraic integers \(O_K\) of biquadratic number fields \(K\) and of universal totally positive quadratic forms with coefficients in \(O_K\). The authors give sufficient conditions for an indecomposable element of a quadratic subfield to remain indecomposable in the biquadratic number field \(K\).
Reviewer: Meinhard Peters (Münster)An analogue of a formula for Chebotarev densities.https://zbmath.org/1456.111762021-04-16T16:22:00+00:00"Wang, Biao"https://zbmath.org/authors/?q=ai:wang.biaoCyclic symmetry on complex tori and Bagnera-De Franchis manifolds.https://zbmath.org/1456.320072021-04-16T16:22:00+00:00"Catanese, Fabrizio"https://zbmath.org/authors/?q=ai:catanese.fabrizioSummary: We describe the possible linear actions of a cyclic group \(G=\mathbb{Z}/n\) on a complex torus, using the cyclotomic exact sequence for the group algebra \(\mathbb{Z}[G]\). The main application is devoted to a structure theorem for Bagnera-De Franchis Manifolds (these are the quotients of a complex torus by a free action of a cyclic group), but we also give an application to hypergeometric integrals, namely, we describe the intersection product and Hodge structures for the homology of fully ramified cyclic coverings of the projective line.
For the entire collection see [Zbl 07237934].Multi-point codes from the GGS curves.https://zbmath.org/1456.941362021-04-16T16:22:00+00:00"Hu, Chuangqiang"https://zbmath.org/authors/?q=ai:hu.chuangqiang"Yang, Shudi"https://zbmath.org/authors/?q=ai:yang.shudiAlgebraic curves over finite fields can be used to obtain error correcting codes since the seminal work of Goppa in the early 1980s. Algebraic geometric (AG) codes have ``good'' parameters when the underlying curve has many rational points with respect to its genus. For this reason, maximal curves (i.e. curves attaining the upper bound in the Hasse-Weil bound) have been widely investigated. Recently, AG codes from Hermitian, Suzuki, Klein quartic, GK, and GGS curves and their quotients attracted a lot of attention. Most of the constructions of AG codes are one-point. In the case of multi-point AG codes, the main problem is a suitable description of Riemann-Roch spaces associated with divisors having a large support.
This paper deals with the construction of AG codes defined from GGS curves, a generalization of the GK curve. In particular, the authors describe bases for the Riemann-Roch spaces associated with some rational places, and characterize explicitly the Weierstrass semigroups and pure gaps (a generalization of gaps) by an exhaustive computation for the basis of Riemann-Roch spaces from GGS curves. As a byproduct, multi-point codes with parameters achieving new records are obtained.
Reviewer: Daniele Bartoli (Perugia)A note on trace of powers of algebraic numbers.https://zbmath.org/1456.111322021-04-16T16:22:00+00:00"Philippon, Patrice"https://zbmath.org/authors/?q=ai:philippon.patrice"Rath, Purusottam"https://zbmath.org/authors/?q=ai:rath.purusottamThe paper under review deals with the problem of detecting algebraic integers amongst algebraic numbers, by looking at the traces of their powers. These questions were studied by \textit{G. Pólya} [Rend. Circ. Mat. Palermo 40, 1--16 (1915; JFM 45.0655.02)] and \textit{B. de Smit} [J. Number Theory 45, No. 1, 112--116 (1993; Zbl 0782.11027)], who showed that if \(\alpha\) is an algebraic number and \(\mathrm{Tr}_{\mathbb{Q}(\alpha)/\mathbb{Q}}(\alpha^i) \in \mathbb{Z}\) for some finite, explicit sequence of integers \(i\) (depending on the degree of \(\alpha\)), then \(\alpha\) is an algebraic integer. As a corollary, one sees that \(\alpha\) is an algebraic integer as soon as \(\mathrm{Tr}_{\mathbb{Q}(\alpha)/\mathbb{Q}}(\alpha^i) \in \mathbb{Z}\) for all but finitely many \(i \in \mathbb{N}\).
This last assertion is generalised by Theorem 2 of the paper under review, which shows that an algebraic number \(\alpha\) is an algebraic integer as soon as, for any fixed algebraic number \(\lambda\), the trace \(\mathrm{Tr}_{\mathbb{Q}(\alpha, \lambda)/\mathbb{Q}}(\lambda \alpha^i)\) is integral and non-zero for all but finitely many \(i \in \mathbb{N}\). It is necessary that these traces are non-zero, as the examples \(\alpha = 1/\sqrt{2}\) or \(\alpha = (1 + \sqrt{5})/\sqrt{2}\) show.
The rest of the paper under review (see in particular Theorem 5 and Theorem 12) aims to study pairs of non-zero algebraic numbers \(\alpha, \lambda\) such that \(\mathrm{Tr}_{F/K}(\lambda \alpha^i) = 0\) for infinitely many \(i \in \mathbb{N}\), where \(K \subseteq F\) are two number fields such that \(\mathbb{Q}(\alpha,\lambda) \subseteq F\). We mention in particular that Corollary 6 shows that \(\mathrm{Tr}_{\mathbb{Q}(\alpha)/\mathbb{Q}}(\alpha^i) = 0\) for infinitely many \(i \in \mathbb{N}\) if and only if \(\alpha \not\in \mathbb{Q}(\alpha^{h})\), where \(h\) denotes the order of the torsion part of the Galois group of \(\alpha\) (that is, the Galois group of the splitting field of the minimal polynomial of \(\alpha\)). Moreover, Corollary 13 proves the analogous result with the extension \(\mathbb{Q} \subseteq \mathbb{Q}(\alpha)\) replaced by \(\mathbb{Q}(\zeta) \subseteq \mathbb{Q}(\alpha,\zeta)\), where \(\zeta\) is a primitive root of unity of order \(h\).
Perhaps surprisingly, the proofs of the paper under review use some deep results from Diophantine approximation, such as the theorem of Skolem, Mahler and Lech (see the work of \textit{G. Hansel} [Theor. Comput. Sci. 43, 91--98 (1986; Zbl 0605.10007)] for an elementary proof). Such a theorem does not have a literal analogue for function fields of positive characteristic (see however the work of \textit{H. Derksen} [Invent. Math. 168, No. 1, 175--224 (2007; Zbl 1205.11030)] for a possible analogue). The last section of the paper under review shows that also some of the aforementioned results do not have a literal analogue over function fields of positive characteristic.
Reviewer: Riccardo Pengo (Lyon)The irreducibility of some Wronskian Hermite polynomials.https://zbmath.org/1456.112062021-04-16T16:22:00+00:00"Grosu, Codruţ"https://zbmath.org/authors/?q=ai:grosu.codrut"Grosu, Corina"https://zbmath.org/authors/?q=ai:grosu.corinaSummary: We study the irreducibility in \(\mathbb{Z}[x]\) of Wronskian Hermite polynomials labelled by partitions. It is known that these polynomials factor as a power of \(x\) times a remainder polynomial. We show that the remainder polynomial is irreducible for the partitions \((n,m)\) with \(m\leq 2\), and \((n,n)\) when \(n+1\) is a square.
Our main tools are two theorems that we prove for all partitions. The first result gives a sharp upper bound for the slope of the edges of the Newton polygon for the remainder polynomial. The second result is a Schur-type congruence for Wronskian Hermite polynomials.
We also explain how irreducibility determines the number of real zeros of Wronskian Hermite polynomials, and prove Veselov's conjecture for partitions of the form \((n,k,k-1,\ldots,1)\).On the local behavior of specializations of function field extensions.https://zbmath.org/1456.112202021-04-16T16:22:00+00:00"König, Joachim"https://zbmath.org/authors/?q=ai:konig.joachim"Legrand, François"https://zbmath.org/authors/?q=ai:legrand.francois"Neftin, Danny"https://zbmath.org/authors/?q=ai:neftin.dannyThe aim of the paper is to study the local behavior at primes of a field \(k\) of characteristic \(0\), of a finite Galois extension of \(k\) arising as specializations of finite Galois extensions \(E/k(T)\), where \(E/k\) is regular, at points \(t_0\in{\mathbb P}_1(k)\). The Grunwald problem is the following. Let \(k\) be a number field. Given a finite set \(S\) of primes of \(k\), and given finite Galois extensions \(L^{(\mathfrak p)}/k_{\mathfrak p}\), \(\mathfrak p\in S\), the Grunwald problem \((G,(L^{(\mathfrak p)}/k_{\mathfrak p})_{\mathfrak p\in S})\) asks whether there exists a \(G\)-extension \(L/k\) whose completion \(L_{\mathfrak p}\) at \(\mathfrak p\) is \(k_{\mathfrak p}\)-isomorphic to \(L^{(\mathfrak p)}\) for all \(\mathfrak p\in S\). There are several examples of Grunwald problems with no solution. One important technique for realizations of non-solvable groups \(G\) over \(k\) is via \(k\)-regular \(G\)-extensions, that is, by using Hilbert's irreducible theorem that establishes that every non-trivial \(k\)-regular \(G\)-extension \(E/k(T)\) has infinitely many linearly disjoint specializations \(E_{t_0}/k\), \(t_0\in {\mathbb P}^1(k)\) with Galois group \(G\). This specialization process provides a natural way to study Grunwald problems for finite groups \(G\) admitting \(k\)-regular \(G\)-extensions of \(k(T)\).
The first main result is the following. Let \(E_{t_0}/k\) be a specialization of \(E/k(T)\), \(t_0\in {\mathbb P}^1(k)\). There exists a finite set \(S_{\text{exc}}\) of primes of \(k\) such that for \(\mathfrak p\notin S_{\text{exc}}\) and for a given branch point \(t_i\) which is \(k\)-rational and such that \(t_i\) and \(t_0\) meet modulo \(\mathfrak p\), and that the exponent \(a_{\mathfrak p}:=v_{\mathfrak p}(t_0-t_i)\) is coprime to \(|I_{t_i}|\), where \(I_{t_i}\) is the inertia group, the decomposition group of \(E_{t_0}/k\) at \(\mathfrak p\) is conjugate by an element of \(G\) to \(\varphi_i^{-1}(D_{t_i,\mathfrak p})\), where \(\varphi_i\colon D_{t_i}\longrightarrow D_{t_i}/I_{t_i}\) is the natural map. That is, the decomposition group of \(E_{t_0}/k\) at \(\mathfrak p\) is determined by the local data \((\varphi_i,D_{i,\mathfrak p})\) at \(t_i\), when \(t_0\) and \(t_i\) meet modulo \(\mathfrak p\).
The second main result is that the only constraint on completions of \(E_{t_0}/k\) at \(\mathfrak p\) for such specialization points \(t_0\) is that \(t_i\) and \(t_0\) meet modulo \(\mathfrak p\). More precisely, suppose that \(S\) is a finite set of primes of \(k\) disjoint from some finite set of primes \(S'_{\text{exc}}\). For each \(\mathfrak p\in S\), fix a \(k\)-rational branch point \(t_{i(\mathfrak p)}\) of \(E/k(T)\) and a finite extension \(L^{(\mathfrak p)}/k_{\mathfrak p}\) with Galois group \(\varphi_{i(\mathfrak p)}^{-1}(D_{t_i(\mathfrak p),\mathfrak p})\) and inertia group \(I_{t_i(\mathfrak p)}\). Then
there exists \(t_0\in k\) such that \(E_{t_0}/k\) is a solution to the Grunwald problem \((G,(L^{(\mathfrak p)}/k_{\mathfrak p})_{\mathfrak p\in S})\).
The principal tool for the proof of the main results is to consider the fraction field \(F\) of \(R_{\mathfrak p}[[T-t_i]]\), where \(R\) is a Dedekind domain of characteristic \(0\) and to use a result of Eisenstein to show that, for all but finitely many primes \(\mathfrak p\) of \(k\), the group \(\text{Gal}(EF/F)\) is determined by the local data \((\varphi_i,D_{t_i,
\mathfrak p})\) at \(t_i\).
Among the applications of the main results are the study of crossed product division algebras, the Hilbert-Grunwald property and finite parametric sets. An applications for crossed products is that given a branch point \(t_i\) of \(E/k(T)\), there exist infinitely many primes \(\mathfrak p\) of \(k\), and, for each such prime \(\mathfrak p\), infinitely many \(t_0\in k\) such that \(E_{t_0}/k\) is a \(G\)-extension with decomposition group \(\langle I_{t_i},\tau\rangle\) at \(\mathfrak p\), where \(D_{t_i}\) and \(I_{t_i}\) denote the decomposition and the inertia groups at \(t_i\) respectively and \(\tau\in D_{t_i}\). With respect to the \(G\)-crossed products, it is proved that if \(p\) is a prime \(\equiv 3\) or \(5\bmod 8\), then there exists a \(\text{
PSL}_2({\mathbb F}_p)\) crossed product division algebra with center \({\mathbb Q}\). It is also proved that if \(G\) has a non--cyclic abelian subgroup, then no \(k\)-regular \(G\)-extension of \(k(T)\) has the Hilbert-Grunwald property.
Reviewer: Gabriel D. Villa Salvador (Ciudad de México)A class number calculation of the \(5^{\mathrm{th}}\) layer of the cyclotomic \(\mathbb{Z}_2\)-extension of \(\mathbb{Q}(\sqrt{5})\).https://zbmath.org/1456.112152021-04-16T16:22:00+00:00"Aoki, Takuya"https://zbmath.org/authors/?q=ai:aoki.takuyaSummary: For a positive integer \(n\), let \(K_n\) be the \(n\)-th layer of the the cyclotomic \(\mathbb{Z}_2\)-extension of \(\mathbb{Q}(\sqrt{5})\), which is the real quadratic field with the minimal discriminant. We prove that the class number of \(K_5\) is 1.Computation of the fundamental units of number rings using a generalized continued fraction.https://zbmath.org/1456.111262021-04-16T16:22:00+00:00"Bruno, A. D."https://zbmath.org/authors/?q=ai:bruno.alexander-dSummary: A global generalization of continued fraction is proposed. It is based on computer algebra and can be used to find the best Diophantine approximations. This generalization provides a basis for computing the fundamental units of algebraic rings and for finding all solutions of a class of Diophantine equations. Examples in dimensions two, three, and four are given.On the arithmetic of modified idèle class groups.https://zbmath.org/1456.112112021-04-16T16:22:00+00:00"Lee, Wan"https://zbmath.org/authors/?q=ai:lee.wan"Seo, Soogil"https://zbmath.org/authors/?q=ai:seo.soogilOn congruence half-factorial Krull monoids with cyclic class group.https://zbmath.org/1456.200152021-04-16T16:22:00+00:00"Plagne, Alain"https://zbmath.org/authors/?q=ai:plagne.alain"Schmid, Wolfgang A."https://zbmath.org/authors/?q=ai:schmid.wolfgang-alexanderThe authors carry out a detailed investigation of congruence half-factorial Krull monoids of various orders with finite cyclic class group and related problems. Specifically, they determine precisely all relatively large values that can occur as a minimal distance of a Krull monoid with finite cyclic class group, as well as the exact distribution of prime divisors over the ideal classes in these cases. Their results apply to various classical objects, including maximal orders and certain semigroups of modules. In addition, they present applications to quantitative problems in factorization theory. More specifically, they determine exponents in the asymptotic formulas for the number of algebraic integers whose sets of lengths have a large difference.
Reviewer: C. P. Anil Kumar (Chennai)The Lind-Lehmer constant for \(\mathbb{Z}_2^r\times\mathbb{Z}_4^s\).https://zbmath.org/1456.112042021-04-16T16:22:00+00:00"Mossinghoff, Michael J."https://zbmath.org/authors/?q=ai:mossinghoff.michael-j"Pigno, Vincent"https://zbmath.org/authors/?q=ai:pigno.vincent"Pinner, Christopher"https://zbmath.org/authors/?q=ai:pinner.christopher-gFor a finite abelian group \(G=\mathbb{Z}_{n_{1}}\times \cdots \times \mathbb{Z}_{n_{k}}\), where \(\mathbb{Z}_{n_{j}}\) \((1\leq j\leq k)\) denotes the cyclic group with order \(n_{j}\), define
\[
\lambda (G)=\min \left( \left\{ \prod_{j_{1}=1}^{n_{1}}\dots\prod_{j_{k}=1}^{n_{k}}\left\vert F(e^{i2\pi j_{1}/n_{1}},\dots,e^{i2\pi j_{k}/n_{k}})\right\vert \mid F\in \mathbb{Z}[x_{1},\dots,x_{k}]\right\} \cap \lbrack 2,\infty )\right) .
\]
According to [\textit{D. Lind} et al., Proc. Am. Math. Soc. 133, No. 5, 1411--1416 (2005; Zbl 1056.43005); \textit{D. Desilva} and \textit{C. Pinner}, Proc. Am. Math. Soc. 142, No. 6, 1935--1941 (2014; Zbl 1294.11185)], if \(G\neq \mathbb{Z}_{2}\), then
\[
\lambda (G)\leq \operatorname{card}(G)-1, \tag{*}
\]
\((\lambda (\mathbb{Z}_{p^{n}}),\lambda (\mathbb{Z}_{2^{n}}))=(2,3)\) for any natural number \(n\) and any odd prime \(p\), and (*) is sharp when \(G=\mathbb{Z}_{3}^{n}\), or when \(G=\mathbb{Z}_{2}^{n}\) and \(n\geq 2\).
In the paper under review, the authors continue to investigate the values of \(\lambda (G)\) for \(G\) running through certain families of \(p\)-groups, where \(p\in \{2,3\}\). Mainly, they show that \(\lambda (\mathbb{Z}_{3}\times \mathbb{Z}_{3^{n}})=8\), \(n\geq 3\Rightarrow \lambda (\mathbb{Z}_{2}\times \mathbb{Z}_{2^{n}})=9\), and equality occurs (again) in (*) whenever \(G\neq \mathbb{Z}_{2}\) and the factors of \(G\) are all \(\mathbb{Z}_{2}\) or \(\mathbb{Z}_{4}\).
The proofs of these results are based on a generalization of Lemma 2.1 of the last mentioned reference about a congruence satisfied by the rational integers defining \(\lambda (G)\), when \(G\) is a \(p\)-group.
Reviewer: Toufik Zaïmi (Riyadh)Arithmetic topology in Ihara theory. II: Milnor invariants, dilogarithmic Heisenberg coverings and triple power residue symbols.https://zbmath.org/1456.112162021-04-16T16:22:00+00:00"Hirano, Hikaru"https://zbmath.org/authors/?q=ai:hirano.hikaru"Morishita, Masanori"https://zbmath.org/authors/?q=ai:morishita.masanoriIn this paper, a wide class of triple quadratic (resp., cubic) residue symbols \([p_1,p_2,p_3]\) of primes \(p_i\) (\(i=1,2,3\)
in \(\mathbb{Q}\) (resp., \(\mathbb{Q}(\sqrt{-3})\)) is connected to the mod \(\ell\) Milnor invariants introduced in previous work [\textit{H. Kodani} et al., Publ. Res. Inst. Math. Sci. 53, No. 4, 629--688 (2017; Zbl 1430.11082)] as certain coefficients of Magnus series of Frobenius elements arising from Ihara theory on Galois representations in the pro-\(\ell\) fundamental groups of punctured projective lines. Dilogarithmic mod \(\ell\) Heisenberg ramified covering \(D(\ell)\) of \(\mathbb{P}^1\) plays a central role ``as a higher analog of the dilogarithmic function for the gerbe associated to the mod \(\ell\) Heisenberg group''.
The monodromy transformations of certain functions on \(D(\ell)\) along the pro-\(\ell\) longitudes of Frobenius elements
turn out to capture the aimed power residue symbols via Wojtkowiak's work on the \(\ell\)-adic Galois polylogarithms.
Reviewer: Hiroaki Nakamura (Osaka)On the Selmer group of a certain \(p\)-adic Lie extension.https://zbmath.org/1456.112092021-04-16T16:22:00+00:00"Bhave, Amala"https://zbmath.org/authors/?q=ai:bhave.amala"Bora, Lachit"https://zbmath.org/authors/?q=ai:bora.lachitSummary: Let \(E\) be an elliptic curve over \(\mathbb{Q}\) without complex multiplication. Let \(p\geq 5\) be a prime in \(\mathbb{Q}\) and suppose that \(E\) has good ordinary reduction at \(p\). We study the dual Selmer group of \(E\) over the compositum of the \(\mathrm{GL}_2\) extension and the anticyclotomic \(\mathbb{Z}_p\)-extension of an imaginary quadratic extension as an Iwasawa module.The Galois action and cohomology of a relative homology group of Fermat curves.https://zbmath.org/1456.112172021-04-16T16:22:00+00:00"Davis, Rachel"https://zbmath.org/authors/?q=ai:davis.rachel"Pries, Rachel"https://zbmath.org/authors/?q=ai:pries.rachel-j"Stojanoska, Vesna"https://zbmath.org/authors/?q=ai:stojanoska.vesna"Wickelgren, Kirsten"https://zbmath.org/authors/?q=ai:wickelgren.kirsten-gLet \(p\) be a prime satisfying Vandiver's conjecture, i.e., such that \(p\) does not divide the order of \(h^+\) of the class group of \(\mathbb{Q}(\zeta+\zeta^{-1})\), where \(\zeta\) is a \(p\)-th root of unity. Let \(X\) be the degree \(p\) Fermat curve \(x^p+y^p=z^p\). Let \(U\subset X\) be the affine open given by \(z\neq 0\). Consider the closed subscheme \(Y\subset U\) defined by \(xy=0\). Let \(H_1(U,Y;\mathbb{Z}/p)\) denote the étale homology group with \(\mathbb{Z}/p \) coefficients, of the pair \((U\otimes \bar{K},Y\otimes\bar{K})\). By [\textit{G. W. Anderson}, Duke Math. J. 54, 501--561 (1987; Zbl 1370.11069)], the group \(H_1(U,Y;\mathbb{Z}/p)\) is a free rank-one \(\mathbb{Z}/p[\mu_p\times\mu_p]\)-module with generator \(\beta\). The Galois action of \(\sigma\in G_{\mathbb{Q}(\zeta)}\) is then determined by \(\sigma\beta=B_\sigma\beta\), for some \(B_\sigma\in \mathbb{Z}/p[\mu_p\times\mu_p]\). Anderson theoretically described \(B_\sigma\). In this paper, a closed form formula for \(B_\sigma\) is given. Intermediate results by the same authors [\textit{R. Davis} et al., Assoc. Women Math. Ser. 3, 57--86 (2016; Zbl 1416.11045)] about the isomorphism class of the Galois group of the field extension through the action of \(G_{\mathbb{Q}(\zeta)}\) factors, are strongly used.
The first application of this formula is that the norm of \(B_\sigma\) is \(0\) for almost all \(\sigma\). This is important in computing Galois cohomology as in Section 6 where a method for the efficient computation of the first cohomology group \(H^1(G_{\mathbb{Q}(\eta)}, H_1(U,Y;\mathbb{Z}/p))\) is given. This will eventually play a key role in understanding obstructions for rational points on Fermat curves as Ellenberg's obstruction related to the non-abelian Chabauty method.
A second application of the main formula is a proof of the fact that \(H_1(U;\mathbb{Z}/p)\) is trivialized by the product of \(\lfloor 2p/3\rfloor\) terms of the form \((B_\sigma-1)\).
Reviewer: Elisa Lorenzo García (Rennes)On pro-\(p\) link groups of number fields.https://zbmath.org/1456.112142021-04-16T16:22:00+00:00"Mizusawa, Yasushi"https://zbmath.org/authors/?q=ai:mizusawa.yasushiSummary: As an analogue of a link group, we consider the Galois group of the maximal pro-\( p\)-extension of a number field with restricted ramification which is cyclotomically ramified at \(p\), i.e., tamely ramified over the intermediate cyclotomic \(\mathbb{Z}_p\)-extension of the number field. In some basic cases, such a pro-\( p\) Galois group also has a Koch type presentation described by linking numbers and \(\mod{2}\) Milnor numbers (Rédei symbols) of primes. Then the pro-2 Fox derivative yields a calculation of Iwasawa polynomials analogous to Alexander polynomials.Geometric Waldspurger periods. II.https://zbmath.org/1456.110912021-04-16T16:22:00+00:00"Lysenko, Sergey"https://zbmath.org/authors/?q=ai:lysenko.sergeySummary: In this paper we extend the calculation of the geometric Waldspurger periods from our paper [Part I, Compos. Math. 144, No. 2, 377--438 (2008; Zbl 1209.14010)] to the case of ramified coverings. We give some applications to the study of Whittaker coeffcients of the theta-lifting of automorphic sheaves from \(\operatorname{PGL}_2\) to the metaplectic group \(\widetilde{\operatorname{SL}}_2\); they agree with our conjectures from [``Geometric Whittaker models and Eisenstein series for \(\mathrm{Mp}_2\)'', Preprint, \url{arXiv:1221.1596}]. In the process of the proof, we construct some new automorphic sheaves for \({\operatorname{GL}_2}\) in the ramified setting. We also formulate stronger conjectures about Waldspurger periods and geometric theta-lifting for the dual pair \((\widetilde{\operatorname{SL}}_2, \operatorname{PGL}_2)\).On the GIT stratification of prehomogeneous vector spaces. II.https://zbmath.org/1456.112252021-04-16T16:22:00+00:00"Tajima, Kazuaki"https://zbmath.org/authors/?q=ai:tajima.kazuaki"Akihiko, Yukie"https://zbmath.org/authors/?q=ai:akihiko.yukieSummary: We determine all orbits of two prehomogeneous vector spaces rationally over an arbitrary perfect field in this paper.
This is part two of a series of four papers. For Part I see [``On the GIT stratification of prehomogeneous vector spaces. I'', Preprint, \url{arXiv:1902.04274}].Certain \(\ast\)-homomorphisms acting on unital \(C^\ast\)-probability spaces and semicircular elements induced by \(p\)-adic number fields over primes \(p\).https://zbmath.org/1456.460532021-04-16T16:22:00+00:00"Cho, Ilwoo"https://zbmath.org/authors/?q=ai:cho.ilwooSummary: In this paper, we study the Banach \(*\)-probability space
\((A\otimes_{\mathbb{C}}\mathbb{LS}, \tau_A^0)\) generated by a fixed unital \(C^*\)-probability space \((A, \varphi_A)\), and the semicircular elements \(\Theta_{p,j}\) induced by \(p\)-adic number fields \(\mathbb{Q}_p\), for all \(p \in \mathcal{P}\), \(j\in\mathbb{Z}\), where \(\mathcal{P}\) is the set of all primes, and \(\mathbb{Z}\) is the set of all integers. In particular, from the order-preserving shifts \(g\times h_\pm\) on \(\mathcal{P} \times \mathbb{Z}\), and \(*\)-homomorphisms \(\theta\) on \(A\), we define the corresponding \(*\)-homomorphisms \(\sigma_{(\pm ,1)}^{1:\theta}\) on \(A\otimes_{\mathbb{C}} \mathbb{LS}\), and consider free-distributional data affected by them.Hermitian \(K\)-theory, Dedekind \(\zeta \)-functions, and quadratic forms over rings of integers in number fields.https://zbmath.org/1456.112212021-04-16T16:22:00+00:00"Kylling, Jonas Irgens"https://zbmath.org/authors/?q=ai:kylling.jonas-irgens"Röndigs, Oliver"https://zbmath.org/authors/?q=ai:rondigs.oliver"Østvær, Paul Arne"https://zbmath.org/authors/?q=ai:ostvaer.paul-arneSummary: We employ the slice spectral sequence, the motivic Steenrod algebra, and Voevodsky's solutions of the Milnor and Bloch-Kato conjectures to calculate the hermitian \(K\)-groups of rings of integers in number fields. Moreover, we relate the orders of these groups to special values of Dedekind \(\zeta \)-functions for totally real abelian number fields. Our methods apply more readily to the examples of algebraic \(K\)-theory and higher Witt-theory, and give a complete set of invariants for quadratic forms over rings of integers in number fields.Zeta functions of reductive groups and their zeros.https://zbmath.org/1456.110042021-04-16T16:22:00+00:00"Weng, Lin"https://zbmath.org/authors/?q=ai:weng.linLet \({\mathcal{M}}_{F,n}^{\mathrm{tot}}\) be the moduli space of \(({\mathcal{O}}_{F^-})\)lattices \(\Lambda\) of rank \(n\) (\({\mathcal{O}}_F\) is the ring of integers of a number field \(F\)), and \(\Theta(\Lambda)\) and \({\mathrm{vol}(\Lambda)}\) denote the theta series and the co-volume of \(\Lambda\), respectively. As a generalization of the Dedekind zeta-function of field \(F\), there exists an integral
\[
\int_{{\mathcal{M}}_{F,n}^{\mathrm{tot}}} \Theta(\Lambda){\mathrm{vol}}(\Lambda)^s
d \mu(\Lambda), \quad \Re (s) >1.\]
Using the Mellin transform, for \(T>0\), it can be rewritten as
\[
\int_{0}^{\infty}\int_{{\mathcal{M}}_{F,n}^{\mathrm{tot}}[T]} \Theta(\Lambda){\mathrm{vol}}(\Lambda)^s
d \mu(\Lambda)\frac{dT}{T}=\int_{{\mathcal{M}}_{F,n}^{\mathrm{tot}}[1]} {\widehat{E}}(\Lambda,s)d\mu(\Lambda),
\]
where \({\mathcal{M}}_{F,n}^{\mathrm{tot}}[T]\) denotes the moduli space rank \(n\) lattices of co-volumes \(T>0\), and \({\widehat{E}}(\Lambda,s)\) is the complete Epstein zeta-functions of \(\Lambda\). Because Epstein zeta-functions are special cases of Eisenstein series, in the case \(F=\mathbb{Q}\), \({\mathcal{M}}_{F,n}^{\mathrm{tot}}[1]\) is isomorphic to \({\mathrm{SL}}_n(\mathbb{Z})\setminus {\mathrm{SL}}_n(\mathbb{R})/{\mathrm{SO}}_n\), and it is known that integrals over moduli spaces diverge.
The book is devoted to the study of non-abelian zeta-function \({\widehat{\zeta}}_{F,n}(s)\) of a number field \(F\), for which the rank \(n\) is defined by
\[
{\widehat{\zeta}}_{F,n}(s):=\int_{{\mathcal{M}}_{F,n}}\Theta(\Lambda){\mathrm{vol}(\Lambda)^s}d\mu(\Lambda), \quad \Re (s)>1,
\]
or, in view of the Mellin transform,
\[
{\widehat{\zeta}}_{F,n}(s)=\int_{{\mathcal{M}}_{F,n}[1]}{\widehat{E}}(\Lambda,s)d\mu(\Lambda).
\]
The main direction of the content of this book is to study algebraic, analytic and geometric structures of these functions.
Of particular interest is a weak Riemann hypothesis for the rank \(n\) non-abelian zeta-functions, i.e., it is established, ensuring that all but finitely many zeros of \({\widehat{\zeta}}_{{\mathbb{Q}},n}(s)\) lie on the line \(\Re s=\frac{1}{2}\) when \(n\geq2\). In the book, a new theory for zeta-functions of split reductive groups \(G\) and their maximal parabolic subgroups \(P\) over number fields is developed, also.
Much of the work shows the author's own influence and interests. All six parts (Non-abelian zeta-functions, Rank two zeta-functions, Eisenstein periods and multiple \(L\)-functions, Zeta-functions for reductive groups, Algebraic, analytic structures and Riemann hypothesis, Geometric structures and Riemann hypothesis) are well written and give a good ground for further studies to researchers who are interested in this field. The book ends with five appendices based on author's joint works with prof. K.~Sugahara.
This book is a fine piece of work which gives a more deeper and wider look to the theory of zeta-functions.
Reviewer: Roma Kačinskaitė (Kaunas)Ramanujan denesting formulas for cubic radicals.https://zbmath.org/1456.112072021-04-16T16:22:00+00:00"Antipov, M. A."https://zbmath.org/authors/?q=ai:antipov.m-a"Pimenov, K. I."https://zbmath.org/authors/?q=ai:pimenov.konstantinSummary: This paper contains an explanation of Ramanujan-type formulas with cubic radicals of cubic irrationalities in the situation when these radicals are contained in a pure cubic extension. We give a complete description of formulas of such type, answering the Zippel's question. It turns out that Ramanujan-type formulas are in some sense unique in this situation. In particular, there must be no more than three summands in the right-hand side and the norm of the irrationality in question must be a cube. In this situation we associate cubic irrationalities with a cyclic cubic polynomial, which is reducible if and only if one can simplify the corresponding cubic radical. This correspondence is inverse to the so-called Ramanujan correspondence defined in the preceding papers, where one associates a pure cubic extension to some cyclic polynomial.Explicit counting of ideals and a Brun-Titchmarsh inequality for the Chebotarev density theorem.https://zbmath.org/1456.112182021-04-16T16:22:00+00:00"Debaene, Korneel"https://zbmath.org/authors/?q=ai:debaene.korneelOn distribution formulas for complex and \(l\)-adic polylogarithms.https://zbmath.org/1456.111212021-04-16T16:22:00+00:00"Nakamura, Hiroaki"https://zbmath.org/authors/?q=ai:nakamura.hiroaki.1"Wojtkowiak, Zdzisław"https://zbmath.org/authors/?q=ai:wojtkowiak.zdzislawSummary: We study an \(l\)-adic Galois analogue of the distribution formulas for polylogarithms with special emphasis on path dependency and arithmetic behaviors. As a goal, we obtain a notion of certain universal Kummer-Heisenberg measures that enable interpolating the \(l\)-adic polylogarithmic distribution relations for all degrees.
For the entire collection see [Zbl 1446.81002].Diophantine equation generated by the maximal subfield of a circular field.https://zbmath.org/1456.112032021-04-16T16:22:00+00:00"Galyautdinov, I. G."https://zbmath.org/authors/?q=ai:galyautdinov.ildarkhan-galyautdinovich"Lavrentyeva, E. E."https://zbmath.org/authors/?q=ai:lavrentyeva.elena-evgenevnaSummary: Using the fundamental basis of the field \(L_9=\mathbb{Q} (2\cos(\pi/9))\), the form \(N_{L_9}(\gamma)=f(x, y, z)\) is found and the Diophantine equation \(f(x,y,z)=a\) is solved. A similar scheme is used to construct the form \(N_{L_7}(\gamma)=g(x,y,z)\). The Diophantine equation \(g (x, y, z)=a\) is solved.Codimension two cycles in Iwasawa theory and elliptic curves with supersingular reduction.https://zbmath.org/1456.112122021-04-16T16:22:00+00:00"Lei, Antonio"https://zbmath.org/authors/?q=ai:lei.antonio"Palvannan, Bharathwaj"https://zbmath.org/authors/?q=ai:palvannan.bharathwajLet us fix an odd prime \(p\) and \(\mathcal{R}\) be a Noetherian, complete, integrally closed, local domain of characteristic zero with Krull dimension \(n+1\) and whose residue field has characteristic \(p\). To a continuous Galois representation \(\rho_{d, n}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow\mathrm{Gal}_d(\mathcal{R})\) satisfying the `Panchishkin condition', which is a type of `ordinariness' assumption for Galois deformations introduced by Greenberg in Section 4 of [Proc. Symp. Pure Math. 55, 193--223 (1994; Zbl 0819.11046)], \textit{R. Greenberg} has formulated a main conjecture in Iwasawa theory. The Iwasawa main conjecture provides a relation involving codimension one cycles in the divisor group of the ring \(\mathcal{R}\), relating a \(p\)-adic \(L\)-function, satisfying suitable interpolation properties, to a Selmer group. The divisor group, denoted by \(Z^1(\mathcal{R})\), is the free abelian group on the set of height \(1\) prime ideals of the ring \(\mathcal{R}\).
One could consider \(Z^2(\mathcal{R})\), the free abelian group on the set of height \(2\) prime ideals of the ring \(\mathcal{R}\). Many standard conjectures in Iwasawa theory expect that pseudonull modules are ubiquitous. For example, see Conjecture 3.5 in \textit{R. Greenberg}'s article [Adv. Stud. Pure Math. 30, 335--385 (2001; Zbl 0998.11054)] and Conjecture B in the work of \textit{J. Coates} and \textit{R. Sujatha} [Math. Ann. 331, No. 4, 809--839 (2005; Zbl 1197.11142)]. These pseudonull \(\mathcal{R}\)-modules are supported in codimension at least two. One desirable extension of the Iwasawa main conjecture is an answer to the following question: Can we use codimension two cycles from \(Z^2(\mathcal{R})\) to associate analytic invariants to pseudonull modules in Iwasawa theory ?
A result of \textit{F. M. Bleher} et al. [Am. J. Math. 142, No. 2, 627--682 (2020; Zbl 07208784)] has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz's 2-variable \(p\)-adic \(L\)-functions) and algebraic objects (two `everywhere unramified' Iwasawa modules) involving codimension two cycles in a \(2\)-variable Iwasawa algebra. In the current work under review, the authors prove a result by considering the restriction to an imaginary quadratic field \(K\) (where an odd prime \(p\) splits) of an elliptic curve \(E\), defined over \(\mathbb{Q}\), with good supersingular reduction at \(p\). On the analytic side, we consider eight pairs of \(2\)-variable \(p\)-adic \(L\)-functions in this setup. On the algebraic side, they consider modifications of fine Selmer groups over the \(\mathbb{Z}^2_p\)-extension of \(K\). The authors also provide numerical evidence, using algorithms of Pollack [\url{http://math.bu.edu/people/rpollack/Data/data.html}], towards a pseudonullity conjecture of \textit{J. Coates} and \textit{R. Sujatha} [Math. Ann. 331, No. 4, 809--839 (2005; Zbl 1197.11142)].
Reviewer: Wei Feng (Beijing)Hida duality and the Iwasawa main conjecture.https://zbmath.org/1456.110722021-04-16T16:22:00+00:00"Lafferty, Matthew J."https://zbmath.org/authors/?q=ai:lafferty.matthew-jBased on authors' abstract: The central result of this paper is a refinement of Hida's duality theorem between ordinary \(\Lambda\)-adic modular forms and the universal ordinary Hecke algebra. In particular, the author gives a sufficient condition for this duality to be integral with respect to particular submodules of the space of ordinary \(\Lambda\)-adic modular forms. This refinement allows us to give a simple proof that the universal ordinary cuspidal Hecke algebra modulo Eisenstein ideal is isomorphic to the Iwasawa algebra modulo an ideal related to the Kubota-Leopoldt \(p\)-adic \(L\)-function. The motivation behind these results stems from a proof of the Iwasawa main conjecture over \(\mathbb{Q}\) by \textit{M. Ohta} [Ann. Sci. Éc. Norm. Supér. (4) 36, No. 2, 225--269 (2003; Zbl 1047.11046)]. Ohta's argument in [loc. cit.] employs results on congruence modules that require a Euler's totient function condition and a non-exceptional hypotheses. While simple and elegant, Ohta's proof requires some restrictive hypotheses which we could be removed using the author's results. The results in this paper were obtained in an effort to extend Ohta's proof in [loc. cit.] by circumventing the obstructions arising from his congruence module argument.
Reviewer: Wei Feng (Beijing)An application of the Hasse-Weil bound to rational functions over finite fields.https://zbmath.org/1456.112282021-04-16T16:22:00+00:00"Hou, Xiang-Dong"https://zbmath.org/authors/?q=ai:hou.xiang-dong"Iezzi, Annamaria"https://zbmath.org/authors/?q=ai:iezzi.annamariaLet \(\mathbb{F}_q\) be the finite field of \(q\) elements. In the paper under review, an application of the Aubry-Perret bound (a generalization of the Hasse-Weil bound for smooth curves) for an algebraic projective, absolutely irreducible and non-singular curve defined over \(\mathbb{F}_q\) is studied [\textit{Y. Aubry} and \textit{M. Perret}, in: Arithmetic, geometry, and coding theory. Proceedings of the international conference held at CIRM, Luminy, France, 1993. Berlin: Walter de Gruyter, 1--7 (1996; Zbl 0873.11037)]. More precisely, the authors prove that if \(f, g \in\mathbb{F}_q(X)\) are non-constant rational functions with degree \(d\) and \(\delta\) respectively such that \(q \geq (d+ \delta)^4\), \(f(\mathbb{F}_q) \subset g (\mathbb{F}_q \cup \{ \infty\})\) and for all \(a \in \mathbb{F}_q \cup \{ \infty\}\) with at most \(8(d+ \delta)\) exceptions, \(|\{x \in \mathbb{F}_q: g(x)=g(a)\}| > \frac{\delta}{2}\), then there exists a rational function \(h \in \mathbb{F}_q(X)\) such that \(f=g \circ h.\)
This application of the Aubry-Perret bound to rational functions on finite fields is motivated by two special cases related to the study of permutation polynomials: \(g(X)=X^2+X\) when the characteristic of \(\mathbb{F}_q\) is \(2\) and \(g(X)=X^2\) when the characteristic of \(\mathbb{F}_q\) is \(3\) [\textit{X.-D. Hou}, Cryptogr. Commun. 11, No. 6, 1199--1210 (2019; Zbl 1446.11206)] and [\textit{X.-D. Hou} et al., Finite Fields Appl. 61, Article ID 101596, 27 p. (2020; Zbl 07160767)].
Finally, using an explicit estimate for absolutely irreducible multivariate polynomials with coefficients in \(\mathbb{F}_q\) [\textit{A. Cafure} and \textit{G. Matera}, Finite Fields Appl. 12, No. 2, 155--185 (2006; Zbl 1163.11329)], a generalization to multivariate rational functions is also included.
Reviewer: Mariana Pérez (Buenos Aires)On the scalar complexity of Chudnovsky\(^2\) multiplication algorithm in finite fields.https://zbmath.org/1456.112352021-04-16T16:22:00+00:00"Ballet, Stéphane"https://zbmath.org/authors/?q=ai:ballet.stephane"Bonnecaze, Alexis"https://zbmath.org/authors/?q=ai:bonnecaze.alexis"Dang, Thanh-Hung"https://zbmath.org/authors/?q=ai:dang.thanh-hungThe paper under review deals with the multiplicative complexity of multiplication in a finite field \(F_{q^n}\) which is the number of multiplications required to multiply in the \(F_q\)-vector space \(F_{q^n}\). The types of multiplications in \(F_q\) are the scalar multiplication and the bilinear one. The scalar multiplication is
the multiplication by a constant in \(F_q\). The bilinear multiplication is a multiplication that depends on the elements of \(F_{q^n}\) that are multiplied. D. V. and G. V. Chudnovsky, generalizing interpolation algorithms on the projective line over \(F_q\) to algebraic curves of higher genus over \(F_q\), provided a method which enabled to prove the linearity of the bilinear complexity of multiplication in finite extensions
of a finite field [\textit{D. V. Chudnovsky} and \textit{G. V. Chudnovsky}, J. Complexity 4, No. 4, 285--316 (1988; Zbl 0668.68040)]. This is the so-called Chudnovsky\(^2\) algorithm.
In this paper, a new method of construction with an objective to reduce the scalar complexity of Chudnovsky\(^2\) multiplication algorithms is proposed. An optimized basis representation of the Riemann-Roch space \(L(2D)\) is sought in order to minimize the number of scalar multiplications in the algorithm.
In particular, the Baum-Shokrollahi construction for multiplication in \(F_{256}/F_4\) based on the elliptic Fermat curve \(x^3 + y^3 = 1\) is improved.
For the entire collection see [Zbl 1428.68013].
Reviewer: Dimitros Poulakis (Thessaloniki)An effective bound for the cyclotomic Loxton-Kedlaya rank.https://zbmath.org/1456.112082021-04-16T16:22:00+00:00"Beli, Constantin N."https://zbmath.org/authors/?q=ai:beli.constantin-nicolae"Stan, Florin"https://zbmath.org/authors/?q=ai:stan.florin"Zaharescu, Alexandru"https://zbmath.org/authors/?q=ai:zaharescu.alexandruFor a positive integer \(m\), an \(m\)-Weil number is an algebraic integer all of whose conjugates have absolute value \(\sqrt{m}\). Let \(H_m\) denote the set of elements \(x=x_1^{e_1}x_2^{e_2}\cdots x_n^{e_n}\), where \(x_1,\ldots, x_n\) are \(m\)-Weil numbers in \(\mathbb{Q}^{\mathrm{ab}}\), the maximal abelian extension of \(\mathbb{Q}\) and the integer exponents \(e_1,\ldots,e_n\) satisfy \(e_1+\cdots+e_n=0\). Let \(\mu_\infty\) denote the group of all roots of unit in \(\mathbb{Q}^{\mathrm{ab}}\). The quotient group \(H_m/\mu_\infty\) is free of finite rank, \(r_m\), called the Loxton-Kedlaya rank by \textit{F. Stan} and \textit{A. Zaharescu} [Trans. Am. Math. Soc. 367, No. 6, 4359--4376 (2015; Zbl 1322.11109)]. The main result of the paper is an explicit but very large bound for \(r_m\), of rough order of magnitude \(2^{2^{2^{2m+2}}}\).
For an algebraic number \(\alpha\), define \(A(\alpha)=(1/n)\sum_{i=1}^n|\alpha_i|^2\), where \(n\) is the degree of \(\alpha\) and \(\alpha_1,\ldots,\alpha_n\) are its conjugates. A cyclotomic integer \(\alpha\) is a sum of roots of unity; define its length \(\ell(\alpha)\) to be the smallest number \(\ell\) such that \(\alpha\) can be written as a sum of \(\ell\) roots of unity. For any positive integer \(n\), there is an integer \(k_n\) such that any integer in \(\mathbb{Q}^{\mathrm{ab}}\) with \(A(\alpha)\le n/2\) has \(\ell(\alpha)\le k_n\); one can take \(k_n=n!2^{\pi(n^2+n-2)}\) where \(\pi(x)\) denotes the number of primes less than or equal to \(x\). The proof is by induction based on the structure of cyclotomic extensions as a tower of extensions.
The proof of the main result uses two combinatorial lemmas. From [Stan and Zaharescu, loc. cit.], if \(k\ge4\) is an integer, \(p>6^{k/2}\) is a prime and \(a_1,\ldots,a_k\) are distinct elements of \(\mathbb{Z}/p\mathbb{Z}\), then at least one of the differences \(a_i-a_j \ (1\le i\ne j\le k)\) occurs only once. The other ingredient is a \(p\)-adic extension of a result from an unpublished manuscript of Kedlaya (see [Stan and Zaharescu, loc. cit.]): if \(a_1,\ldots,a_n\) are integers mutually distinct in \(\mathbb{Z}/p\mathbb{Z}\), then
\[\max_{i,j} \min_{(k,l)\ne(i,j)} |(a_i-a_j)-(a_k-a_l)|_p \ge 6^{-(n-1)/2}.\]
The proof given in the paper is an ingenious piece of linear algebra.
These preparations allow the construction of a finite set \(T_m\) consisting of cyclotomic \(m\)-Weil numbers and an explicit bound on \(|T_m|\) in terms of \(m\) such that \(T_m\mu_\infty\) is the set of all cyclotomic \(m\)-Weil numbers. Suppose \(\beta\) is a cyclotomic \(m\)-Weil number in \(\mathbb{Q}(\zeta_q)\) where \(\zeta_q\) denotes a primitive \(q\)th root of unity. For the simplest case, suppose \(p\) is a prime and \(p\nmid q\). Write \(\beta = a_1\zeta_p^{j_1}+\ldots+a_k\zeta_p^{j_k}\) where \(0\le j_1<\cdots<j_k\le p-1\) and the \(a_j\in\mathbb{Z}[\zeta_{q/p}]\) are nonzero and, one can show, \(k\le 2m\). If \(p>6^m\), there is a pair, \((j_1,j_2)\) say, such that the difference \(j_1-j_2\) occurs only once. Expanding \(m=\beta\overline{\beta} =
(a_1\zeta_p^{j_1}+\ldots+a_k\zeta_p^{j_k})(\overline{a_1}\zeta_p^{-j_1}+\ldots+\overline{a_k}\zeta_p^{-j_k})\) produces a term \(\alpha_2\overline{\alpha_1}\zeta_p^{j_2-j_1}\) which cannot be cancelled by any other terms. This contradiction gives the bound \(p<6^m\). A similar argument works in case \(p^2\mid q\). Using Kedlaya's lemma and a similar but more involved argument leads to a bound in terms of \(m\) for the exponent \(r\) where \(p^r\nmid q\). Putting these pieces together gives the bound on \(|T_m|\).
Reviewer: John H. Loxton (Greenwich)On a type of permutation rational functions over finite fields.https://zbmath.org/1456.112192021-04-16T16:22:00+00:00"Hou, Xiang-dong"https://zbmath.org/authors/?q=ai:hou.xiang-dong"Sze, Christopher"https://zbmath.org/authors/?q=ai:sze.christopherSummary: Let \(p\) be a prime and \(n\) be a positive integer. Let \(f_b(X)=X+(X^p-X+b)^{-1}\), where \(b\in\mathbb{F}_{p^n}\) is such that \(\text{Tr}_{p^n/p}(b)\neq 0\). In 2008, \textit{J. Yuan} et al. [Finite Fields Appl. 14, No. 2, 482--493 (2008; Zbl 1211.11136)] showed that for \(p=2,3,f_b\) permutes \(\mathbb{F}_{p^n}\) for all \(n\geq 1\). Using the Hasse-Weil bound, we show that when \(p>3\) and \(n\geq 5, f_b\) does not permute \(\mathbb{F}_{p^n} \). For \(p>3\) and \(n=2\), we prove that \(f_b\) permutes \(\mathbb{F}_{p^2}\) if and only if \(\text{Tr}_{p^2/p}(b)=\pm 1\). We conjecture that for \(p>3\) and \(n=3,4,f_b\) does not permute \(\mathbb{F}_{p^n}\).Exceptional zero formulae for anticyclotomic \(p\)-adic \(L\)-functions of elliptic curves in the ramified case.https://zbmath.org/1456.112132021-04-16T16:22:00+00:00"Longo, Matteo"https://zbmath.org/authors/?q=ai:longo.matteo"Pati, Maria Rosaria"https://zbmath.org/authors/?q=ai:pati.maria-rosariaLet \(E\) be an elliptic curve over \(\mathbb Q\) of conductor \(N\) and let \(p\) be a prime of multiplicative reduction for \(E\). Let \(K\) be an imaginary quadratic field, not equal to \(\mathbb Q(\sqrt{-1})\) or \(\mathbb Q(\sqrt{-3})\), in which \(p\) is ramified and all other primes dividing \(N\) are unramified.
Assume \(E\) has multiplicative reduction at the primes dividing \(N\) that are inert in \(K\) and assume that the number of such primes is odd. Using the eigenform of weight 2 on \(\Gamma_0(N)\) attached to \(E\), the authors, following the work of \textit{M. Bertolini} et al. [Am. J. Math. 124, No. 2, 411--449 (2002; Zbl 1079.11036)], construct
a \(p\)-adic \(L\)-function \(L_p(E/K, \chi, s)\) for all finite order ramified characters \(\chi\) of ring class fields of \(K\) of \(p\)-power conductor. Let \(H\) be the Hilbert class field of \(K\) and let \(H_p\) be the maximal subextension in which the prime of \(K\) above \(p\)
splits completely. The main result of the paper is that for characters \(\chi\) that factor through \(H_p\),
\[
L_p'(E/K,\chi, 1)=\frac{2}{[H : H_p]} \log_E(y_{\chi}-\overline{y}_{\chi}),
\]
where \(\log_E\) is the logarithm of the formal group of \(E\) and where \(y_{\chi}\) is a twisted Heegner point constructed from a uniformization of \(E\) by a Shimura curve and \(\overline{y}_{\chi}\) is its complex conjugate.
This result is an analogue of the work of \textit{M. Bertolini} and \textit{H. Darmon} [Invent. Math. 131 No. 2, 453--491 (1998; Zbl 0899.11029)] ], who considered the case where \(\chi\) is trivial and \(p\) is inert in \(K\).
Reviewer: Lawrence C. Washington (College Park)Counting integer reducible polynomials with bounded measure.https://zbmath.org/1456.112052021-04-16T16:22:00+00:00"Dubickas, Artūras"https://zbmath.org/authors/?q=ai:dubickas.arturasSummary: In this paper, we give an asymptotic formula for the number of integer reducible polynomials with fixed degree \(d \ge 2\) and Mahler measure bounded above by \(T\) and also for the number of such monic polynomials as \(T\to\infty\). We also consider the case of monic polynomials which have all their roots in the disc \(|z|\le R\) and find asymptotics for the number of such reducible polynomials too as \(T\to\infty\). In all cases the constants in the main terms are related to the constants of the corresponding counting formulas for the number of such irreducible polynomials due to \textit{S. J. Chern} and \textit{J. D. Vaaler} [J. Reine Angew. Math. 540, 1--47 (2001; Zbl 0986.11017)] (in case of Mahler measure) and \textit{S. Akiyama} and \textit{A. Pethő} [J. Math. Soc. Japan 66, No. 3, 927--949 (2014; Zbl 1305.33006); Unif. Distrib. Theory 9, No. 1, 5--19 (2014; Zbl 1384.11081)] (in case of a disc).