Recent zbMATH articles in MSC 14Ghttps://zbmath.org/atom/cc/14G2024-04-15T15:10:58.286558ZWerkzeugSiegel modular forms of degree two and three and invariant theoryhttps://zbmath.org/1530.110472024-04-15T15:10:58.286558Z"van der Geer, Gerard"https://zbmath.org/authors/?q=ai:van-der-geer.gerardSummary: This is a survey based on the construction of Siegel modular forms of degree 2 and 3 using invariant theory in joint work with \textit{F. Cléry} and \textit{C. Faber} [Math. Ann. 369, No. 3--4, 1649--1669 (2017; Zbl 1427.11042); Math. Comput. 88, No. 319, 2423--2441 (2019; Zbl 1470.11070)].
For the entire collection see [Zbl 1515.14011].The equation \(y^2 = x^6+x^2+1\) revisitedhttps://zbmath.org/1530.110582024-04-15T15:10:58.286558Z"Tho, Nguyen Xuan"https://zbmath.org/authors/?q=ai:nguyen-xuan-tho.Summary: We give a new proof that all rational points on \(y^2 = x^6+x^2+1\) are \(\pm\infty\), \((0, \pm 1)\), \((\pm\frac{1}{2}\), \(\pm\frac{9}{8})\). Our approach combines the two descent map on elliptic curves with the elliptic curve Chabauty method over certain quartic number fields.On the density of \(S\)-adic integers near some projective \(G\)-varietieshttps://zbmath.org/1530.110652024-04-15T15:10:58.286558Z"Lazar, Youssef"https://zbmath.org/authors/?q=ai:lazar.youssefLet \(S\) be a finite set of valuations of \({\mathbb{Q}}\) containing the Archimedean one \(\infty\). Let \((f_{i,p})_{p\in S}\) (\(1\le i\le r)\) be a finite family of homogeneous polynomials in \(n\) variables, where \(f_{i,p}\) has coefficients in the completion \({\mathbb{Q}}_p\) of \({\mathbb{Q}}\) relative to the place \(p\in S\). The author considers the following problem: given any real number \(\varepsilon>0\), does there exist a nonzero \(S\)--integral vector \(x\) in \({\mathbb{Z}}_S^n\) such that \(0<|f_{i,p}(x)|_p\le \varepsilon\) for every \(p\in S\) and \(i=1,\dots,r\)? Let \(S_f\) be the set of finite places in \(S\), so that \(S=S_f\cup\{\infty\}\). Define \({\mathbb{Q}}_S=\prod_{p\in S}{\mathbb{Q}}_p\), \({\mathbb{A}}_S=\prod_{p\in S}{\mathbb{Q}}_p\times \prod_{p\not\in S}{\mathbb{Z}}_p\), so that the ring of \(S\)-integers of \({\mathbb{Q}}\) is \({\mathbb{Z}}_S={\mathbb{A}}_S\cap {\mathbb{Q}}\). When \(X\) is an algebraic variety over \({\mathbb{Q}}\), embedding it diagonally in a finite product of completions relative to the places in \(S\) gives rise for each \(p\in S\) to an affine variety \(X_p\) over \({\mathbb{Q}}_p\), zero set of a family of polynomials \((f_{i,p})\) (\(1\le i\le r\)); let \(X_S\) denote the direct product of the completions of \(X_p\) for \(p\in S\) in \(\prod_{p\in S}\overline{\mathbb{Q}}_{p}^n\). For \(\varepsilon>0\), let \(X_S^\varepsilon\) denotes the set of \(x\in {\mathbb{Q}}_S^n\) such that \(|f_{i,p}(x)|_p\le \varepsilon\) for every \(p\in S\) and \(i=1,\dots,r\). Here is the main result of the paper under review. For each \(s\in S\), let \(X_s\) be a projective variety over \({\mathbb{Q}}_s\) defined by a homogeneous prime ideal \(I_s\) of \({\mathbb{Q}}_s[x_1,\dots,x_n]\) and let \(H_s\) be the algebraic \({\mathbb{Q}}_s\)-subgroup of \({\mathrm {SL}}_n({\mathbb{Q}}_s)\) leaving invariant every generator of \(I_s\). Let \(S_1\) be the set of \(p\in S_f\) such that \(H_p\) is rational over \({\mathbb{Q}}\); assume \(S_1\) is not empty. Assume that there is a connected algebraic subgroup \(H_0\) of \({\mathrm {SL}}_n({\mathbb{Q}})\) rational over \({\mathbb{Q}}\) and a hypersurface \(X_0=V(f_0)\) defined over \({\mathbb{Q}}\) such that (1) \(H_0\) is a semisimple absolutely almost simple algebraic \({\mathbb{Q}}\)-group, (2) For every prime \(p\in S_1\), \(X_p=X_0\) and \(H_p=H_0\), (3) Every \({\mathbb{Q}}\)-simple factor of \(H_0\) is isotropic over \(S_1\). Then for every \(\varepsilon>0\), we have \(X_S^\varepsilon\cap {\mathbb{Z}}_S^n\not=\{0\}\). As an application, the author deals with the intersection of a nondegenerate quadric defined by a quadratic form cut out by hyperplanes. The method of proof is an adaptation of the work of \textit{A. Borel} and \textit{G. Prasad} [Compos. Math. 83, No. 3, 347--372 (1992; Zbl 0777.11008)]. It is based on the strong approximation theorem for algebraic groups.
Reviewer: Michel Waldschmidt (Paris)On the number of rational points of certain algebraic varieties over finite fieldshttps://zbmath.org/1530.110952024-04-15T15:10:58.286558Z"Zhu, Guangyan"https://zbmath.org/authors/?q=ai:zhu.guangyan"Hong, Siao"https://zbmath.org/authors/?q=ai:hong.siao.1|hong.siaoSummary: Let \(\mathbb{F}_q\) be the finite field of odd characteristic \(p\) with \(q\) elements \((q=p^n\), \(n \in \mathbb{N})\) and let \(\mathbb{F}_q^*\) represent the set of nonzero elements of \(\mathbb{F}_q\). By making use of the Smith normal form of exponent matrices, we obtain an explicit formula for the number of rational points on the variety defined by the following system of equations over \(\mathbb{F}_q\):
\[
\begin{cases}
\displaystyle\sum_{i=1}^r a^{(1)}_i x_1^{e_{i1}^{(1)}} \cdots x_n^{e_{in}^{(1)}}=b_1, \\
\displaystyle\sum^{t-1}_{j'=0} \sum^{r_{j'+1}-r_{j'}}_{i'=1} a^{(2)}_{r_{j'}+i'} x_1^{e_{r_{j'}+i',1}^{(2)}} \cdots x_{n_{{j'}+1}}^{e_{r_{j'}+i', n_{{j'}+1}}^{(2)}} = b_2,
\end{cases}
\]
where \(b_i \in \mathbb{F}_q\) \((i=1, 2)\), \(t \in \mathbb{N}\),
\[
0=n_0<n_1<n_2<\cdots<n_t,
\]
\(n_{k-1} < n \leq n_k\) for some \(1 \leq k \leq t\),
\[
0=r_0 < r_1 < r_2 < \cdots < r_t,
\]
\(a^{(1)}_i \in \mathbb{F}_q^*\) for \(i \in \{1, \dots, r\}, a^{(2)}_{i'} \in \mathbb{F}_q^*\) for \(i' \in \{1, \dots, r_t\}\), and the exponent of each variable is a positive integer. This generalizes the results obtained previously by Wolfmann, Sun, Cao, and others. Our result also gives a partial answer to an open problem raised by \textit{S. Hu} et al. [J. Number Theory 156, 135--153 (2015; Zbl 1375.11075)].Field arithmetichttps://zbmath.org/1530.120032024-04-15T15:10:58.286558Z"Fried, Michael D."https://zbmath.org/authors/?q=ai:fried.michael-david"Jarden, Moshe"https://zbmath.org/authors/?q=ai:jarden.mosheThis is the fourth edition of Fried and Jarden's monumental work on \textit{Field Arithmetic}. (Refer to Zbl 0625.12001, Zbl 1055.12003, and Zbl 1145.12001 for previous editions.) It covers field extensions and their Galois groups and uses both classical results as well as methods from logic and model theory due to James Ax and Abraham Robinson. The first part of the book provides the necessary background such as infinite Galois theory, valuation theory, algebraic function fields and number fields and plane algebraic curves, and it discusses gems like the Riemann Hypothesis for function fields, the Chebotarev density theorem or the inequality of Golod-Shafarevich. The main part of the book deals with pseudo-algebraically closed fields and Hilbertian fields.
The present edition contains several new results, and a new chapter on Hilbertian subfields of Galois extensions. Chapters 2, 13 and 16 were split into two chapters. Other than that, typographical errors and a few mathematical inaccuracies have been corrected.
Reviewer: Franz Lemmermeyer (Jagstzell)Finite generation of nilpotent quotients of fundamental groups of punctured spectrahttps://zbmath.org/1530.140062024-04-15T15:10:58.286558Z"Suzuki, Takashi"https://zbmath.org/authors/?q=ai:suzuki.takashi.1|suzuki.takashi|suzuki.takashi.2Summary: In SGA 2, Grothendieck conjectured that the étale fundamental group of the punctured spectrum of a complete noetherian local domain of dimension at least two with algebraically closed residue field is topologically finitely generated. In this paper, we prove a weaker statement, namely that the maximal pro-nilpotent quotient of the fundamental group is topologically finitely generated. The proof uses \(p\)-adic nearby cycles and negative definiteness of intersection pairings over resolutions of singularities as well as some analysis of Lie algebras of certain algebraic group structures on deformation cohomology.Vanishing theorems and adjoint linear systems on normal surfaces in positive characteristichttps://zbmath.org/1530.140132024-04-15T15:10:58.286558Z"Enokizono, Makoto"https://zbmath.org/authors/?q=ai:enokizono.makotoSummary: We prove the Kawamata-Viehweg vanishing theorem for a large class of divisors on surfaces in positive characteristic. By using this vanishing theorem, Reider-type theorems and extension theorems of morphisms for normal surfaces are established. As an application of the extension theorems, we characterize nonsingular rational points on any plane curve over an arbitrary base field in terms of rational functions on the curve.On Galois descent of complete intersectionshttps://zbmath.org/1530.140142024-04-15T15:10:58.286558Z"Pieropan, Marta"https://zbmath.org/authors/?q=ai:pieropan.martaSummary: We introduce a notion of strict complete intersections with respect to Cox rings and we prove Galois descent for this new notion.On the integral Tate conjecture for the product of a curve and a \(CH_0\)-trivial surface over a finite fieldhttps://zbmath.org/1530.140162024-04-15T15:10:58.286558Z"Colliot-Thélène, Jean-Louis"https://zbmath.org/authors/?q=ai:colliot-thelene.jean-louis"Scavia, Federico"https://zbmath.org/authors/?q=ai:scavia.federicoSummary: We investigate a strong version of the integral Tate conjecture for 1-cycles on the product of a curve and a surface over a finite field, under the assumption that the surface is geometrically \(CH_0\)-trivial. By this we mean that over any algebraically closed field extension, the degree map on the zero-dimensional Chow group of the surface is an isomorphism. This applies to Enriques surfaces. When the Néron-Severi group has no torsion, we recover earlier results of A. Pirutka. The results rely on a detailed study of the third unramified cohomology group of specific products of varieties.Notes on Frobenius stable direct imageshttps://zbmath.org/1530.140232024-04-15T15:10:58.286558Z"Ejiri, Sho"https://zbmath.org/authors/?q=ai:ejiri.shoSummary: In this note, we prove the coherence of Frobenius stable direct images in a new case. We also show a generation theorem regarding to it. Furthermore, we prove a corresponding theorem in characteristic zero.\(p\)-adic GKZ hypergeometric complexhttps://zbmath.org/1530.140412024-04-15T15:10:58.286558Z"Fu, Lei"https://zbmath.org/authors/?q=ai:fu.lei"Li, Peigen"https://zbmath.org/authors/?q=ai:li.peigen"Wan, Daqing"https://zbmath.org/authors/?q=ai:wan.daqing"Zhang, Hao"https://zbmath.org/authors/?q=ai:zhang.hao.14|zhang.hao|zhang.hao.2|zhang.hao.3|zhang.hao-helen|zhang.hao.4Authors' abstract: To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, which are now called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the \(p\)-adic counterpart of the GKZ hypergeometric system. The \(p\)-adic GKZ hypergeometric complex is a twisted relative de Rham complex of overconvergent differential forms with logarithmic poles. It is an over-holonomic object in the derived category of arithmetic \(\mathcal{D}\)-modules with Frobenius structures. Traces of Frobenius on fibers at Techmüller points of the GKZ hypergeometric complex define the hypergeometric function over the finite field introduced by Gelfand and Graev. Over the non-degenerate locus, the GKZ hypergeometric complex defines an overconvergent \(F\)-isocrystal. It is the crystalline companion of the \(\ell\)-adic GKZ hypergeometric sheaf that we constructed before. Our method is a combination of Dwork's theory and the theory of arithmetic \(\mathcal{D}\)-modules of Berthelot.
Reviewer: Nikolai L. Manev (Sofia)Local solubility for a family of quadrics over a split quadric surfacehttps://zbmath.org/1530.140442024-04-15T15:10:58.286558Z"Browning, Tim"https://zbmath.org/authors/?q=ai:browning.timothy-daniel"Lyczak, Julian"https://zbmath.org/authors/?q=ai:lyczak.julian"Sarapin, Roman"https://zbmath.org/authors/?q=ai:sarapin.romanSummary: We study the density of everywhere locally soluble diagonal quadric surfaces, parametrised by rational points that lie on a split quadric surface.Rational points on a certain genus 2 curvehttps://zbmath.org/1530.140452024-04-15T15:10:58.286558Z"Nguyen, Xuan Tho"https://zbmath.org/authors/?q=ai:nguyen.xuan-thoSummary: We give a correct proof to the fact that all rational points on the curve
\[
y^2=(x^2+1)(x^2+3)(x^2+7)
\]
are \(\pm\infty\) and \((\pm 1,\,\pm 8)\). This corrects previous works of \textit{H. Cohen} [Number theory. II: Analytic and modern tools. New York, NY: Springer (2007; Zbl 1119.11002)] and \textit{S. Duquesne} [Calculs effectifs des points entiers et rationnels sur les courbes. 155 p. (2001); J. Théor. Nombres Bordx. 15, No. 1, 99-113 (2003; Zbl 1097.11014)].On Liu morphisms in non-Archimedean geometryhttps://zbmath.org/1530.140462024-04-15T15:10:58.286558Z"Xia, Mingchen"https://zbmath.org/authors/?q=ai:xia.mingchenThe article studies a new notion of Stein space in non-Archimedean analytic geometry. These spaces are the analytic counterpart of the affine spaces of algebraic geometry as they are completely determined by their algebras of analytic functions and satisfy the celebrated Theorems A and B of Cartan. The author calls such spaces Liu spaces, in honor of the paper [\textit{Q. Liu}, C. R. Acad. Sci., Paris, Sér. I 307, No. 2, 83--86 (1988; Zbl 0682.14012)] where this class of spaces first appeared. He also introduces a relative version of this concept via the notion of Liu morphisms. The author shows that the resulting class of spaces and morphisms behaves very similarly to affine spaces and affine morphisms of algebraic geometry. Remarkably, he shows that for a separated analytic space \(X\) there is a relative spectrum construction that realizes a duality between the category of Liu spaces over \(X\) and the category of analytic quasi-coherent sheaves of Liu algebras over \(X\). This construction is very interesting because there is no general well-behaved notion of quasi-coherent sheaf on analytic spaces, not even for complex analytic spaces, and recent works (see [\textit{F. Bambozzi} and \textit{K. Kremnizer}, ``On the Sheafyness Property of Spectra of Banach Rings'', Preprint, \url{arXiv:2009.13926}; \textit{O. Ben-Bassat} and \textit{K. Kremnizer}, Ann. Fac. Sci. Toulouse, Math. (6) 26, No. 1, 49--126 (2017; Zbl 1401.14128)], and Clausen and Scholze work on condensed mathematics) have so far provided only the correct notion of derived quasi-coherent sheaf. The author is able to show that the transversality property of sheaves of Liu algebras yields derived quasi-coherent sheaves that are concentrated in degree \(0\), showing that the relative spectrum construction works without the need for derived geometry in this case. The paper ends with the study of quasi-Liu morphisms which are the analytic analog of quasi-affine morphisms of algebraic geometry.
Reviewer: Federico Bambozzi (Padova)Uniruledness of some low-dimensional ball quotientshttps://zbmath.org/1530.140472024-04-15T15:10:58.286558Z"Maeda, Yota"https://zbmath.org/authors/?q=ai:maeda.yota\textit{V. Gritsenko} and \textit{K. Hulek} [J. Algebr. Geom. 23, No. 4, 711--725 (2014; Zbl 1309.14030)] showed that some orthogonal modular varieties are uniruled. They used reflective modular forms and the numerical criterion of uniruledness.
In the present paper, the author applies this method to ball quotients associated with Hermitian forms of signatures \((1, 3)\), \((1, 4)\), and \((1, 5)\), and shows that some of them are uniruled. In particular, he gives examples of Hermitian lattices over the rings of integers of imaginary quadratic fields \(\mathbb{Q}(\sqrt{-1})\) and \(\mathbb{Q}(\sqrt{-2})\) for which the associated ball quotients are uniruled.
Reviewer: Lei Yang (Beijing)Arakelov theory on arithmetic surfaces over a trivially valued fieldhttps://zbmath.org/1530.140482024-04-15T15:10:58.286558Z"Chen, Huayi"https://zbmath.org/authors/?q=ai:chen.huayi"Moriwaki, Atsushi"https://zbmath.org/authors/?q=ai:moriwaki.atsushiThe paper under review introduces an alternative version of Arakelov theory based on replacing the archimedean valued field \(\mathbb C\) at infinite places of a number field with a trivially valued field. (A \textit{trivially valued field} is a field \(k\), together with the trivial absolute value \(|\cdot|_0\) on \(k\), which is defined by \(|0|_0=0\) and \(|x|_0=1\) for all \(x\in k\setminus\{0\}\).)
A key reason for considering this approach is as follows. In Arakelov theory, when working for example with a metrized vector bundle \(\overline{\mathscr E}\) on an arithmetic surface \(X\), proper and flat over the ring of integers of a number field \(k\), one focuses on sections in \(H^0(X,\overline{\mathscr E})\) of small norm, but this set is not in general closed under addition. This is in contrast to the corresponding geometric situation of a vector bundle \(\mathscr E\) on a nonsingular surface \(X\), proper and flat over a smooth projective curve \(C\) over a field \(F\), in which \(H^0(X,\mathscr E)\) is a vector space over \(F\). This difference prevents one from directly carrying over methods from algebraic geometry when studying problems in Arakelov geometry.
A way around this difficulty is suggested by the observation that giving a norm on a vector space over a field \(k\) compatible with the trivial absolute value on \(k\) is equivalent to giving an \(\mathbb R\)-filtration on that vector space. Therefore, if one were working in a version of Arakelov theory based on trivially valued fields, one would again have a situation in which \(H^0(X,\overline{\mathscr E})\) was closed under addition.
This version of Arakelov theory differs from the classical version, in that the arithmetic surface is not a scheme of dimension \(2\), but rather an infinite tree of length \(1\) based on the Berkovich analytic space \(X^{\text{an}}\) associated to an irreducible smooth projective curve \(X\) over the trivially valued field \(k\). On such a curve \(X/k\), the paper under review develops a theory of metrised \(\mathbb R\)-divisors \(\overline D=(D,g)\) on \(X\), where \(D\) is an \(\mathbb R\)-divisor on \(X\) and \(g\) is a Green function of \(D\). The set of such divisors forms a real vector space \(\widehat{\operatorname{Div}}_{\mathbb R}(X)\).
Further, the paper develops an intersection pairing \((\overline D_1\cdot\overline D_2)\) of metrised \(\mathbb R\)-divisors \(\overline D_i=(D_i,g_i)\) (\(i=1,2\)) with certain conditions on the Green functions \(g_i\), and proves an analogue of the arithmetic Hilbert-Samuel theorem for metrised \(\mathbb R\)-divisors \(\overline D=(D,g)\) on \(X\) when \(\deg D>0\) and \(g\) is plurisubharmonic.
The paper also studies the effectivity up to \(\mathbb R\)-linear equivalence of pseudoeffective metrised \(\mathbb R\)-divisors. The authors note that the analogue of this problem in algebraic geometry is related to the non-vanishing conjecture, which in turn is used to show the existence of log minimal models -- see [\textit{C. Birkar}, J. Reine Angew. Math. 658, 99--113 (2011; Zbl 1226.14021)]. This analogue is also related to Keel's conjecture -- see Question 0.9 of \textit{S. Keel} [Commun. Algebra 31, No. 8, 3955--3982 (2003; Zbl 1051.14017)] and Question 0.3 of [\textit{A. Moriwaki}, Kyoto J. Math. 55, No. 4, 799--817 (2015; Zbl 1349.14094)].
Reviewer: Paul Vojta (Berkeley)Monge-Ampère measures for toric metrics on abelian varietieshttps://zbmath.org/1530.140492024-04-15T15:10:58.286558Z"Gubler, Walter"https://zbmath.org/authors/?q=ai:gubler.walter"Stadlöder, Stefan"https://zbmath.org/authors/?q=ai:stadloder.stefanSummary: Toric metrics on a line bundle of an abelian variety \(A\) are the invariant metrics under the natural torus action coming from Raynaud's uniformization theory. We compute here the associated Monge-Ampère measures for the restriction to any closed subvariety of \(A\). This generalizes the computation of canonical measures done by the first author from canonical metrics to toric metrics and from discrete valuations to arbitrary non-Archimedean fields.
For the entire collection see [Zbl 1530.11001].Analytic torsion forms for fibrations by projective curveshttps://zbmath.org/1530.140502024-04-15T15:10:58.286558Z"Köhler, Kai"https://zbmath.org/authors/?q=ai:kohler.kaiAnalytic torsion forms, which have been constructed by \textit{J.-M. Bismut} and the author in [J. Algebr. Geom. 1, No. 4, 647--684 (1992; Zbl 0784.32023)], are differential forms on the base \(B\) associated to Hermitian holomorphic vector bundles \(\overline{E}\) over fibrations \( \pi: M\to B\) of complex manifolds equipped with a certain Kähler structure. While their degree 0 part equals Ray-Singer's complex analytic torsion, there are only few explicitly known values of analytic torsion forms in higher degree. In this paper the author gives an explicit formula for analytic torsion forms for fibrations by projective curves. He also uses this to obtain a formula for direct images in Arakelov geometry. The proof mainly uses a new description of Bismut's equivariant Bott-Chern current in the case of isolated fixed points.
Reviewer: Guoquan Gao (Beijing)On the number of points of given order on odd-degree hyperelliptic curveshttps://zbmath.org/1530.140552024-04-15T15:10:58.286558Z"Boxall, John"https://zbmath.org/authors/?q=ai:boxall.john-lSummary: For integers \(N \ge 2\) and \(g \ge 1\), we study bounds on the cardinality of the set of points of order dividing \(N\) lying on a hyperelliptic curve of genus \(g\) embedded in its Jacobian using a Weierstrass point as base point. This leads us to revisit division polynomials introduced by \textit{D. G. Cantor} [J. Reine Angew. Math. 447, 91--145 (1994; Zbl 0788.14026)] and strengthen a divisibility result proved by him. Several examples are discussed.Algorithmic study of superspecial hyperelliptic curves over finite fieldshttps://zbmath.org/1530.141042024-04-15T15:10:58.286558Z"Kudo, Momonari"https://zbmath.org/authors/?q=ai:kudo.momonari"Harashita, Sushi"https://zbmath.org/authors/?q=ai:harashita.sushiThis paper is a full version of the conference paper [\textit{M. Kudo} and \textit{S. Harashita}, Lect. Notes Comput. Sci. 11321, 58--73 (2018; Zbl 1446.11120)]. The paper studies the question of enumerating superspecial hyperelliptic curves of genus \(g\) over a finite field with \(q > 2g+1\) elements. The curves are counted in two ways: by \(\mathbb{F}_q\)-isomorphism classes and by \(\overline{\mathbb{F}_q}\)-isomorphism classes. The enumeration has been implemented and carried out for some small \(q\) and \(g\) and applications to maximal and minimal hyperelliptic curves are discussed.
The algorithm uses the Cartier-Manin matrix of a hyperelliptic curve to give algebraic conditions in terms of its coefficients to detect whether the curve is superspecial. These equations are reduced so that a Gröbner basis computation can be used to find all the solutions. Then isomorphism testing is used to remove curves that have been counted multiple times.
The paper also discusses an algorithm to compute the automorphism group and geometric automorphism group of hyperelliptic curves, and how these can be used in combination with Galois cohomology to determine the number of \(\mathbb{F}_q\)-forms of a given curve over \(\overline{\mathbb{F}_q}\). As an application the authors compute the automorphism groups for the superspecial curves that they enumerated.
Reviewer: Raymond van Bommel (Cambridge, MA)Most words are geometrically almost uniformhttps://zbmath.org/1530.202012024-04-15T15:10:58.286558Z"Larsen, Michael Jeffrey"https://zbmath.org/authors/?q=ai:larsen.michael-jSummary: If \(w\) is a word in \(d>1\) letters and \(G\) is a finite group, evaluation of \(w\) on a uniformly randomly chosen \(d\)-tuple in \(G\) gives a random variable with values in \(G\), which may or may not be uniform. It is known that if \(G\) ranges over finite simple groups of given root system and characteristic, a positive proportion of words \(w\) give a distribution which approaches uniformity in the limit as \(|G|\to\infty\). In this paper, we show that the proportion is in fact \(1\).Free products and AQFThttps://zbmath.org/1530.811182024-04-15T15:10:58.286558Z"Tanimoto, Yoh"https://zbmath.org/authors/?q=ai:tanimoto.yohSummary: We review the free product construction of von Neumann algebras, its application to a question in Algebraic Quantum Field Theory (AQFT) and an application of AQFT techniques to a question of free products. We show the existence of half-sided modular inclusions with trivial relative commutant and nontriviality of relative commutant for an inclusion of free product von Neumann algebras.
For the entire collection see [Zbl 1492.47001].Protecting the most significant bits in scalar multiplication algorithmshttps://zbmath.org/1530.940182024-04-15T15:10:58.286558Z"Alpirez Bock, Estuardo"https://zbmath.org/authors/?q=ai:alpirez-bock.estuardo"Chmielewski, Lukasz"https://zbmath.org/authors/?q=ai:chmielewski.lukasz"Miteloudi, Konstantina"https://zbmath.org/authors/?q=ai:miteloudi.konstantinaSummary: The Montgomery Ladder is widely used for implementing the scalar multiplication in elliptic curve cryptographic designs. This algorithm is efficient and provides a natural robustness against (simple) side-channel attacks. Previous works however showed that implementations of the Montgomery Ladder using Lopez-Dahab projective coordinates easily leak the value of the most significant bits of the secret scalar, which led to a full key recovery in an attack known as \textit{LadderLeak} [\textit{D. F. Aranha} et al., in: CCS '20: Proceedings of the 2020 ACM SIGSAC conference on computer and communications security, October 2020. New York, NY: Association for Computing Machinery (ACM). 225--242 (2020; \url{doi.org/10.1145/3372297.3417268})]. In light of such leakage, we analyse further popular methods for implementing the Montgomery Ladder. We first consider open source software implementations of the X25519 protocol which implement the Montgomery Ladder based on the ladderstep algorithm from \textit{M. Düll} et al. [Des. Codes Cryptography 77, No. 2--3, 493--514 (2015; Zbl 1327.94042)]. We confirm via power measurements that these implementations also easily leak the most significant scalar bits, even when implementing Z-coordinate randomisations. We thus propose simple modifications of the algorithm and its handling of the most significant bits and show the effectiveness of our modifications via experimental results. Particularly, our re-designs of the algorithm do not incurring significant efficiency penalties. As a second case study, we consider open source hardware implementations of the Montgomery Ladder based on the complete addition formulas for prime order elliptic curves, where we observe the exact same leakage. As we explain, the most significant bits in implementations of the complete addition formulas can be protected in an analogous way as we do for Curve25519 in our first case study.
For the entire collection see [Zbl 1516.68007].