Recent zbMATH articles in MSC 14Ghttps://www.zbmath.org/atom/cc/14G2021-04-16T16:22:00+00:00WerkzeugRational points on Shimura curves and the Manin obstruction.https://www.zbmath.org/1456.111092021-04-16T16:22:00+00:00"Arai, Keisuke"https://www.zbmath.org/authors/?q=ai:arai.keisukeSummary: In a previous article, we proved that Shimura curves have no points rational over number fields under a certain assumption. In this article, we give another criterion of the nonexistence of rational points on Shimura curves and obtain new counterexamples to the Hasse principle for Shimura curves. We also prove that such counterexamples obtained from the above results are accounted for by the Manin obstruction.Rational points in their fiber: complements to a theorem of Poonen.https://www.zbmath.org/1456.140282021-04-16T16:22:00+00:00"Moret-Bailly, Laurent"https://www.zbmath.org/authors/?q=ai:moret-bailly.laurentSummary: Let \(f:X\rightarrow S\) be a morphism of integral schemes, which is surjective and of finite presentation. When \(S\) is of finite type over a field or over \(\mathbb{Z}\), of positive dimension, \textit{B. Poonen} [J. Algebra 243, No. 1, 224--227 (2001; Zbl 1077.14505)] has shown that there is a point \(x\in X\) whose residue field \(\kappa (x)\) is purely inseparable over \(\kappa (f(x))\). In this paper, we extend this result in several ways. First we prove that we can take \(x\) such that \(f(x)\) is a codimension 1 point of \(S\); if \(S\) is smooth over a field \(k\), we can require \(\kappa (f(x))\) to be separable over \(k\). In another direction, we prove that similar results hold for other schemes \(S\), such as noetherian integral schemes of dimension \(\geq 2\).A note on the Brauer group and the Brauer-Manin set of a product.https://www.zbmath.org/1456.140252021-04-16T16:22:00+00:00"Lv, Chang"https://www.zbmath.org/authors/?q=ai:lv.changThe article under review examines Brauer groups and Brauer-Manin sets for products \(X \times Y\) of varieties over number fields. Consider the natural map
\[\Phi: \text{Br} (X) \oplus \text{Br} (Y) \rightarrow \text{Br} (X \times Y). \]
\textit{A. N. Skorobogatov} and \textit{Y. G. Zarhin} [J. Eur. Math. Soc. (JEMS) 16, No. 4, 749--769 (2014; Zbl 1295.14021)] proved that the cokernel of \(\Phi\) is finite provided that \(X\) and \(Y\) are smooth, projective, geometrically integral varieties. In the paper under review, the author generalizes this statement to the case when \(\left( X \times Y \right)(k) \neq 0 \) or \(H^3 \left( k, \overline{k}^\times\right) =0\). The proof relies on the result in the projective case and a comparison of cohomology groups of the varieties \(X\) and \(Y\) and their smooth compactifications.
Furthermore, the author proves that if \(X\) and \(Y\) are smooth, geometrically integral varieties over a number field, then the Brauer-Manin set
\(\left( X \times Y \right) \left( \mathbb{A}_k\right)^{\text{Br}(X\times Y)}\) coincides with the product of \(X \left( \mathbb{A}_k \right)^{\text{Br}(X)}\) and \(Y \left( \mathbb{A}_k \right)^{\text{Br}(Y)}\), relaxing the projectivity requirement in a similar result in [loc. cit.]. The proof uses similar methods to the proof in the projective case.
Reviewer: Charlotte Ure (Charlottesville)LCD codes and self-orthogonal codes in generalized dihedral group algebras.https://www.zbmath.org/1456.140332021-04-16T16:22:00+00:00"Gao, Yanyan"https://www.zbmath.org/authors/?q=ai:gao.yanyan"Yue, Qin"https://www.zbmath.org/authors/?q=ai:yue.qin"Wu, Yansheng"https://www.zbmath.org/authors/?q=ai:wu.yanshengA group code is a right ideal in a group ring \(R[G]\) where \(R\) is a commutative ring and \(G\) a finite group. In this paper, the authors consider a finite field \(\mathbb{F}_q\) and a generalized dihedral group \(D_{2n,r}\), \(\gcd(2n,q)=1\). As a first main result, they explicitly describe the primitive idempotents of the group algebra \(\mathbb{F}_q[D_{2n,r}]\). Given a code \(C\), it is named LCD whenever \(C \cap C^\perp = \{0\}\) and it is self-orthogonal iff \(C \subseteq C^\perp\), where \(C^\perp\) means dual of \(C\). LCD codes have cryptographic applications and self-orthogonal codes provide quantum codes. The second main result of the paper describes and counts LCD and self-orthogonal group codes in \(\mathbb{F}_q[D_{2n,r}]\).
Reviewer: Carlos Galindo (Castellón)Algebraic geometry codes over abelian surfaces containing no absolutely irreducible curves of low genus.https://www.zbmath.org/1456.140562021-04-16T16:22:00+00:00"Aubry, Yves"https://www.zbmath.org/authors/?q=ai:aubry.yves"Berardini, Elena"https://www.zbmath.org/authors/?q=ai:berardini.elena"Herbaut, Fabien"https://www.zbmath.org/authors/?q=ai:herbaut.fabien"Perret, Marc"https://www.zbmath.org/authors/?q=ai:perret.marcIn this paper, a theoretical study of algebrao-geometric codes constructed from abelian surfaces \(A\) defined over finite fields \(F_q\) is provided. Let \(m = \lfloor 2 \sqrt{q}\rfloor\), \(H\) be an ample divisor on \(A\) rational over \(F_q\) and \(r\) be a positive integer such that \(rH\) is very ample. We denote by \(Tr(A)\) the trace of \(A\) and by \(C(A, rH)\) the generalised evaluation code. Then, it is proved that the distance \(d(A, rH)\) of the code \(C(A, rH)\) satisfies the inequality: \[d(A, rH) \geq \# A(F_q) - rH^2(q + 1 -Tr(A) + m) - r^2m \frac{H^2}{2}.\] Furthermore, if \(A\) is simple and contains no absolutely irreducible curves of arithmetic genus \(\leq \ell\), then \[d(A, rH) \geq \# A(F_q)-\max\left( r\left\lfloor \sqrt{\frac{H^2}{2}}\right \rfloor (\ell-1), \varphi(1), \varphi\left(\left\lfloor r \sqrt{\frac{H^2}{2 \ell}}\right\rfloor \right) \right), \] where \[ \varphi(x) := m \left( r \sqrt{\frac{H^2}{2}} - x\sqrt{\ell} \right)^2 +2m\ell \left( r \sqrt{\frac{H^2}{2}} - x\sqrt{\ell} \right)+\] \[ x\Big(q+1-Tr(A)+(\ell-1)(m-\sqrt{\ell})\Big)+r\sqrt{\frac{H^2}{2}}\ (\ell-1).\] The proof of this result is relied on a characterisation of isogeny classes of abelian surfaces containing Jacobians of curves of genus 2 obtained by \textit{E. W. Howe} et al. [Ann. Inst. Fourier 59, No. 1, 239--289 (2009; Zbl 1236.11058)].
Reviewer: Dimitros Poulakis (Thessaloniki)On the bielliptic and bihyperelliptic loci.https://www.zbmath.org/1456.140302021-04-16T16:22:00+00:00"Frediani, Paola"https://www.zbmath.org/authors/?q=ai:frediani.paola"Porru, Paola"https://www.zbmath.org/authors/?q=ai:porru.paolaLet \(\mathcal {A}_g\) be the moduli space of \(g\)-dimensional principallly polarized Abelian varieties and \(j: \mathcal {M}_g\to \mathcal {A}_g\) the Torelli map. \(\mathcal{A}_g\) has a metric, the Siegel metric, and a conjecture of Coleman and Oort is related to totally geodesic subvarieties for the Siegel metric. \(\mathcal {M}_g\) get the metric and the second Gaussian map may be used to detect its totally geodesic subvarieties [\textit{F. Colombo} and \textit{P. Frediani}, Mich. Math. J. 58, No. 3, 745--758 (2009; Zbl 1191.14030); Trans. Amer. Math. Soc. 362, No. 3, 745--758 (2010; Zbl. 1191.14030); \textit{E. Colombo} et al., Int. J. Math. 26, No. 1, Article ID 15500005 (2015; Zbl. 1312.14076)]). In this paper the authors prove that for \(g\ge 4\) the locus of bielliptic curves is not totally geodesic in \(\mathcal {A}_f\), contrary to the case \(g=3\), and that it is not totally geodesic the locus \(\mathcal {BH}_{g,g'}\) of double coverings of genus \(g\) of hyperelliptic curves of genus \(g'\) (double hyperelliptic curves) for \(g \ge 3g'\ge 6\).
Reviewer: Edoardo Ballico (Povo)On the work of Peter Scholze.https://www.zbmath.org/1456.140012021-04-16T16:22:00+00:00"Wedhorn, T."https://www.zbmath.org/authors/?q=ai:wedhorn.torstenThe article surveys the mathematical work of Peter Scholze giving an exposition suitable for non-experts. So, no previous knowledge of the subjects touched by the paper is necessary but for basic knowledge of algebraic geometry. Because of the vastness of the topics of Scholze's research, it is impossible to introduce all of them in detail in a short paper, therefore only basic definitions are recalled and more advanced notions are described informally and references are given to specialized literature for details.
The paper is divided into five parts. The first part surveys some basic notions of arithmetic geometry mainly related to the interplay between geometry over fields of characteristic zero and fields over characteristic \(p\), for \(p\) a prime number. The second part describes how the theory of perfectoid spaces, introduced by Peter Scholze, permits a novel way of passing between characteristic \(0\) and \(p\). The main applications of the theory of perfectoid spaces are also discussed: the proof of new important cases of the weight monodromy conjecture, obtained by Scholze himself, and the full proof of Hochster's conjecture obtained by Yves Anré using perfectoid spaces. The third part deals with the work of Scholze on \(p\)-adic Hodge theory until the most recent developments of prismatic cohomology. The fourth part is about the work of Scholze on the Langlands program. This part is the one that relies more on references to the literature as the Langlands program is such an intricate web of ideas, most of which of high technical nature, that it is impossible to summarize them in few pages. So, the author bounds the discussion in describing the main ideas of Scholze on the topic giving precise references for those who want to understand more. The last part briefly describes other topics of Scholze's research: topological Hochschild homology and the new topic of condensed mathematics.
Reviewer: Federico Bambozzi (Oxford)On the scalar complexity of Chudnovsky\(^2\) multiplication algorithm in finite fields.https://www.zbmath.org/1456.112352021-04-16T16:22:00+00:00"Ballet, Stéphane"https://www.zbmath.org/authors/?q=ai:ballet.stephane"Bonnecaze, Alexis"https://www.zbmath.org/authors/?q=ai:bonnecaze.alexis"Dang, Thanh-Hung"https://www.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)Arithmetic Chern-Simons theory. I.https://www.zbmath.org/1456.140292021-04-16T16:22:00+00:00"Kim, Minhyong"https://www.zbmath.org/authors/?q=ai:kim.minhyongSummary: In this paper, we apply ideas of \textit{R. Dijkgraaf} and \textit{E. Witten} [Commun. Math. Phys. 129, No. 2, 393--429 (1990; Zbl 0703.58011)] and \textit{E. Witten} [Commun. Math. Phys. 121, No. 3, 351--399 (1989; Zbl 0667.57005)] on \(2+1\) dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fields. In the first three sections, we define classical Chern-Simons functionals on spaces of Galois representations. In the highly speculative Sect. 6, we consider the far-fetched possibility of using Chern-Simons theory to construct \(L\)-functions.
For the entire collection see [Zbl 07237934].Second descent and rational points on Kummer varieties.https://www.zbmath.org/1456.111072021-04-16T16:22:00+00:00"Harpaz, Yonatan"https://www.zbmath.org/authors/?q=ai:harpaz.yonatanSummary: A powerful method, pioneered by Swinnerton-Dyer, allows one to study rational points on pencils of curves of genus 1 by combining the fibration method with a sophisticated form of descent. A variant of this method, first used by \textit{A. Skorobogatov} and \textit{P. Swinnerton-Dyer} in [Adv. Math. 198, No. 2, 448--483 (2005; Zbl 1085.14021)], can be applied to the study of rational points on Kummer varieties. In this paper we extend the method to include an additional step of second descent. Assuming finiteness of the relevant Tate-Shafarevich groups, we use the extended method to show that the Brauer-Manin obstruction is the only obstruction to the Hasse principle on Kummer varieties associated to abelian varieties with all rational 2-torsion, under relatively mild additional hypotheses.Representability theorem in derived analytic geometry.https://www.zbmath.org/1456.140182021-04-16T16:22:00+00:00"Porta, Mauro"https://www.zbmath.org/authors/?q=ai:porta.mauro"Yu, Tony Yue"https://www.zbmath.org/authors/?q=ai:yu.tony-yueIn the paper under review, the authors prove a representability theorem in derived analytic geometry, analogous to
Lurie's generalization of Artin's representablility criteria to derived algebraic geometry.
This is an important, standard type result for the study of moduli problems and
a crucial step towards a solid theory of derived analytic geometry.
More specifically, the authors show that a derived stack for the étale site of derived analytic spaces is a derived analytic stack
if and only if
it is compatible with Postnikov towers, has a global analytic cotangent complex, and its truncation is an analytic stack in the classical
(underived) sense.
The result applies both to complex analytic geometry and non-archimedean analytic geometry.
Central to representability results as in the present paper is deformation theory, which the authors develop here for the derived analytic setup.
The authors define an analytic version of the cotangent complex which controls the deformation theory of the derived stack.
As in the algebraic setting, the cotangent complex represents a functor of derivations.
One key step in order to define the analytic cotangent complex is the elegant description of the \(\infty\)-category of modules over a
derived analytic space \(X\) as the \(\infty\)-category of spectrum objects of a certain \(\infty\)-category associated with \(X\).
Another important construction is the analytification functor which they establish in the derived setting.
To apply derived geometry to classical moduli problems, one may try to enrich classical moduli spaces with derived structures. The paper under review is an important tool in verifying when such enrichments are indeed the correct ones.
Reviewer: Eric Ahlqvist (Stockholm)Intersections between the norm-trace curve and some low degree curves.https://www.zbmath.org/1456.140322021-04-16T16:22:00+00:00"Bonini, Matteo"https://www.zbmath.org/authors/?q=ai:bonini.matteo"Sala, Massimiliano"https://www.zbmath.org/authors/?q=ai:sala.massimilianoThe affine plane curve \(\mathcal{X}\) defined over the finite field \(\mathbb{F}_{q^r}\) by the equation \[x^{\frac{q^r-1}{q-1}}= y^{q^{r-1}}+y^{q^{r-2}}+\cdots +y^q+y\] is called {\em norm-trace curve}. The norm-trace curve is a natural generalization of the Hermitian curve to any extension field \(\mathbb{F}_{q^r}\). The determination of the intersection of a given curve \(\mathcal{X}\) and low degree curves is often useful for the determination of the weight distribution of the AG code arising from \(\mathcal{X}\).
In this paper, the intersection of \(\mathcal{X}\) with cubics defined by equations of the form \(y = ax^3 + bx^2 + cx + d\), where \(a, b, c, d \in \mathbb{F}_{q^3}\), is studied. Especially, the intersection between the norm-trace curve and parabolas is characterised, and tools are provided in order to get sharp bounds for the number of intersections in the other cases. For this purpose specific irreducible surfaces over finite fields are studied. Furthermore, the weight distribution of the corresponding one-point codes are obtained.
Reviewer: Dimitros Poulakis (Thessaloniki)Arithmetic Teichmuller theory.https://www.zbmath.org/1456.110952021-04-16T16:22:00+00:00"Rastegar, Arash"https://www.zbmath.org/authors/?q=ai:rastegar.arashSummary: By Grothendieck's anabelian conjectures, Galois representations landing in outer automorphism group of the algebraic fundamental group which are associated to hyperbolic smooth curves defined over number-fields encode all the arithmetic information of these curves. The goal of this paper is to develop an arithmetic Teichmuller theory, by which we mean, introducing arithmetic objects summarizing the arithmetic information coming from all curves of the same topological type defined over number-fields. We also introduce the Hecke-Teichmüller Lie algebra which plays the role of Hecke algebra in the anabelian framework.On the \(\ell\)-adic cohomology of some \(p\)-adically uniformized Shimura varieties.https://www.zbmath.org/1456.111122021-04-16T16:22:00+00:00"Shen, Xu"https://www.zbmath.org/authors/?q=ai:shen.xuIn the present paper, the author determines the Galois
representations inside the \(\ell\)-adic cohomology of some
unitary Shimura varieties at split places where they admit
uniformization by finite products of Drinfeld upper half
spaces. His results confirm Langlands-Kottwitz's description
of the cohomology of Shimura varieties in new cases. In fact,
\textit{P. Scholze} [J. Am. Math. Soc. 26, No. 1, 227--259 (2013; Zbl 1383.11082)] has developed the Langlands-Kottwitz approach for some
PEL Shimura varieties with arbitrary level at \(p\). The key new
ingredient is to define some test functions by deformation spaces
of \(p\)-divisible groups with some additional structures. The author
defines the test functions by Scholze's method and proves the
vanishing property of these test functions by global method. He
establishes the trace formula as a sum over the set of equivalent
Kottwitz triples and deduces the character identity of the transfers
of test functions. In the end, he gives the local semi-simple zeta
functions of Shimura varieties.
Reviewer: Lei Yang (Beijing)Modular polynomials on Hilbert surfaces.https://www.zbmath.org/1456.110782021-04-16T16:22:00+00:00"Milio, Enea"https://www.zbmath.org/authors/?q=ai:milio.enea"Robert, Damien"https://www.zbmath.org/authors/?q=ai:robert.damienSummary: We describe an evaluation/interpolation approach to compute modular polynomials on a Hilbert surface, which parametrizes abelian surfaces with maximal real multiplication. Under some heuristics we obtain a quasi-linear algorithm. The corresponding modular polynomials are much smaller than the ones on the Siegel threefold. We explain how to compute even smaller polynomials by using pullbacks of theta functions to the Hilbert surface.On the elliptic Stark conjecture in higher weight.https://www.zbmath.org/1456.111202021-04-16T16:22:00+00:00"Gatti, Francesca"https://www.zbmath.org/authors/?q=ai:gatti.francesca"Guitart, Xavier"https://www.zbmath.org/authors/?q=ai:guitart.xavierSummary: We study the special values of the triple product \(p\)-adic \(L\)-function constructed by \textit{H. Darmon} and \textit{V. Rotger} [Ann. Sci. Éc. Norm. Supér. (4) 47, No. 4, 779--832 (2014; Zbl 1356.11039)] at all classical points outside the region of interpolation. We propose conjectural formulas for these values that can be seen as extending the Elliptic Stark Conjecture, and we provide theoretical evidence for them by proving some particular cases.On the degeneracy of integral points and entire curves in the complement of nef effective divisors.https://www.zbmath.org/1456.111192021-04-16T16:22:00+00:00"Heier, Gordon"https://www.zbmath.org/authors/?q=ai:heier.gordon"Levin, Aaron"https://www.zbmath.org/authors/?q=ai:levin.aaronSummary: As a consequence of the divisorial case of our recently established generalization of Schmidt's subspace theorem, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel aspect of our result is the attainment of a strong degeneracy conclusion (arithmetic quasi-hyperbolicity) under weak positivity assumptions on the divisors. The proof hinges on applying our recent theorem with a well-situated ample divisor realizing a certain lexicographical minimax. We also explore the connections with earlier work by other authors and make a conjecture regarding bounds for the numbers of divisors necessary, including consideration of the question of arithmetic hyperbolicity. Under the standard correspondence between statements in Diophantine approximation and Nevanlinna theory, one obtains analogous degeneration statements for entire curves.Some explicit computations in Arakelov geometry of abelian varieties.https://www.zbmath.org/1456.111062021-04-16T16:22:00+00:00"Gaudron, Éric"https://www.zbmath.org/authors/?q=ai:gaudron.ericSummary: Given a polarized complex abelian variety \((\mathsf{A}, \mathsf{L})\), a Gromov lemma makes a comparison between the sup and \(L^2\) norms of a global section of \(\mathsf{L}\). We give here an explicit bound which depends on the dimension, degree and injectivity diameter of \((\mathsf{A}, \mathsf{L})\). It rests on a more general estimate for the jet of a global section of \(\mathsf{L}\). As an application we deduce some estimates of the maximal slope of the tangent and cotangent spaces of a polarized abelian variety defined over a number field. These results are effective versions of previous works by Masser and Wüstholz on one hand and
\textit{J. B. Bost} [Duke Math. J. 82, No. 1, 21--70 (1996; Zbl 0867.14010)] on the other. They also improve some similar statements established by \textit{P. Graftieaux} in [Duke Math. J. 106, No. 1, 81--121 (2001; Zbl 1064.14045)].The finite subgroups of \(\mathrm{SL}(3,\overline{F})\).https://www.zbmath.org/1456.200562021-04-16T16:22:00+00:00"Flicker, Yuval Z."https://www.zbmath.org/authors/?q=ai:flicker.yuval-zThe \textit{special linear group} \(\mathrm{SL}(n,F)\) of degree \(n\) over a field \(F\) is the set of all \(n\times n\) matrices with determinant \(1\), with the group operations of ordinary matrix multiplication and matrix inversion. In the paper under review, the author gives a complete exposition of the finite subgroups of \(\mathrm{SL}(3,\bar{F})\), where \(\bar{F}\) is a separably closed field of characteristic not dividing the order of the finite subgroup. This completes the earlier work of \textit{H. F. Blichfeldt} [Math. Ann. 63, 552--572 (1907; JFM 38.0192.03)]. Several elementary examples are illustrated by the author in the exposition.
Reviewer: Mahadi Ddamulira (Kampala)Arithmetic diagonal cycles on unitary Shimura varieties.https://www.zbmath.org/1456.140312021-04-16T16:22:00+00:00"Rapoport, M."https://www.zbmath.org/authors/?q=ai:rapoport.michael"Smithling, B."https://www.zbmath.org/authors/?q=ai:smithling.brian-d"Zhang, W."https://www.zbmath.org/authors/?q=ai:zhang.wei.1The paper under review has three main objectives, clearly and explicitly stated by the authors in its introduction. The first one of these, achieved in Section 6, is to give a detailed and explicit variant of the arithmetic Gan-Gross-Prasad conjecture for Shimura varieties of PEL type associated to certain unitary groups. This conjecture concerns the order of vanishing at the point \(s = 1/2\) of the \(L\)-functions \(L(s,\pi,R)\) associated to certain automorphic representations \(\pi\) of the adelic points of an algebraic group \(\widetilde{HG}\) defined over \(\mathbb{Q}\), which is introduced in Section 2.1. More precisely, Conjecture 6.10 of the paper under review relates the aforementioned order of vanishing to the dimension of the module of Hecke-equivariant homomorphisms between the \(K\)-invariant finite part \(\pi_f^K\) of the automorphic representation \(\pi\), where \(K \subseteq \widetilde{HG}(\mathbb{A}_f)\) is some open and compact subgroup of the finite-adelic points of \(\widetilde{HG}\), and the cyclic Hecke module \(\mathcal{Z}_{K,0} \subseteq \mathrm{CH}^{n - 1}(M_K(\widetilde{HG}))_{\mathbb{Q},0}\) generated by a suitable cohomologically trivial algebraic cycle constructed on the canonical model \(M_K(\widetilde{HG})\) of the Shimura variety \(\mathrm{Sh}_K(\widetilde{HG})\) associated to \(K\) and \(\widetilde{HG}\). Note that \(\dim(M_K(\widetilde{HG})) = 2 n - 3\), so that the module \(\mathcal{Z}_{K,0}\) is defined by an algebraic cycle of degree just above the middle dimension of the Shimura variety \(M_K(\widetilde{HG})\). Moreover, Conjecture 6.12 predicts that \(L(s,\pi,R)\) vanishes at \(s = 1/2\) with order exactly one if and only if a suitable linear form \(\ell_K\), defined on the Chow group \(\mathrm{CH}^{n - 1}(M_K(\widetilde{HG}))_{\mathbb{Q},0}\), is non-trivial when restricted to the \(\pi_f^K\)-isotypic component of \(\mathcal{Z}_{K,0}\). Note that the linear form \(\ell_K\) is defined using the Beilinson-Bloch height pairing for algebraic cycles, whose construction is recalled in Section 6.1 of the paper under review, and is conditional on certain conjectures concerning the existence of suitable integral models (Conjecture 6.1) and the possibility to lift algebraic cycles from the generic fibre of these integral models (Conjecture 6.2). However, as the authors observe in Remark 6.18, the Beilinson-Bloch height pairing is unconditionally defined when \(n = 2\), and it coincides with the Néron-Tate height pairing. In particular, the arithmetic Gan-Gross-Prasad conjecture should be closely related in this case to the \textit{Gross-Zagier formula on Shimura curves}, studied in the foundational book [\textit{X. Yuan} et al., The Gross-Zagier formula on Shimura curves. Princeton, NJ: Princeton University Press (2013; Zbl 1272.11082)].
The second goal of the paper under review, achieved in Section 8, is to formulate a global analogue of the \textit{arithmetic fundamental lemma} and \textit{arithmetic transfer} conjectures, which are local conjectures proposed respectively by the third author of the paper under review [\textit{W. Zhang}, Invent. Math. 188, No. 1, 197--252 (2012; Zbl 1247.14031)] and by the same authors of the paper under review [\textit{M. Rapoport} et al., Math. Ann. 370, No. 3--4, 1079--1175 (2018; Zbl 1408.14143)]. The advantage of these new conjectures over the arithmetic Gan-Gross-Prasad conjecture is that they bypass the use of the Beilinson-Bloch height pairing, which is only conjecturally defined, by working with specific integral models of the Shimura varieties in question, and using the arithmetic intersection pairing defined in [\textit{H. Gillet} and \textit{C. Soulé}, Publ. Math., Inst. Hautes Étud. Sci. 72, 93--174 (1990; Zbl 0741.14012)]. We note that Theorem 8.14 of the paper under review reduces the global Conjectures 8.2 and 8.8 to the semi-global Conjecture 8.13. Moreover, Theorem 8.15 of the paper under review proves that half of Conjecture 8.13 holds true if \(n \leq 3\), where again \(n := (\dim(M_K(\widetilde{HG})) + 3)/2\).
The third main goal of the paper under review, achieved in Sections 4 and 5, is precisely to define the integral model of the Shimura variety \(M_K(\widetilde{HG})\) which appears in the aforementioned global and semi-global conjectures. Note that \(M_K(\widetilde{HG})\) is associated to a suitable moduli problem for abelian varieties with level structure, whose definition is spelled out in Section 3.2. Moreover, \(M_K(\widetilde{HG})\) is closely related to the Shimura varieties considered in [\textit{R. E. Kottwitz}, J. Am. Math. Soc. 5, No. 2, 373--444 (1992; Zbl 0796.14014)] and [\textit{M. Harris} and \textit{R. Taylor}, The geometry and cohomology of some simple Shimura varieties. With an appendix by Vladimir G. Berkovich. Princeton, NJ: Princeton University Press (2001; Zbl 1036.11027)].
To conclude, the paper under review deepens considerably our understanding of the arithmetic Gan-Gross-Prasad conjectures for Shimura varieties associated to unitary groups, and makes clear the links between the local and global pictures. The notation and terminologies employed in the paper under review can easily seem overwhelmingly tecnical, as it is often the case in this research area. However, the authors make a great effort of clarity, and the most important notations are summarized when needed.
Reviewer: Riccardo Pengo (Lyon)A hypergeometric version of the modularity of rigid Calabi-Yau manifolds.https://www.zbmath.org/1456.110732021-04-16T16:22:00+00:00"Zudilin, Wadim"https://www.zbmath.org/authors/?q=ai:zudilin.wadimThis paper considers the fourteen one-parameter families of Calabi-Yau
threefolds whose periods are expressed in terms of hypergeometric functions.
For these fourteen families, periods are solutions of hypergeometric equations
with parameter \((r, 1-r, t, 1-t)\), where
\begin{multline*}
(r,t)=\Big(\frac{1}{2},\frac{1}{2}\Big),\Big(\frac{1}{2},\frac{1}{3}\Big),\Big(\frac{1}{2},\frac{1}{4}\Big),
\Big(\frac{1}{2},\frac{1}{6}\Big),\Big(\frac{1}{3}\Big),\Big(\frac{1}{3},\frac{1}{4}\Big),\Big(\frac{1}{3},\frac{1}{6}\Big),\\
\Big(\frac{1}{4},\frac{1}{4}\Big),\Big(\frac{1}{4},\frac{1}{6}\Big),\Big(\frac{1}{6},\frac{1}{6}\Big),\Big(\frac{1}{5},\frac{2}{5}\Big),
\Big(\frac{1}{8},\frac{3}{8}\Big),\Big(\frac{1}{10},\frac{3}{10}\Big),\Big(\frac{1}{12},\frac{5}{12}\Big).
\end{multline*}
At a conifold point, any of these Calabi-Yau threefolds becomes rigid,
and the \(p\)-th coefficient \(a(p)\) of the corresponding modular form of weight \(4\)
can be recovered from the truncated partial sums of the corresponding
hypergeometric series modulo a higher power of \(p\), where \(p\) is any good prime \(>5\).
This paper discusses relationships between the critical values of the \(L\)-series of the modular form
and the values of a related basis of solutions to the hypergeometric differential equation.
It is numerically observed that the critical \(L\)-values are \(\mathbb{Q}\)-proportional to the
hypergeometric values \(F_1(1), F_2(1), F_3(1)\), where \(F_j(z)\) are solutions of the hypergeometric
equation for the hypergeometric function \(F_0(z)=_4F_3(z)\) with parameters \((r, 1-r, t, 1-t)\).
This confirms the prediction of Golyshev concerning gamma structures [\textit{V. Golyshev} and \textit{A. Mellit}, J. Geom. Phys. 78, 12--18 (2014; Zbl 1284.33001)].
Reviewer: Noriko Yui (Kingston)Counting Richelot isogenies between superspecial abelian surfaces.https://www.zbmath.org/1456.140552021-04-16T16:22:00+00:00"Katsura, Toshiyuki"https://www.zbmath.org/authors/?q=ai:katsura.toshiyuki"Takashima, Katsuyuki"https://www.zbmath.org/authors/?q=ai:takashima.katsuyukiSummary: \textit{W. Castryck} et al. [``Hash functions from superspecial genus-2 curves using Richelot isogenies'', J. Math. Crypt. 14, 268--292 (2020; \url{doi:10.1515/jmc-2019-0021})] used superspecial genus-2 curves and their Richelot isogeny graph for basing genus-2 isogeny cryptography, and recently, \textit{C. Costello} and \textit{B. Smith} [``The supersingular isogeny problem in genus 2 and beyond'', in: Post-quantum cryptography. Cham: Springer. 151--168 (2020)], devised an improved isogeny path-finding algorithm in the genus-2 setting. In order to establish a firm ground for the cryptographic construction and analysis, we give a new characterization of decomposed Richelot isogenies in terms of involutive reduced automorphisms of genus-2 curves over a finite field, and explicitly count such decomposed (and nondecomposed) Richelot isogenies between superspecial principally polarized abelian surfaces. As a corollary, we give another algebraic geometric proof of Theorem 2 in the paper of Castryck et al. [loc. cit.].
For the entire collection see [Zbl 1452.11005].Globally analytic \(p\)-adic representations of the pro-\(p\) Iwahori subgroup of \(\mathrm{GL}(2)\) and base change. II: A Steinberg tensor product theorem.https://www.zbmath.org/1456.112102021-04-16T16:22:00+00:00"Clozel, Laurent"https://www.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)Ordinary K3 surfaces over a finite field.https://www.zbmath.org/1456.140442021-04-16T16:22:00+00:00"Taelman, Lenny"https://www.zbmath.org/authors/?q=ai:taelman.lennyThe author, building on [\textit{N. O. Nygaard}, Progr. Math. No 35, p. 267--276 (1983; Zbl 0574.14031)] and [\textit{J.-D. Yu} Pure Appl. Math. Q. 8, No. 3, 805--824 (2012; Zbl 1252.14008)], studies the category of ordinary \(K3\) surfaces over a finite field.
A \(K3\) surface \(X\) over a finite field \(k\) of characteristic \(p>0\) is called \emph{ordinary} if \(| X(k) | \not\equiv 1 \bmod p\). Nygaard and Yu already exhibited the existence of a fully faithful functor between the groupoids of
\begin{itemize}
\item[1.] the ordinary \(K3\) surfaces over a finite field \(\mathbb{F}_q\) and
\item[2.] the triples \((M, F, \mathcal{K})\), consisting of
\begin{itemize}
\item[(a)] an integral lattice \(M\),
\item[(b)] an endomorphism \(F\) of \(M\), and
\item[(c)] a convex subset of \(\mathcal{K} \subset \mathbb{R}\otimes M\)
\end{itemize}
satisfying certain conditions.
\end{itemize}
The main result of the paper under review consists in studying the image of this functor and, in particular, in showing that if every \(K3\) surface over the fraction field of the ring of Witt vectors \(W(\mathbb{F}_q)\) satisfies a strong form of ``potential semi-stable reduction'', then the functor is essentially surjective. Finally, using this property, the author describes three sub-grupoids on which the functor restricts to an equivilance.
Reviewer: Dino Festi (Mainz)Evaluation codes and their basic parameters.https://www.zbmath.org/1456.140342021-04-16T16:22:00+00:00"Jaramillo, Delio"https://www.zbmath.org/authors/?q=ai:jaramillo.delio"Vaz Pinto, Maria"https://www.zbmath.org/authors/?q=ai:vaz-pinto.maria"Villarreal, Rafael H."https://www.zbmath.org/authors/?q=ai:villarreal.rafael-heraclioLet \(K=GF(q)\) denote the finite field with \(q\) elements, where \(q\) is a
prime power, and let \(S=K[t_1,\dots,t_s]\) be the polynomial ring associated
with the affine space \({\mathbb{A}}^s\).
If \(X=\{P_1,\dots,P_m\}\subset {\mathbb{A}}^s\) is a
finite set of points then we can define a \(K\)-linear
evaluation map
\[
mathrm{ev}_X:S\to K^m, \ \ mathrm{ev}_X(f)=(f(P_1),\dots,f(P_m)).
\]
If \(L\subset S\) is a finite dimensional space then
its image under \(mathrm{ev}_X\), \(L_X=mathrm{ev}_X(L)\), is called an
\textit{evaluation code} on \(X\).
In the paper under review, the authors study the basic parameters of the
family of evaluation codes and those of certain interesting
subfamilies. The parameters investigated are
(a) the length \(|X|\),
(b) the dimension \(\dim_K(L_X)\),
(c) the generalized Hamming weights \(\delta_r(L_X)\), \(1\leq r \leq \dim_K(L_X)\).
The first main result of the authors is a formula for
\(\delta_r(L_X)\) in terms of ring-theoretic data involving
\(S\), the vanishing ideal of \(X\) and related ideals. The second
main result is a lower bound for the
\(\delta_r(L_X)\) in terms of what they call the footprint of \(L_X\), a
quantity expressed in terms of ring-theoretic data.
These two results are applied to
toric codes and to ``squarefree'' evaluation codes,
when \(X\) is taken to be the set of all \(K\)-rational
points of the affine torus.
For those families of codes, the authors obtain explicit formulas for both the
minimum distance (namely, \(\delta_1(L_X)\)) and the second
generalized Hamming weight \(\delta_2(L_X)\).
The exact statements are too technical to state in this review,
so the reader is referred to the paper for details.
Reviewer: David Joyner (Annapolis)