Algebra & Number Theory Articles (Project Euclid)
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The latest articles from Algebra & Number Theory on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 19 Oct 2017 12:49 EDTThu, 19 Oct 2017 12:49 EDThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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The umbral moonshine module for the unique unimodular Niemeier root system
https://projecteuclid.org/euclid.ant/1508431771
<strong>John Duncan</strong>, <strong>Jeffrey Harvey</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 11, Number 3, 505--535.</p><p><strong>Abstract:</strong><br/>
We use canonically twisted modules for a certain super vertex operator algebra to construct the umbral moonshine module for the unique Niemeier lattice that coincides with its root sublattice. In particular, we give explicit expressions for the vector-valued mock modular forms attached to automorphisms of this lattice by umbral moonshine. We also characterize the vector-valued mock modular forms arising, in which four of Ramanujan’s fifth-order mock theta functions appear as components.
</p>projecteuclid.org/euclid.ant/1508431771_20171019124943Thu, 19 Oct 2017 12:49 EDTA finiteness theorem for specializations of dynatomic polynomialshttps://projecteuclid.org/euclid.ant/1558144826<strong>David Krumm</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 4, 963--993.</p><p><strong>Abstract:</strong><br/>
Let [math] and [math] be indeterminates, let [math] , and for every positive integer [math] let [math] denote the [math] -th dynatomic polynomial of [math] . Let [math] be the Galois group of [math] over the function field [math] , and for [math] let [math] be the Galois group of the specialized polynomial [math] . It follows from Hilbert’s irreducibility theorem that for fixed [math] we have [math] for every [math] outside a thin set [math] . By earlier work of Morton (for [math] ) and the present author (for [math] ), it is known that [math] is infinite if [math] . In contrast, we show here that [math] is finite if [math] . As an application of this result we show that, for these values of [math] , the following holds with at most finitely many exceptions: for every [math] , more than [math] of prime numbers [math] have the property that the polynomial [math] does not have a point of period [math] in the [math] -adic field [math] .
</p>projecteuclid.org/euclid.ant/1558144826_20190517220029Fri, 17 May 2019 22:00 EDTSurjectivity of Galois representations in rational families of abelian varietieshttps://projecteuclid.org/euclid.ant/1563328820<strong>Aaron Landesman</strong>, <strong>Ashvin A. Swaminathan</strong>, <strong>James Tao</strong>, <strong>Yujie Xu</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 5, 995--1038.</p><p><strong>Abstract:</strong><br/>
In this article, we show that for any nonisotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density- [math] subset. Our results can be applied to a number of interesting families of abelian varieties, such as rational families dominating the moduli of Jacobians of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we prove that for any dimension [math] , there are infinitely many abelian varieties over [math] with adelic Galois representation having image equal to all of [math] .
</p>projecteuclid.org/euclid.ant/1563328820_20190716220048Tue, 16 Jul 2019 22:00 EDTA unified and improved Chebotarev density theoremhttps://projecteuclid.org/euclid.ant/1563328821<strong>Jesse Thorner</strong>, <strong>Asif Zaman</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 5, 1039--1068.</p><p><strong>Abstract:</strong><br/>
We establish an unconditional effective Chebotarev density theorem that improves uniformly over the well-known result of Lagarias and Odlyzko. As a consequence, we give a new asymptotic form of the Chebotarev density theorem that can count much smaller primes with arbitrary log-power savings, even in the case where a Landau–Siegel zero is present. Our main theorem also interpolates the strongest unconditional upper bound for the least prime ideal with a given Artin symbol as well as the Chebotarev analogue of the Brun–Titchmarsh theorem proved by the authors.
</p>projecteuclid.org/euclid.ant/1563328821_20190716220048Tue, 16 Jul 2019 22:00 EDTOn the Brauer–Siegel ratio for abelian varieties over function fieldshttps://projecteuclid.org/euclid.ant/1563328822<strong>Douglas Ulmer</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 5, 1069--1120.</p><p><strong>Abstract:</strong><br/>
Hindry has proposed an analog of the classical Brauer–Siegel theorem for abelian varieties over global fields. Roughly speaking, it says that the product of the regulator of the Mordell–Weil group and the order of the Tate–Shafarevich group should have size comparable to the exponential differential height. Hindry–Pacheco and Griffon have proved this for certain families of elliptic curves over function fields using analytic techniques. Our goal in this work is to prove similar results by more algebraic arguments, namely by a direct approach to the Tate–Shafarevich group and the regulator. We recover the results of Hindry–Pacheco and Griffon and extend them to new families, including families of higher-dimensional abelian varieties.
</p>projecteuclid.org/euclid.ant/1563328822_20190716220048Tue, 16 Jul 2019 22:00 EDTA five-term exact sequence for Kac cohomologyhttps://projecteuclid.org/euclid.ant/1563328823<strong>César Galindo</strong>, <strong>Yiby Morales</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 5, 1121--1144.</p><p><strong>Abstract:</strong><br/>
We use relative group cohomologies to compute the Kac cohomology of matched pairs of finite groups. This cohomology naturally appears in the theory of abelian extensions of finite dimensional Hopf algebras. We prove that Kac cohomology can be computed using relative cohomology and relatively projective resolutions. This allows us to use other resolutions, besides the bar resolution, for computations. We compute, in terms of relative cohomology, the first two pages of a spectral sequence which converges to the Kac cohomology and its associated five-term exact sequence. Through several examples, we show the usefulness of the five-term exact sequence in computing groups of abelian extensions.
</p>projecteuclid.org/euclid.ant/1563328823_20190716220048Tue, 16 Jul 2019 22:00 EDTOn the paramodularity of typical abelian surfaceshttps://projecteuclid.org/euclid.ant/1563328827<strong>Armand Brumer</strong>, <strong>Ariel Pacetti</strong>, <strong>Cris Poor</strong>, <strong>Gonzalo Tornaría</strong>, <strong>John Voight</strong>, <strong>David S. Yuen</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 5, 1145--1195.</p><p><strong>Abstract:</strong><br/>
Generalizing the method of Faltings–Serre, we rigorously verify that certain abelian surfaces without extra endomorphisms are paramodular. To compute the required Hecke eigenvalues, we develop a method of specialization of Siegel paramodular forms to modular curves.
</p>projecteuclid.org/euclid.ant/1563328827_20190716220048Tue, 16 Jul 2019 22:00 EDTContragredient representations over local fields of positive characteristichttps://projecteuclid.org/euclid.ant/1563328828<strong>Wen-Wei Li</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 5, 1197--1242.</p><p><strong>Abstract:</strong><br/>
It was conjectured bsy Adams, Vogan and Prasad that under the local Langlands correspondence, the [math] -parameter of the contragredient representation equals that of the original representation composed with the Chevalley involution of the [math] -group. We verify a variant of their prediction for all connected reductive groups over local fields of positive characteristic, in terms of the local Langlands parametrization of A. Genestier and V. Lafforgue. We deduce this from a global result for cuspidal automorphic representations over function fields, which is in turn based on a description of the transposes of Lafforgue’s excursion operators.
</p>projecteuclid.org/euclid.ant/1563328828_20190716220048Tue, 16 Jul 2019 22:00 EDTPositivity functions for curves on algebraic varietieshttps://projecteuclid.org/euclid.ant/1566353005<strong>Brian Lehmann</strong>, <strong>Jian Xiao</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 6, 1243--1279.</p><p><strong>Abstract:</strong><br/>
This is the second part of our work on Zariski decomposition structures, where we compare two different volume type functions for curve classes. The first function is the polar transform of the volume for divisor classes. The second function captures the asymptotic geometry of curves analogously to the volume function for divisors. We prove that the two functions coincide, generalizing Zariski’s classical result for surfaces to all varieties. Our result confirms the log concavity conjecture of the first named author for weighted mobility of curve classes in an unexpected way, via Legendre–Fenchel type transforms. During the course of the proof, we obtain a refined structure theorem for the movable cone of curves.
</p>projecteuclid.org/euclid.ant/1566353005_20190820220338Tue, 20 Aug 2019 22:03 EDTThe congruence topology, Grothendieck duality and thin groupshttps://projecteuclid.org/euclid.ant/1566353008<strong>Alexander Lubotzky</strong>, <strong>Tyakal Nanjundiah Venkataramana</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 6, 1281--1298.</p><p><strong>Abstract:</strong><br/>
This paper answers a question raised by Grothendieck in 1970 on the “Grothendieck closure” of an integral linear group and proves a conjecture of the first author made in 1980. This is done by a detailed study of the congruence topology of arithmetic groups, obtaining along the way, an arithmetic analogue of a classical result of Chevalley for complex algebraic groups. As an application we also deduce a group theoretic characterization of thin subgroups of arithmetic groups.
</p>projecteuclid.org/euclid.ant/1566353008_20190820220338Tue, 20 Aug 2019 22:03 EDTOn the ramified class field theory of relative curveshttps://projecteuclid.org/euclid.ant/1566353009<strong>Quentin Guignard</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 6, 1299--1326.</p><p><strong>Abstract:</strong><br/>
We generalize Deligne’s approach to tame geometric class field theory to the case of a relative curve, with arbitrary ramification.
</p>projecteuclid.org/euclid.ant/1566353009_20190820220338Tue, 20 Aug 2019 22:03 EDTBlow-ups and class field theory for curveshttps://projecteuclid.org/euclid.ant/1566353010<strong>Daichi Takeuchi</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 6, 1327--1351.</p><p><strong>Abstract:</strong><br/>
We propose another proof of geometric class field theory for curves by considering blow-ups of symmetric products of curves.
</p>projecteuclid.org/euclid.ant/1566353010_20190820220338Tue, 20 Aug 2019 22:03 EDTAlgebraic monodromy groups of $l$-adic representations of Gal$(\overline{\mathbb{Q}} /\mathbb{Q})$https://projecteuclid.org/euclid.ant/1566353011<strong>Shiang Tang</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 6, 1353--1394.</p><p><strong>Abstract:</strong><br/>
In this paper we prove that for any connected reductive algebraic group [math] and a large enough prime [math] , there are continuous homomorphisms
Gal
(
ℚ
̄
∕
ℚ
)
→
G
(
ℚ
̄
l
)
with Zariski-dense image, in particular we produce the first such examples for [math] and [math] . To do this, we start with a mod- [math] representation of [math] related to the Weyl group of [math] and use a variation of Stefan Patrikis’ generalization of a method of Ravi Ramakrishna to deform it to characteristic zero.
</p>projecteuclid.org/euclid.ant/1566353011_20190820220338Tue, 20 Aug 2019 22:03 EDTWeyl bound for $p$-power twist of $\mathrm{GL}(2)$ $L$-functionshttps://projecteuclid.org/euclid.ant/1566353012<strong>Ritabrata Munshi</strong>, <strong>Saurabh Kumar Singh</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 6, 1395--1413.</p><p><strong>Abstract:</strong><br/>
Let [math] be a cuspidal eigenform (holomorphic or Maass) for the congruence group [math] with [math] square-free. Let [math] be a prime and let [math] be a primitive character of modulus [math] . We shall prove the Weyl-type subconvex bound
L
(
1
2
+
i
t
,
f
⊗
χ
)
≪
f
,
t
,
ε
p
r
+
ε
,
where [math] is any positive real number.
</p>projecteuclid.org/euclid.ant/1566353012_20190820220338Tue, 20 Aug 2019 22:03 EDTExamples of hypergeometric twistor $\mathcal{D}$-moduleshttps://projecteuclid.org/euclid.ant/1566353013<strong>Alberto Castaño Domínguez</strong>, <strong>Thomas Reichelt</strong>, <strong>Christian Sevenheck</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 6, 1415--1442.</p><p><strong>Abstract:</strong><br/>
We show that certain one-dimensional hypergeometric differential systems underlie objects of the category of irregular mixed Hodge modules, which was recently introduced by Sabbah, and compute the irregular Hodge filtration for them. We also provide a comparison theorem between two different types of Fourier–Laplace transformation for algebraic integrable twistor [math] -modules.
</p>projecteuclid.org/euclid.ant/1566353013_20190820220338Tue, 20 Aug 2019 22:03 EDTUlrich bundles on K3 surfaceshttps://projecteuclid.org/euclid.ant/1566353014<strong>Daniele Faenzi</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 6, 1443--1454.</p><p><strong>Abstract:</strong><br/>
We show that any polarized K3 surface supports special Ulrich bundles of rank [math] .
</p>projecteuclid.org/euclid.ant/1566353014_20190820220338Tue, 20 Aug 2019 22:03 EDTUnlikely intersections in semiabelian surfaceshttps://projecteuclid.org/euclid.ant/1566353015<strong>Daniel Bertrand</strong>, <strong>Harry Schmidt</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 6, 1455--1473.</p><p><strong>Abstract:</strong><br/>
We consider a family, depending on a parameter, of multiplicative extensions of an elliptic curve with complex multiplications. They form a 3-dimensional variety [math] which admits a dense set of special curves, known as Ribet curves, which strictly contains the torsion curves. We show that an irreducible curve [math] in [math] meets this set Zariski-densely only if [math] lies in a fiber of the family or is a translate of a Ribet curve by a multiplicative section. We further deduce from this result a proof of the Zilber–Pink conjecture (over number fields) for the mixed Shimura variety attached to the threefold [math] , when the parameter space is the universal one.
</p>projecteuclid.org/euclid.ant/1566353015_20190820220338Tue, 20 Aug 2019 22:03 EDTCongruences of parahoric group schemeshttps://projecteuclid.org/euclid.ant/1566353016<strong>Radhika Ganapathy</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 6, 1475--1499.</p><p><strong>Abstract:</strong><br/>
Let [math] be a nonarchimedean local field and let [math] be a torus over [math] . With [math] denoting the Néron–Raynaud model of [math] , a result of Chai and Yu asserts that the model [math] is canonically determined by [math] for [math] , where [math] with [math] denoting the natural projection of [math] on [math] , and [math] . In this article we prove an analogous result for parahoric group schemes attached to facets in the Bruhat–Tits building of a connected reductive group over [math] .
</p>projecteuclid.org/euclid.ant/1566353016_20190820220338Tue, 20 Aug 2019 22:03 EDTAn improved bound for the lengths of matrix algebrashttps://projecteuclid.org/euclid.ant/1566353017<strong>Yaroslav Shitov</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 6, 1501--1507.</p><p><strong>Abstract:</strong><br/>
Let [math] be a set of [math] matrices over a field [math] . We show that the [math] -linear span of the words in [math] of length at most
2
n
log
2
n
+
4
n
is the full [math] -algebra generated by [math] . This improves on the [math] bound by Paz (1984) and an [math] bound of Pappacena (1997).
</p>projecteuclid.org/euclid.ant/1566353017_20190820220338Tue, 20 Aug 2019 22:03 EDTModuli of stable maps in genus one and logarithmic geometry, IIhttps://projecteuclid.org/euclid.ant/1572314504<strong>Dhruv Ranganathan</strong>, <strong>Keli Santos-Parker</strong>, <strong>Jonathan Wise</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 8, 1765--1805.</p><p><strong>Abstract:</strong><br/>
This is the second in a pair of papers developing a framework to apply logarithmic methods in the study of stable maps and singular curves of genus [math] . This volume focuses on logarithmic Gromov–Witten theory and tropical geometry. We construct a logarithmically nonsingular and proper moduli space of genus [math] curves mapping to any toric variety. The space is a birational modification of the principal component of the Abramovich–Chen–Gross–Siebert space of logarithmic stable maps and produces logarithmic analogues of Vakil and Zinger’s genus one reduced Gromov–Witten theory. We describe the nonarchimedean analytic skeleton of this moduli space and, as a consequence, obtain a full resolution to the tropical realizability problem in genus [math] .
</p>projecteuclid.org/euclid.ant/1572314504_20191028220203Mon, 28 Oct 2019 22:02 EDTMultiplicity one for wildly ramified representationshttps://projecteuclid.org/euclid.ant/1572314505<strong>Daniel Le</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 8, 1807--1827.</p><p><strong>Abstract:</strong><br/>
Let [math] be a totally real field in which [math] is unramified. Let [math] be a modular Galois representation which satisfies the Taylor–Wiles hypotheses and is generic at a place [math] above [math] . Let [math] be the corresponding Hecke eigensystem. We show that the [math] -torsion in the [math] cohomology of Shimura curves with full congruence level at [math] coincides with the [math] -representation [math] constructed by Breuil and Paškūnas. In particular, it depends only on the local representation [math] , and its Jordan–Hölder factors appear with multiplicity one. This builds on and extends work of the author with Morra and Schraen and, independently, Hu–Wang, which proved these results when [math] was additionally assumed to be tamely ramified. The main new tool is a method for computing Taylor–Wiles patched modules of integral projective envelopes using multitype tamely potentially Barsotti–Tate deformation rings and their intersection theory.
</p>projecteuclid.org/euclid.ant/1572314505_20191028220203Mon, 28 Oct 2019 22:02 EDTTheta operators on unitary Shimura varietieshttps://projecteuclid.org/euclid.ant/1572314506<strong>Ehud de Shalit</strong>, <strong>Eyal Z. Goren</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 8, 1829--1877.</p><p><strong>Abstract:</strong><br/>
We define a theta operator on [math] -adic vector-valued modular forms on unitary groups of arbitrary signature, over a quadratic imaginary field in which [math] is inert. We study its effect on Fourier–Jacobi expansions and prove that it extends holomorphically beyond the [math] -ordinary locus, when applied to scalar-valued forms.
</p>projecteuclid.org/euclid.ant/1572314506_20191028220203Mon, 28 Oct 2019 22:02 EDTInfinitely generated symbolic Rees algebras over finite fieldshttps://projecteuclid.org/euclid.ant/1572314507<strong>Akiyoshi Sannai</strong>, <strong>Hiromu Tanaka</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 8, 1879--1891.</p><p><strong>Abstract:</strong><br/>
For the polynomial ring over an arbitrary field with twelve variables, there exists a prime ideal whose symbolic Rees algebra is not finitely generated.
</p>projecteuclid.org/euclid.ant/1572314507_20191028220203Mon, 28 Oct 2019 22:02 EDTManin's $b$-constant in familieshttps://projecteuclid.org/euclid.ant/1572314508<strong>Akash Kumar Sengupta</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 8, 1893--1905.</p><p><strong>Abstract:</strong><br/>
We show that the [math] -constant (appearing in Manin’s conjecture) is constant on very general fibers of a family of algebraic varieties. If the fibers of the family are uniruled, then we show that the [math] -constant is constant on general fibers.
</p>projecteuclid.org/euclid.ant/1572314508_20191028220203Mon, 28 Oct 2019 22:02 EDTEquidimensional adic eigenvarieties for groups with discrete serieshttps://projecteuclid.org/euclid.ant/1572314509<strong>Daniel R. Gulotta</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 8, 1907--1940.</p><p><strong>Abstract:</strong><br/>
We extend Urban’s construction of eigenvarieties for reductive groups [math] such that [math] has discrete series to include characteristic [math] points at the boundary of weight space. In order to perform this construction, we define a notion of “locally analytic” functions and distributions on a locally [math] -analytic manifold taking values in a complete Tate [math] -algebra in which [math] is not necessarily invertible. Our definition agrees with the definition of locally analytic distributions on [math] -adic Lie groups given by Johansson and Newton.
</p>projecteuclid.org/euclid.ant/1572314509_20191028220203Mon, 28 Oct 2019 22:02 EDTCohomological and numerical dynamical degrees on abelian varietieshttps://projecteuclid.org/euclid.ant/1572314510<strong>Fei Hu</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 8, 1941--1958.</p><p><strong>Abstract:</strong><br/>
We show that for a self-morphism of an abelian variety defined over an algebraically closed field of arbitrary characteristic, the second cohomological dynamical degree coincides with the first numerical dynamical degree.
</p>projecteuclid.org/euclid.ant/1572314510_20191028220203Mon, 28 Oct 2019 22:02 EDTA comparison between pro-$p$ Iwahori–Hecke modules and mod $p$ representationshttps://projecteuclid.org/euclid.ant/1572314511<strong>Noriyuki Abe</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 8, 1959--1981.</p><p><strong>Abstract:</strong><br/>
We give an equivalence of categories between certain subcategories of modules of pro- [math] Iwahori–Hecke algebras and modulo [math] representations.
</p>projecteuclid.org/euclid.ant/1572314511_20191028220203Mon, 28 Oct 2019 22:02 EDTProof of a conjecture of Colliot-Thélène and a diophantine excision theoremhttps://projecteuclid.org/euclid.ant/1576292479<strong>Jan Denef</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 1983--1996.</p><p><strong>Abstract:</strong><br/>
We prove a conjecture of Colliot-Thélène that implies the Ax–Kochen Theorem on [math] -adic forms. We obtain it as an easy consequence of a diophantine excision theorem whose proof forms the body of the present paper.
</p>projecteuclid.org/euclid.ant/1576292479_20191213220142Fri, 13 Dec 2019 22:01 ESTIrreducible characters with bounded root Artin conductorhttps://projecteuclid.org/euclid.ant/1576292483<strong>Amalia Pizarro-Madariaga</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 1997--2004.</p><p><strong>Abstract:</strong><br/>
We prove that the best possible lower bound for the Artin conductor is exponential in the degree.
</p>projecteuclid.org/euclid.ant/1576292483_20191213220142Fri, 13 Dec 2019 22:01 ESTFrobenius–Perron theory of endofunctorshttps://projecteuclid.org/euclid.ant/1576292484<strong>Jianmin Chen</strong>, <strong>Zhibin Gao</strong>, <strong>Elizabeth Wicks</strong>, <strong>James J. Zhang</strong>, <strong>Xiaohong Zhang</strong>, <strong>Hong Zhu</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 2005--2055.</p><p><strong>Abstract:</strong><br/>
We introduce the Frobenius–Perron dimension of an endofunctor of a [math] -linear category and provide some applications.
</p>projecteuclid.org/euclid.ant/1576292484_20191213220142Fri, 13 Dec 2019 22:01 ESTPositivity of anticanonical divisors and $F$-purity of fibershttps://projecteuclid.org/euclid.ant/1576292485<strong>Sho Ejiri</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 2057--2080.</p><p><strong>Abstract:</strong><br/>
We prove that given a flat generically smooth morphism between smooth projective varieties with [math] - pure closed fibers, if the source space is Fano, weak Fano or a variety with nef anticanonical divisor, respectively, then so is the target space. We also show that, in arbitrary characteristic, a generically smooth surjective morphism between smooth projective varieties cannot have nef and big relative anticanonical divisor, if the target space has positive dimension.
</p>projecteuclid.org/euclid.ant/1576292485_20191213220142Fri, 13 Dec 2019 22:01 ESTA probabilistic approach to systems of parameters and Noether normalizationhttps://projecteuclid.org/euclid.ant/1576292486<strong>Juliette Bruce</strong>, <strong>Daniel Erman</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 2081--2102.</p><p><strong>Abstract:</strong><br/>
We study systems of parameters over finite fields from a probabilistic perspective and use this to give the first effective Noether normalization result over a finite field. Our central technique is an adaptation of Poonen’s closed point sieve, where we sieve over higher dimensional subvarieties, and we express the desired probabilities via a zeta function-like power series that enumerates higher dimensional varieties instead of closed points. This also yields a new proof of a recent result of Gabber, Liu and Lorenzini (2015) and Chinburg, Moret-Bailly, Pappas and Taylor (2017) on Noether normalizations of projective families over the integers.
</p>projecteuclid.org/euclid.ant/1576292486_20191213220142Fri, 13 Dec 2019 22:01 ESTThe structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectureshttps://projecteuclid.org/euclid.ant/1576292487<strong>Terence Tao</strong>, <strong>Joni Teräväinen</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 2103--2150.</p><p><strong>Abstract:</strong><br/>
We study the asymptotic behaviour of higher order correlations
E
n
≤
X
∕
d
g
1
(
n
+
a
h
1
)
⋯
g
k
(
n
+
a
h
k
)
as a function of the parameters [math] and [math] , where [math] are bounded multiplicative functions, [math] are integer shifts, and [math] is large. Our main structural result asserts, roughly speaking, that such correlations asymptotically vanish for almost all [math] if [math] does not (weakly) pretend to be a twisted Dirichlet character [math] , and behave asymptotically like a multiple of [math] otherwise. This extends our earlier work on the structure of logarithmically averaged correlations, in which the [math] parameter is averaged out and one can set [math] . Among other things, the result enables us to establish special cases of the Chowla and Elliott conjectures for (unweighted) averages at almost all scales; for instance, we establish the [math] -point Chowla conjecture [math] for [math] odd or equal to [math] for all scales [math] outside of a set of zero logarithmic density.
</p>projecteuclid.org/euclid.ant/1576292487_20191213220142Fri, 13 Dec 2019 22:01 ESTVI-modules in nondescribing characteristic, part Ihttps://projecteuclid.org/euclid.ant/1576292488<strong>Rohit Nagpal</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 2151--2189.</p><p><strong>Abstract:</strong><br/>
Let VI be the category of finite dimensional [math] -vector spaces whose morphisms are injective linear maps and let [math] be a noetherian ring. We study the category of functors from VI to [math] -modules in the case when [math] is invertible in [math] . Our results include a structure theorem, finiteness of regularity, and a description of the Hilbert series. These results are crucial in the classification of smooth irreducible [math] -representations in nondescribing characteristic which is contained in Part II of this paper ( VI-modules in nondescribing characteristic , part II, arxiv:1810.04592).
</p>projecteuclid.org/euclid.ant/1576292488_20191213220142Fri, 13 Dec 2019 22:01 ESTDegree of irrationality of very general abelian surfaceshttps://projecteuclid.org/euclid.ant/1576292489<strong>Nathan Chen</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 2191--2198.</p><p><strong>Abstract:</strong><br/>
The degree of irrationality of a projective variety [math] is defined to be the smallest degree of a rational dominant map to a projective space of the same dimension. For abelian surfaces, Yoshihara computed this invariant in specific cases, while Stapleton gave a sublinear upper bound for very general polarized abelian surfaces [math] of degree [math] . Somewhat surprisingly, we show that the degree of irrationality of a very general polarized abelian surface is uniformly bounded above by 4, independently of the degree of the polarization. This result disproves part of a conjecture of Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery.
</p>projecteuclid.org/euclid.ant/1576292489_20191213220142Fri, 13 Dec 2019 22:01 ESTLower bounds for the least prime in Chebotarevhttps://projecteuclid.org/euclid.ant/1576292490<strong>Andrew Fiori</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 2199--2203.</p><p><strong>Abstract:</strong><br/>
In this paper we show there exists an infinite family of number fields [math] , Galois over [math] , for which the smallest prime [math] of [math] which splits completely in [math] has size at least [math] . This gives a converse to various upper bounds, which shows that they are best possible.
</p>projecteuclid.org/euclid.ant/1576292490_20191213220142Fri, 13 Dec 2019 22:01 ESTBrody hyperbolicity of base spaces of certain families of varietieshttps://projecteuclid.org/euclid.ant/1576292491<strong>Mihnea Popa</strong>, <strong>Behrouz Taji</strong>, <strong>Lei Wu</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 9, 2205--2242.</p><p><strong>Abstract:</strong><br/>
We prove that quasi-projective base spaces of smooth families of minimal varieties of general type with maximal variation do not admit Zariski dense entire curves. We deduce the fact that moduli stacks of polarized varieties of this sort are Brody hyperbolic, answering a special case of a question of Viehweg and Zuo. For two-dimensional bases, we show analogous results in the more general case of families of varieties admitting a good minimal model.
</p>projecteuclid.org/euclid.ant/1576292491_20191213220142Fri, 13 Dec 2019 22:01 ESTThe elliptic KZB connection and algebraic de Rham theory for unipotent fundamental groups of elliptic curveshttps://projecteuclid.org/euclid.ant/1579143614<strong>Ma Luo</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 10, 2243--2275.</p><p><strong>Abstract:</strong><br/>
We develop an algebraic de Rham theory for unipotent fundamental groups of once punctured elliptic curves over a field of characteristic zero using the universal elliptic KZB connection of Calaque, Enriquez and Etingof (2009) and Levin and Racinet (2007). We use it to give an explicit version of Tannaka duality for unipotent connections over an elliptic curve with a regular singular point at the identity.
</p>projecteuclid.org/euclid.ant/1579143614_20200115220030Wed, 15 Jan 2020 22:00 ESTMoments of random multiplicative functions, II: High momentshttps://projecteuclid.org/euclid.ant/1579143615<strong>Adam J Harper</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 10, 2277--2321.</p><p><strong>Abstract:</strong><br/>
We determine the order of magnitude of [math] up to factors of size [math] , where [math] is a Steinhaus or Rademacher random multiplicative function, for all real [math] .
In the Steinhaus case, we show that [math] on this whole range. In the Rademacher case, we find a transition in the behavior of the moments when [math] , where the size starts to be dominated by “orthogonal” rather than “unitary” behavior. We also deduce some consequences for the large deviations of [math] .
The proofs use various tools, including hypercontractive inequalities, to connect [math] with the [math] -th moment of an Euler product integral. When [math] is large, it is then fairly easy to analyze this integral. When [math] is close to 1 the analysis seems to require subtler arguments, including Doob’s [math] maximal inequality for martingales.
</p>projecteuclid.org/euclid.ant/1579143615_20200115220030Wed, 15 Jan 2020 22:00 ESTArtin–Mazur–Milne duality for fppf cohomologyhttps://projecteuclid.org/euclid.ant/1579143616<strong>Cyril Demarche</strong>, <strong>David Harari</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 10, 2323--2357.</p><p><strong>Abstract:</strong><br/>
We provide a complete proof of a duality theorem for the fppf cohomology of either a curve over a finite field or a ring of integers of a number field, which extends the classical Artin–Verdier Theorem in étale cohomology. We also prove some finiteness and vanishing statements.
</p>projecteuclid.org/euclid.ant/1579143616_20200115220030Wed, 15 Jan 2020 22:00 ESTBetti numbers of Shimura curves and arithmetic three-orbifoldshttps://projecteuclid.org/euclid.ant/1579143617<strong>Mikołaj Frączyk</strong>, <strong>Jean Raimbault</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 10, 2359--2382.</p><p><strong>Abstract:</strong><br/>
We show that asymptotically the first Betti number [math] of a Shimura curve satisfies the Gauss–Bonnet equality [math] where [math] is hyperbolic volume; equivalently [math] where [math] is the arithmetic genus. We also show that the first Betti number of a congruence hyperbolic 3-orbifold asymptotically vanishes relatively to hyperbolic volume, that is [math] . This generalizes previous results obtained by Frączyk, on which we rely, and uses the same main tool, namely Benjamini–Schramm convergence.
</p>projecteuclid.org/euclid.ant/1579143617_20200115220030Wed, 15 Jan 2020 22:00 ESTCombinatorial identities and Titchmarsh's divisor problem for multiplicative functionshttps://projecteuclid.org/euclid.ant/1579143618<strong>Sary Drappeau</strong>, <strong>Berke Topacogullari</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 10, 2383--2425.</p><p><strong>Abstract:</strong><br/>
Given a multiplicative function [math] which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum [math] , where [math] denotes the divisor function and [math] . We consider in particular the special cases where [math] is the generalized divisor function [math] with [math] , and the characteristic function of sums of two squares (or more generally, ideal norms of abelian extensions). As another application, we deduce a full asymptotic expansion in the generalized Titchmarsh divisor problem [math] , where [math] counts the number of distinct prime divisors of [math] , thus extending a result of Fouvry and Bombieri, Friedlander and Iwaniec.
We present two different proofs: The first relies on an effective combinatorial formula of Heath-Brown’s type for the divisor function [math] with [math] , and an interpolation argument in the [math] -variable for weighted mean values of [math] . The second is based on an identity of Linnik type for [math] and the well-factorability of friable numbers.
</p>projecteuclid.org/euclid.ant/1579143618_20200115220030Wed, 15 Jan 2020 22:00 ESTThe construction problem for Hodge numbers modulo an integerhttps://projecteuclid.org/euclid.ant/1579143619<strong>Matthias Paulsen</strong>, <strong>Stefan Schreieder</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 10, 2427--2434.</p><p><strong>Abstract:</strong><br/>
For any integer [math] and any dimension [math] , we show that any [math] -dimensional Hodge diamond with values in [math] is attained by the Hodge numbers of an [math] -dimensional smooth complex projective variety. As a corollary, there are no polynomial relations among the Hodge numbers of [math] -dimensional smooth complex projective varieties besides the ones induced by the Hodge symmetries, which answers a question raised by Kollár in 2012.
</p>projecteuclid.org/euclid.ant/1579143619_20200115220030Wed, 15 Jan 2020 22:00 ESTCrystalline comparison isomorphisms in $p$-adic Hodge theory:the absolutely unramified casehttps://projecteuclid.org/euclid.ant/1579143649<strong>Fucheng Tan</strong>, <strong>Jilong Tong</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 7, 1509--1581.</p><p><strong>Abstract:</strong><br/>
We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for étale cohomology with nontrivial coefficients, as well as in the relative setting, i.e., for proper smooth morphisms of smooth formal schemes. The proof is formulated in terms of the proétale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the proétale site. Moreover, we need to prove the Poincaré lemma for crystalline period sheaves, for which we adapt the idea of Andreatta and Iovita. Another ingredient for the proof is the geometric acyclicity of crystalline period sheaves, whose computation is due to Andreatta and Brinon.
</p>projecteuclid.org/euclid.ant/1579143649_20200115220053Wed, 15 Jan 2020 22:00 ESTPseudorepresentations of weight one are unramifiedhttps://projecteuclid.org/euclid.ant/1579143650<strong>Frank Calegari</strong>, <strong>Joel Specter</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 7, 1583--1596.</p><p><strong>Abstract:</strong><br/>
We prove that the determinant (pseudorepresentation) associated to the Hecke algebra of Katz modular forms of weight one and level prime to [math] is unramified at [math] .
</p>projecteuclid.org/euclid.ant/1579143650_20200115220053Wed, 15 Jan 2020 22:00 ESTOn the $p$-typical de Rham–Witt complex over $W(k)$https://projecteuclid.org/euclid.ant/1579143651<strong>Christopher Davis</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 7, 1597--1631.</p><p><strong>Abstract:</strong><br/>
Hesselholt and Madsen (2004) define and study the (absolute, [math] -typical) de Rham–Witt complex in mixed characteristic, where [math] is an odd prime. They give as an example an elementary algebraic description of the de Rham–Witt complex over [math] , [math] . The main goal of this paper is to construct, for [math] a perfect ring of characteristic [math] , a Witt complex over [math] with an algebraic description which is completely analogous to Hesselholt and Madsen’s description for [math] . Our Witt complex is not isomorphic to the de Rham–Witt complex; instead we prove that, in each level, the de Rham–Witt complex over [math] surjects onto our Witt complex, and that the kernel consists of all elements which are divisible by arbitrarily high powers of [math] . We deduce an explicit description of [math] for each [math] . We also deduce results concerning the de Rham–Witt complex over certain [math] -torsion-free perfectoid rings.
</p>projecteuclid.org/euclid.ant/1579143651_20200115220053Wed, 15 Jan 2020 22:00 ESTCoherent Tannaka duality and algebraicity of Hom-stackshttps://projecteuclid.org/euclid.ant/1579143652<strong>Jack Hall</strong>, <strong>David Rydh</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 7, 1633--1675.</p><p><strong>Abstract:</strong><br/>
We establish Tannaka duality for noetherian algebraic stacks with affine stabilizer groups. Our main application is the existence of [math] -stacks in great generality.
</p>projecteuclid.org/euclid.ant/1579143652_20200115220053Wed, 15 Jan 2020 22:00 ESTSupercuspidal representations of ${\rm GL}_n({\rm F})$ distinguished by a Galois involutionhttps://projecteuclid.org/euclid.ant/1579143653<strong>Vincent Sécherre</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 7, 1677--1733.</p><p><strong>Abstract:</strong><br/>
Let [math] be a quadratic extension of nonarchimedean locally compact fields of residual characteristic [math] and let [math] denote its nontrivial automorphism. Let [math] be an algebraically closed field of characteristic different from [math] . To any cuspidal representation [math] of [math] , with coefficients in [math] , such that [math] (such a representation is said to be [math] -selfdual) we associate a quadratic extension [math] , where [math] is a tamely ramified extension of [math] and [math] is a tamely ramified extension of [math] , together with a quadratic character of [math] . When [math] is supercuspidal, we give a necessary and sufficient condition, in terms of these data, for [math] to be [math] -distinguished. When the characteristic [math] of [math] is not [math] , denoting by [math] the nontrivial [math] -character of [math] trivial on [math] -norms, we prove that any [math] -selfdual supercuspidal [math] -representation is either distinguished or [math] -distinguished, but not both. In the modular case, that is when [math] , we give examples of [math] -selfdual cuspidal nonsupercuspidal representations which are not distinguished nor [math] -distinguished. In the particular case where [math] is the field of complex numbers, in which case all cuspidal representations are supercuspidal, this gives a complete distinction criterion for arbitrary complex cuspidal representations, as well as a purely local proof, for cuspidal representations, of the dichotomy and disjunction theorem due to Kable and Anandavardhanan, Kable and Tandon, when [math] .
</p>projecteuclid.org/euclid.ant/1579143653_20200115220053Wed, 15 Jan 2020 22:00 ESTA vanishing result for higher smooth dualshttps://projecteuclid.org/euclid.ant/1579143654<strong>Claus Sorensen</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 7, 1735--1763.</p><p><strong>Abstract:</strong><br/>
In this paper we prove a general vanishing result for Kohlhaase’s higher smooth duality functors [math] . If [math] is any unramified connected reductive [math] -adic group, [math] is a hyperspecial subgroup, and [math] is a Serre weight, we show that [math] for [math] , where [math] is a Borel subgroup and the dimension is over [math] . This is due to Kohlhaase for [math] , in which case it has applications to the calculation of [math] for supersingular representations. Our proof avoids explicit matrix computations by making use of Lazard theory, and we deduce our result from an analogous statement for graded algebras via a spectral sequence argument. The graded case essentially follows from Koszul duality between symmetric and exterior algebras.
</p>projecteuclid.org/euclid.ant/1579143654_20200115220053Wed, 15 Jan 2020 22:00 ESTGorenstein-projective and semi-Gorenstein-projective moduleshttps://projecteuclid.org/euclid.ant/1586224818<strong>Claus Michael Ringel</strong>, <strong>Pu Zhang</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 1, 1--36.</p><p><strong>Abstract:</strong><br/>
Let [math] be an artin algebra. An [math] -module [math] will be said to be semi-Gorenstein-projective provided that [math] for all [math] . All Gorenstein-projective modules are semi-Gorenstein-projective and only few and quite complicated examples of semi-Gorenstein-projective modules which are not Gorenstein-projective have been known. One of the aims of the paper is to provide conditions on [math] such that all semi-Gorenstein-projective left modules are Gorenstein-projective (we call such an algebra left weakly Gorenstein). In particular, we show that in case there are only finitely many isomorphism classes of indecomposable left modules which are both semi-Gorenstein-projective and torsionless, then [math] is left weakly Gorenstein. On the other hand, we exhibit a 6-dimensional algebra [math] with a semi-Gorenstein-projective module [math] which is not torsionless (thus not Gorenstein-projective). Actually, also the [math] -dual module [math] is semi-Gorenstein-projective. In this way, we show the independence of the total reflexivity conditions of Avramov and Martsinkovsky, thus completing a partial proof by Jorgensen and Şega. Since all the syzygy-modules of [math] and [math] are 3-dimensional, the example can be checked (and visualized) quite easily.
</p>projecteuclid.org/euclid.ant/1586224818_20200406220023Mon, 06 Apr 2020 22:00 EDTThe 16-rank of $\mathbb{Q}(\sqrt{-p})$https://projecteuclid.org/euclid.ant/1586224819<strong>Peter Koymans</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 1, 37--65.</p><p><strong>Abstract:</strong><br/>
Recently, a density result for the [math] -rank of [math] was established when [math] varies among the prime numbers, assuming a short character sum conjecture. We prove the same density result unconditionally.
</p>projecteuclid.org/euclid.ant/1586224819_20200406220023Mon, 06 Apr 2020 22:00 EDTSupersingular Hecke modules as Galois representationshttps://projecteuclid.org/euclid.ant/1586224820<strong>Elmar Grosse-Klönne</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 1, 67--118.</p><p><strong>Abstract:</strong><br/>
Let [math] be a local field of mixed characteristic [math] , let [math] be a finite extension of its residue field, let [math] be the pro- [math] -Iwahori Hecke [math] -algebra attached to [math] for some [math] . We construct an exact and fully faithful functor from the category of supersingular [math] -modules to the category of [math] -representations over [math] . More generally, for a certain [math] -algebra [math] surjecting onto [math] we define the notion of [math] -supersingular modules and construct an exact and fully faithful functor from the category of [math] -supersingular [math] -modules to the category of [math] -representations over [math] .
</p>projecteuclid.org/euclid.ant/1586224820_20200406220023Mon, 06 Apr 2020 22:00 EDTStability in the homology of unipotent groupshttps://projecteuclid.org/euclid.ant/1586224821<strong>Andrew Putman</strong>, <strong>Steven V Sam</strong>, <strong>Andrew Snowden</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 1, 119--154.</p><p><strong>Abstract:</strong><br/>
Let [math] be a (not necessarily commutative) ring whose additive group is finitely generated and let [math] be the group of upper-triangular unipotent matrices over [math] . We study how the homology groups of [math] vary with [math] from the point of view of representation stability. Our main theorem asserts that if for each [math] we have representations [math] of [math] over a ring [math] that are appropriately compatible and satisfy suitable finiteness hypotheses, then the rule [math] defines a finitely generated [math] -module. As a consequence, if [math] is a field then [math] is eventually equal to a polynomial in [math] . We also prove similar results for the Iwahori subgroups of [math] for number rings [math] .
</p>projecteuclid.org/euclid.ant/1586224821_20200406220023Mon, 06 Apr 2020 22:00 EDTOn the orbits of multiplicative pairshttps://projecteuclid.org/euclid.ant/1586224822<strong>Oleksiy Klurman</strong>, <strong>Alexander P. Mangerel</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 1, 155--189.</p><p><strong>Abstract:</strong><br/>
We characterize all pairs of completely multiplicative functions [math] , where [math] denotes the unit circle, such that
{
(
f
(
n
)
,
g
(
n
+
1
)
)
}
n
≥
1
¯
≠
𝕋
×
𝕋
.
In so doing, we settle an old conjecture of Zoltán Daróczy and Imre Kátai.
</p>projecteuclid.org/euclid.ant/1586224822_20200406220023Mon, 06 Apr 2020 22:00 EDTBirationally superrigid Fano 3-folds of codimension 4https://projecteuclid.org/euclid.ant/1586224823<strong>Takuzo Okada</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 1, 191--212.</p><p><strong>Abstract:</strong><br/>
We determine birational superrigidity for a quasismooth prime Fano [math] -fold of codimension [math] with no projection centers. In particular we prove birational superrigidity for Fano [math] -folds of codimension [math] with no projection centers which were recently constructed by Coughlan and Ducat. We also pose some questions and a conjecture regarding the classification of birationally superrigid Fano [math] -folds.
</p>projecteuclid.org/euclid.ant/1586224823_20200406220023Mon, 06 Apr 2020 22:00 EDTCoble fourfold, $\mathfrak S_6$-invariant quartic threefolds, and Wiman–Edge sexticshttps://projecteuclid.org/euclid.ant/1586224824<strong>Ivan Cheltsov</strong>, <strong>Alexander Kuznetsov</strong>, <strong>Konstantin Shramov</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 14, Number 1, 213--274.</p><p><strong>Abstract:</strong><br/>
We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all [math] -invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman–Edge pencil. As an application, we check that [math] -invariant three-dimensional quartics are unirational, obtain new proofs of rationality of four special quartics among them and irrationality of the others, and describe their Weil divisor class groups as [math] -representations.
</p>projecteuclid.org/euclid.ant/1586224824_20200406220023Mon, 06 Apr 2020 22:00 EDT