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 EDTAkizuki–Witt maps and Kaletha's global rigid inner formshttps://projecteuclid.org/euclid.ant/1532743324<strong>Olivier Taïbi</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 4, 833--884.</p><p><strong>Abstract:</strong><br/>
We give an explicit construction of global Galois gerbes constructed more abstractly by Kaletha to define global rigid inner forms. This notion is crucial to formulate Arthur’s multiplicity formula for inner forms of quasisplit reductive groups. As a corollary, we show that any global rigid inner form is almost everywhere unramified, and we give an algorithm to compute the resulting local rigid inner forms at all places in a given finite set. This makes global rigid inner forms as explicit as global pure inner forms, up to computations in local and global class field theory.
</p>projecteuclid.org/euclid.ant/1532743324_20180727220226Fri, 27 Jul 2018 22:02 EDT$(\varphi,\Gamma)$-modules de de Rham et fonctions $L$ $p$-adiqueshttps://projecteuclid.org/euclid.ant/1532743325<strong>Joaquín Rodrigues Jacinto</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 4, 885--934.</p><p><strong>Abstract:</strong><br/>
On développe une variante des méthodes de Coleman et Perrin-Riou permettant, pour une représentation galoisienne de de Rham, construire des fonctions [math] [math] -adiques à partir d’un système compatible d’éléments globaux. On obtient de la sorte des fonctions analytiques sur un ouvert de l’espace des poids contenant les caractères localement algébriques de conducteur assez grand. Appliqué au système d’Euler de Kato, cela fournit des fonctions [math] [math] -adiques pour les courbes elliptiques à mauvaise réduction additive et, plus généralement, pour les formes modulaires supercuspidales en [math] . En dimension [math] , nous prouvons une équation fonctionnelle pour nos fonctions [math] [math] -adiques.
We develop a variant of Coleman and Perrin-Riou’s methods giving, for a de Rham [math] -adic Galois representation, a construction of [math] -adic [math] -functions from a compatible system of global elements. As a result, we construct analytic functions on an open set of the [math] -adic weight space containing all locally algebraic characters of large enough conductor. Applied to Kato’s Euler system, this gives [math] -adic [math] -functions for elliptic curves with additive bad reduction and, more generally, for modular forms which are supercuspidal at [math] . In the case of dimension [math] , we prove a functional equation for our [math] -adic [math] -functions.
</p>projecteuclid.org/euclid.ant/1532743325_20180727220226Fri, 27 Jul 2018 22:02 EDTInvariant theory of $\bigwedge^3(9)$ and genus-2 curveshttps://projecteuclid.org/euclid.ant/1532743326<strong>Eric M. Rains</strong>, <strong>Steven V Sam</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 4, 935--957.</p><p><strong>Abstract:</strong><br/>
Previous work established a connection between the geometric invariant theory of the third exterior power of a 9-dimensional complex vector space and the moduli space of genus-2 curves with some additional data. We generalize this connection to arbitrary fields, and describe the arithmetic data needed to get a bijection between both sides of this story.
</p>projecteuclid.org/euclid.ant/1532743326_20180727220226Fri, 27 Jul 2018 22:02 EDTSums of two cubes as twisted perfect powers, revisitedhttps://projecteuclid.org/euclid.ant/1532743327<strong>Michael A. Bennett</strong>, <strong>Carmen Bruni</strong>, <strong>Nuno Freitas</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 4, 959--999.</p><p><strong>Abstract:</strong><br/>
We sharpen earlier work (2011) of the first author, Luca and Mulholland, showing that the Diophantine equation
A
3
+
B
3
=
q
α
C
p
,
A
B
C
≠
0
,
gcd
(
A
,
B
)
=
1
,
has, for “most” primes [math] and suitably large prime exponents [math] , no solutions. We handle a number of (presumably infinite) families where no such conclusion was hitherto known. Through further application of certain symplectic criteria , we are able to make some conditional statements about still more values of [math] ; a sample such result is that, for all but [math] primes [math] up to [math] , the equation
A
3
+
B
3
=
q
C
p
.
has no solutions in coprime, nonzero integers [math] , [math] and [math] , for a positive proportion of prime exponents [math] .
</p>projecteuclid.org/euclid.ant/1532743327_20180727220226Fri, 27 Jul 2018 22:02 EDTPseudo-exponential maps, variants, and quasiminimalityhttps://projecteuclid.org/euclid.ant/1532743363<strong>Martin Bays</strong>, <strong>Jonathan Kirby</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 3, 493--549.</p><p><strong>Abstract:</strong><br/>
We give a construction of quasiminimal fields equipped with pseudo-analytic maps, generalizing Zilber’s pseudo-exponential function. In particular we construct pseudo-exponential maps of simple abelian varieties, including pseudo- [math] -functions for elliptic curves. We show that the complex field with the corresponding analytic function is isomorphic to the pseudo-analytic version if and only if the appropriate version of Schanuel’s conjecture is true and the corresponding version of the strong exponential-algebraic closedness property holds. Moreover, we relativize the construction to build a model over a fairly arbitrary countable subfield and deduce that the complex exponential field is quasiminimal if it is exponentially-algebraically closed. This property states only that the graph of exponentiation has nonempty intersection with certain algebraic varieties but does not require genericity of any point in the intersection. Furthermore, Schanuel’s conjecture is not required as a condition for quasiminimality.
</p>projecteuclid.org/euclid.ant/1532743363_20180727220251Fri, 27 Jul 2018 22:02 EDTOn faithfulness of the lifting for Hopf algebras and fusion categorieshttps://projecteuclid.org/euclid.ant/1532743364<strong>Pavel Etingof</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 3, 551--569.</p><p><strong>Abstract:</strong><br/>
We use a version of Haboush’s theorem over complete local Noetherian rings to prove faithfulness of the lifting for semisimple cosemisimple Hopf algebras and separable (braided, symmetric) fusion categories from characteristic [math] to characteristic zero, showing that, moreover, any isomorphism between such structures can be reduced modulo [math] . This fills a gap in our earlier work. We also show that lifting of semisimple cosemisimple Hopf algebras is a fully faithful functor, and prove that lifting induces an isomorphism on Picard and Brauer–Picard groups. Finally, we show that a subcategory or quotient category of a separable multifusion category is separable (resolving an open question from our earlier work), and use this to show that certain classes of tensor functors between lifts of separable categories to characteristic zero can be reduced modulo [math] .
</p>projecteuclid.org/euclid.ant/1532743364_20180727220251Fri, 27 Jul 2018 22:02 EDTMean square in the prime geodesic theoremhttps://projecteuclid.org/euclid.ant/1532743365<strong>Giacomo Cherubini</strong>, <strong>João Guerreiro</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 3, 571--597.</p><p><strong>Abstract:</strong><br/>
We prove upper bounds for the mean square of the remainder in the prime geodesic theorem, for every cofinite Fuchsian group, which improve on average on the best known pointwise bounds. The proof relies on the Selberg trace formula. For the modular group we prove a refined upper bound by using the Kuznetsov trace formula.
</p>projecteuclid.org/euclid.ant/1532743365_20180727220251Fri, 27 Jul 2018 22:02 EDTElliptic quantum groups and Baxter relationshttps://projecteuclid.org/euclid.ant/1532743366<strong>Huafeng Zhang</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 3, 599--647.</p><p><strong>Abstract:</strong><br/>
We introduce a category [math] of modules over the elliptic quantum group of [math] with well-behaved [math] -character theory. We construct asymptotic modules as analytic continuation of a family of finite-dimensional modules, the Kirillov–Reshetikhin modules. In the Grothendieck ring of this category we prove two types of identities: Generalized Baxter relations in the spirit of Frenkel–Hernandez between finite-dimensional modules and asymptotic modules. Three-term Baxter TQ relations of infinite-dimensional modules.
</p>projecteuclid.org/euclid.ant/1532743366_20180727220251Fri, 27 Jul 2018 22:02 EDTDifferential forms in positive characteristic, II: cdh-descent via functorial Riemann–Zariski spaceshttps://projecteuclid.org/euclid.ant/1532743367<strong>Annette Huber</strong>, <strong>Shane Kelly</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 3, 649--692.</p><p><strong>Abstract:</strong><br/>
This paper continues our study of the sheaf associated to Kähler differentials in the cdh-topology and its cousins, in positive characteristic, without assuming resolution of singularities. The picture for the sheaves themselves is now fairly complete. We give a calculation [math] in terms of the seminormalisation. We observe that the category of representable cdh-sheaves is equivalent to the category of seminormal varieties. We conclude by proposing some possible connections to Berkovich spaces and [math] -singularities in the last section. The tools developed for the case of differential forms also apply in other contexts and should be of independent interest.
</p>projecteuclid.org/euclid.ant/1532743367_20180727220251Fri, 27 Jul 2018 22:02 EDTNilpotence order growth of recursion operators in characteristic $p$https://projecteuclid.org/euclid.ant/1532743368<strong>Anna Medvedovsky</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 3, 693--722.</p><p><strong>Abstract:</strong><br/>
We prove that the killing rate of certain degree-lowering “recursion operators” on a polynomial algebra over a finite field grows slower than linearly in the degree of the polynomial attacked. We also explain the motivating application: obtaining a lower bound for the Krull dimension of a local component of a big [math] Hecke algebra in the genus-zero case. We sketch the application for [math] and [math] in level one. The case [math] was first established in by Nicolas and Serre in 2012 using different methods.
</p>projecteuclid.org/euclid.ant/1532743368_20180727220251Fri, 27 Jul 2018 22:02 EDTAlgebraic de Rham theory for weakly holomorphic modular forms of level onehttps://projecteuclid.org/euclid.ant/1532743369<strong>Francis Brown</strong>, <strong>Richard Hain</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 3, 723--750.</p><p><strong>Abstract:</strong><br/>
We establish an Eichler–Shimura isomorphism for weakly modular forms of level one. We do this by relating weakly modular forms with rational Fourier coefficients to the algebraic de Rham cohomology of the modular curve with twisted coefficients. This leads to formulae for the periods and quasiperiods of modular forms.
</p>projecteuclid.org/euclid.ant/1532743369_20180727220251Fri, 27 Jul 2018 22:02 EDTSemistable Chow–Hall algebras of quivers and quantized Donaldson–Thomas invariantshttps://projecteuclid.org/euclid.ant/1534212095<strong>Hans Franzen</strong>, <strong>Markus Reineke</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 5, 1001--1025.</p><p><strong>Abstract:</strong><br/>
The semistable ChowHa of a quiver with stability is defined as an analog of the cohomological Hall algebra of Kontsevich and Soibelman via convolution in equivariant Chow groups of semistable loci in representation varieties of quivers. We prove several structural results on the semistable ChowHa, namely isomorphism of the cycle map, a tensor product decomposition, and a tautological presentation. For symmetric quivers, this leads to an identification of their quantized Donaldson–Thomas invariants with the Chow–Betti numbers of moduli spaces.
</p>projecteuclid.org/euclid.ant/1534212095_20180813220159Mon, 13 Aug 2018 22:01 EDTCertain abelian varieties bad at only one primehttps://projecteuclid.org/euclid.ant/1534212096<strong>Armand Brumer</strong>, <strong>Kenneth Kramer</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 5, 1027--1071.</p><p><strong>Abstract:</strong><br/>
An abelian surface [math] of prime conductor [math] is favorable if its 2-division field [math] is an [math] -extension over [math] with ramification index 5 over [math] . Let [math] be favorable and let [math] be a semistable abelian variety of dimension [math] and conductor [math] with [math] filtered by copies of [math] . We give a sufficient class field theoretic criterion on [math] to guarantee that [math] is isogenous to [math] .
As expected from our paramodular conjecture, we conclude that there is one isogeny class of abelian surfaces for each conductor in [math] . The general applicability of our criterion is discussed in the data section.
</p>projecteuclid.org/euclid.ant/1534212096_20180813220159Mon, 13 Aug 2018 22:01 EDTCharacterization of Kollár surfaceshttps://projecteuclid.org/euclid.ant/1534212097<strong>Giancarlo Urzúa</strong>, <strong>José Ignacio Yáñez</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 5, 1073--1105.</p><p><strong>Abstract:</strong><br/>
Kollár (2008) introduced the surfaces
(
x
1
a
1
x
2
+
x
2
a
2
x
3
+
x
3
a
3
x
4
+
x
4
a
4
x
1
=
0
)
⊂
P
(
w
1
,
w
2
,
w
3
,
w
4
)
where [math] , [math] , and [math] . The aim was to give many interesting examples of [math] -homology projective planes. They occur when [math] . For that case, we prove that Kollár surfaces are Hwang–Keum (2012) surfaces. For [math] , we construct a geometrically explicit birational map between Kollár surfaces and cyclic covers [math] , where [math] are four general lines in [math] . In addition, by using various properties on classical Dedekind sums, we prove that:
For any
w
∗
>
1
, we have
p
g
=
0
if and only if the Kollár surface is rational. This happens when
a
i
+
1
≡
1
or
a
i
a
i
+
1
≡
−
1
(
mod
w
∗
)
for some
i
.
For any
w
∗
>
1
, we have
p
g
=
1
if and only if the Kollár surface is birational to a K3 surface. We classify this
situation.
For
w
∗
≫
0
, we have that the smooth minimal model
S
of a generic Kollár surface is of general type with
K
S
2
∕
e
(
S
)
→
1
.
</p>projecteuclid.org/euclid.ant/1534212097_20180813220159Mon, 13 Aug 2018 22:01 EDTReprésentations de réduction unipotente pour $\mathrm{SO}(2n+1)$, III: Exemples de fronts d'ondehttps://projecteuclid.org/euclid.ant/1534212098<strong>Jean-Loup Waldspurger</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 5, 1107--1171.</p><p><strong>Abstract:</strong><br/>
Let [math] be a group [math] defined over a [math] -adic field. We compute the wave front set of the antitempered irreducible representations of [math] which are of unipotent reduction. The wave front set of such representations is the orthogonal orbit dual to the symplectic orbit appearing in the Arthur’s parametrization of the representation.
</p>projecteuclid.org/euclid.ant/1534212098_20180813220159Mon, 13 Aug 2018 22:01 EDTCorrespondences without a corehttps://projecteuclid.org/euclid.ant/1534212099<strong>Raju Krishnamoorthy</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 5, 1173--1214.</p><p><strong>Abstract:</strong><br/>
We study the formal properties of correspondences of curves without a core, focusing on the case of étale correspondences. The motivating examples come from Hecke correspondences of Shimura curves. Given a correspondence without a core, we construct an infinite graph [math] together with a large group of “algebraic” automorphisms [math] . The graph [math] measures the “generic dynamics” of the correspondence. We construct specialization maps [math] to the “physical dynamics” of the correspondence. Motivated by the abstract structure of the supersingular locus, we also prove results on the number of bounded étale orbits, in particular generalizing a recent theorem of Hallouin and Perret. We use a variety of techniques: Galois theory, the theory of groups acting on infinite graphs, and finite group schemes.
</p>projecteuclid.org/euclid.ant/1534212099_20180813220159Mon, 13 Aug 2018 22:01 EDTLocal topological algebraicity with algebraic coefficients of analytic sets or functionshttps://projecteuclid.org/euclid.ant/1534212100<strong>Guillaume Rond</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 5, 1215--1231.</p><p><strong>Abstract:</strong><br/>
We prove that any complex or real analytic set or function germ is topologically equivalent to a germ defined by polynomial equations whose coefficients are algebraic numbers.
</p>projecteuclid.org/euclid.ant/1534212100_20180813220159Mon, 13 Aug 2018 22:01 EDTPolynomial bound for the nilpotency index of finitely generated nil algebrashttps://projecteuclid.org/euclid.ant/1534212101<strong>Mátyás Domokos</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 5, 1233--1242.</p><p><strong>Abstract:</strong><br/>
Working over an infinite field of positive characteristic, an upper bound is given for the nilpotency index of a finitely generated nil algebra of bounded nil index [math] in terms of the maximal degree in a minimal homogenous generating system of the ring of simultaneous conjugation invariants of tuples of [math] -by- [math] matrices. This is deduced from a result of Zubkov. As a consequence, a recent degree bound due to Derksen and Makam for the generators of the ring of matrix invariants yields an upper bound for the nilpotency index of a finitely generated nil algebra that is polynomial in the number of generators and the nil index. Furthermore, a characteristic free treatment is given to Kuzmin’s lower bound for the nilpotency index.
</p>projecteuclid.org/euclid.ant/1534212101_20180813220159Mon, 13 Aug 2018 22:01 EDTArithmetic functions in short intervals and the symmetric grouphttps://projecteuclid.org/euclid.ant/1534212102<strong>Brad Rodgers</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 5, 1243--1279.</p><p><strong>Abstract:</strong><br/>
We consider the variance of sums of arithmetic functions over random short intervals in the function field setting. Based on the analogy between factorizations of random elements of [math] into primes and the factorizations of random permutations into cycles, we give a simple but general formula for these variances in the large [math] limit for arithmetic functions that depend only upon factorization structure. From this we derive new estimates, quickly recover some that are already known, and make new conjectures in the setting of the integers.
In particular we make the combinatorial observation that any function of this sort can be explicitly decomposed into a sum of functions [math] and [math] , depending on the size of the short interval, with [math] making a negligible contribution to the variance, and [math] asymptotically contributing diagonal terms only.
This variance evaluation is closely related to the appearance of random matrix statistics in the zeros of families of [math] -functions and sheds light on the arithmetic meaning of this phenomenon.
</p>projecteuclid.org/euclid.ant/1534212102_20180813220159Mon, 13 Aug 2018 22:01 EDTCohomology for Drinfeld doubles of some infinitesimal group schemeshttps://projecteuclid.org/euclid.ant/1534212103<strong>Eric Friedlander</strong>, <strong>Cris Negron</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 5, 1281--1309.</p><p><strong>Abstract:</strong><br/>
Consider a field [math] of characteristic [math] , the [math] -th Frobenius kernel [math] of a smooth algebraic group [math] , the Drinfeld double [math] of [math] , and a finite dimensional [math] -module [math] . We prove that the cohomology algebra [math] is finitely generated and that [math] is a finitely generated module over this cohomology algebra. We exhibit a finite map of algebras [math] , which offers an approach to support varieties for [math] -modules. For many examples of interest, [math] is injective and induces an isomorphism of associated reduced schemes. For [math] an irreducible [math] -module, [math] enables us to identify the support variety of [math] in terms of the support variety of [math] viewed as a [math] -module.
</p>projecteuclid.org/euclid.ant/1534212103_20180813220159Mon, 13 Aug 2018 22:01 EDTGeneralized Fourier coefficients of multiplicative functionshttps://projecteuclid.org/euclid.ant/1540432828<strong>Lilian Matthiesen</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 6, 1311--1400.</p><p><strong>Abstract:</strong><br/>
We introduce and analyze a general class of not necessarily bounded multiplicative functions, examples of which include the function [math] , where [math] and where [math] counts the number of distinct prime factors of [math] , as well as the function [math] , where [math] denotes the Fourier coefficients of a primitive holomorphic cusp form.
For this class of functions we show that after applying a [math] -trick, their elements become orthogonal to polynomial nilsequences. The resulting functions therefore have small uniformity norms of all orders by the Green–Tao–Ziegler inverse theorem, a consequence that will be used in a separate paper in order to asymptotically evaluate linear correlations of multiplicative functions from our class. Our result generalizes work of Green and Tao on the Möbius function.
</p>projecteuclid.org/euclid.ant/1540432828_20181024220049Wed, 24 Oct 2018 22:00 EDTA blowup algebra for hyperplane arrangementshttps://projecteuclid.org/euclid.ant/1540432833<strong>Mehdi Garrousian</strong>, <strong>Aron Simis</strong>, <strong>Ştefan O. Tohăneanu</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 6, 1401--1429.</p><p><strong>Abstract:</strong><br/>
It is shown that the Orlik–Terao algebra is graded isomorphic to the special fiber of the ideal [math] generated by the [math] -fold products of the members of a central arrangement of size [math] . This momentum is carried over to the Rees algebra (blowup) of [math] and it is shown that this algebra is of fiber-type and Cohen–Macaulay. It follows by a result of Simis and Vasconcelos that the special fiber of [math] is Cohen–Macaulay, thus giving another proof of a result of Proudfoot and Speyer about the Cohen–Macaulayness of the Orlik–Terao algebra.
</p>projecteuclid.org/euclid.ant/1540432833_20181024220049Wed, 24 Oct 2018 22:00 EDTTorsion in the 0-cycle group with modulushttps://projecteuclid.org/euclid.ant/1540432834<strong>Amalendu Krishna</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 6, 1431--1469.</p><p><strong>Abstract:</strong><br/>
We show, for a smooth projective variety [math] over an algebraically closed field [math] with an effective Cartier divisor [math] , that the torsion subgroup [math] can be described in terms of a relative étale cohomology for any prime [math] . This extends a classical result of Bloch, on the torsion in the ordinary Chow group, to the modulus setting. We prove the Roitman torsion theorem (including [math] -torsion) for [math] when [math] is reduced. We deduce applications to the problem of invariance of the prime-to- [math] torsion in [math] under an infinitesimal extension of [math] .
</p>projecteuclid.org/euclid.ant/1540432834_20181024220049Wed, 24 Oct 2018 22:00 EDTContinuity of the Green function in meromorphic families of polynomialshttps://projecteuclid.org/euclid.ant/1540432835<strong>Charles Favre</strong>, <strong>Thomas Gauthier</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 6, 1471--1487.</p><p><strong>Abstract:</strong><br/>
We prove that along any marked point the Green function of a meromorphic family of polynomials parametrized by the punctured unit disk is the sum of a logarithmic term and a continuous function.
</p>projecteuclid.org/euclid.ant/1540432835_20181024220049Wed, 24 Oct 2018 22:00 EDTLocal root numbers and spectrum of the local descents for orthogonal groups: $p$-adic casehttps://projecteuclid.org/euclid.ant/1540432836<strong>Dihua Jiang</strong>, <strong>Lei Zhang</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 6, 1489--1535.</p><p><strong>Abstract:</strong><br/>
We investigate the local descents for special orthogonal groups over [math] -adic local fields of characteristic zero, and obtain explicit spectral decomposition of the local descents at the first occurrence index in terms of the local Langlands data via the explicit local Langlands correspondence and explicit calculations of relevant local root numbers. The main result can be regarded as a refinement of the local Gan–Gross–Prasad conjecture (2012).
</p>projecteuclid.org/euclid.ant/1540432836_20181024220049Wed, 24 Oct 2018 22:00 EDTBases for quasisimple linear groupshttps://projecteuclid.org/euclid.ant/1540432837<strong>Melissa Lee</strong>, <strong>Martin W. Liebeck</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 6, 1537--1557.</p><p><strong>Abstract:</strong><br/>
Let [math] be a vector space of dimension [math] over [math] , a finite field of [math] elements, and let [math] be a linear group. A base for [math] is a set of vectors whose pointwise stabilizer in [math] is trivial. We prove that if [math] is a quasisimple group (i.e., [math] is perfect and [math] is simple) acting irreducibly on [math] , then excluding two natural families, [math] has a base of size at most 6. The two families consist of alternating groups [math] acting on the natural module of dimension [math] or [math] , and classical groups with natural module of dimension [math] over subfields of [math] .
</p>projecteuclid.org/euclid.ant/1540432837_20181024220049Wed, 24 Oct 2018 22:00 EDTDifference modules and difference cohomologyhttps://projecteuclid.org/euclid.ant/1541732433<strong>Marcin Chałupnik</strong>, <strong>Piotr Kowalski</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 7, 1559--1580.</p><p><strong>Abstract:</strong><br/>
We give some basics about homological algebra of difference representations. We consider both the difference discrete and the difference rational case. We define the corresponding cohomology theories and show the existence of spectral sequences relating these cohomology theories with the standard ones.
</p>projecteuclid.org/euclid.ant/1541732433_20181108220048Thu, 08 Nov 2018 22:00 ESTDensity theorems for exceptional eigenvalues for congruence subgroupshttps://projecteuclid.org/euclid.ant/1541732434<strong>Peter Humphries</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 7, 1581--1610.</p><p><strong>Abstract:</strong><br/>
Using the Kuznetsov formula, we prove several density theorems for exceptional Hecke and Laplacian eigenvalues of Maaß cusp forms of weight [math] or [math] for the congruence subgroups [math] , [math] , and [math] . These improve and extend upon results of Sarnak and Huxley, who prove similar but slightly weaker results via the Selberg trace formula.
</p>projecteuclid.org/euclid.ant/1541732434_20181108220048Thu, 08 Nov 2018 22:00 ESTIrreducible components of minuscule affine Deligne–Lusztig varietieshttps://projecteuclid.org/euclid.ant/1541732435<strong>Paul Hamacher</strong>, <strong>Eva Viehmann</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 7, 1611--1634.</p><p><strong>Abstract:</strong><br/>
We examine the set of [math] -orbits in the set of irreducible components of affine Deligne–Lusztig varieties for a hyperspecial subgroup and minuscule coweight [math] . Our description implies in particular that its number of elements is bounded by the dimension of a suitable weight space in the Weyl module associated with [math] of the dual group.
</p>projecteuclid.org/euclid.ant/1541732435_20181108220048Thu, 08 Nov 2018 22:00 ESTArithmetic degrees and dynamical degrees of endomorphisms on surfaceshttps://projecteuclid.org/euclid.ant/1541732436<strong>Yohsuke Matsuzawa</strong>, <strong>Kaoru Sano</strong>, <strong>Takahiro Shibata</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 7, 1635--1657.</p><p><strong>Abstract:</strong><br/>
For a dominant rational self-map on a smooth projective variety defined over a number field, Kawaguchi and Silverman conjectured that the (first) dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is well-defined and Zariski dense. We prove this conjecture for surjective endomorphisms on smooth projective surfaces. For surjective endomorphisms on any smooth projective varieties, we show the existence of rational points whose arithmetic degrees are equal to the dynamical degree. Moreover, if the map is an automorphism, there exists a Zariski dense set of such points with pairwise disjoint orbits.
</p>projecteuclid.org/euclid.ant/1541732436_20181108220048Thu, 08 Nov 2018 22:00 ESTBig Cohen–Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristichttps://projecteuclid.org/euclid.ant/1541732437<strong>Raymond Heitmann</strong>, <strong>Linquan Ma</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 7, 1659--1674.</p><p><strong>Abstract:</strong><br/>
We prove a version of weakly functorial big Cohen–Macaulay algebras that suffices to establish Hochster and Huneke’s vanishing conjecture for maps of Tor in mixed characteristic. As a corollary, we prove an analog of Boutot’s theorem that direct summands of regular rings are pseudorational in mixed characteristic. Our proof uses perfectoid spaces and is inspired by the recent breakthroughs on the direct summand conjecture by André and Bhatt.
</p>projecteuclid.org/euclid.ant/1541732437_20181108220048Thu, 08 Nov 2018 22:00 ESTBlocks of the category of smooth $\ell$-modular representations of GL$(n,F)$ and its inner forms: reduction to level 0https://projecteuclid.org/euclid.ant/1541732438<strong>Gianmarco Chinello</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 7, 1675--1713.</p><p><strong>Abstract:</strong><br/>
Let [math] be an inner form of a general linear group over a nonarchimedean locally compact field of residue characteristic [math] , let [math] be an algebraically closed field of characteristic different from [math] and let [math] be the category of smooth representations of [math] over [math] . In this paper, we prove that a block (indecomposable summand) of [math] is equivalent to a level- [math] block (a block in which every simple object has nonzero invariant vectors for the pro- [math] -radical of a maximal compact open subgroup) of [math] , where [math] is a direct product of groups of the same type of [math] .
</p>projecteuclid.org/euclid.ant/1541732438_20181108220048Thu, 08 Nov 2018 22:00 ESTAlgebraic dynamics of the lifts of Frobeniushttps://projecteuclid.org/euclid.ant/1541732439<strong>Junyi Xie</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 7, 1715--1748.</p><p><strong>Abstract:</strong><br/>
We study the algebraic dynamics of endomorphisms of projective spaces with coefficients in a [math] -adic field whose reduction in positive characteristic is the Frobenius. In particular, we prove a version of the dynamical Manin–Mumford conjecture and the dynamical Mordell–Lang conjecture for the coherent backward orbits of such endomorphisms. We also give a new proof of a dynamical version of the Tate–Voloch conjecture in this case. Our method is based on the theory of perfectoid spaces introduced by P. Scholze. In the appendix, we prove that under some technical condition on the field of definition, a dynamical system for a polarized lift of Frobenius on a projective variety can be embedded into a dynamical system for some endomorphism of a projective space.
</p>projecteuclid.org/euclid.ant/1541732439_20181108220048Thu, 08 Nov 2018 22:00 ESTA dynamical variant of the Pink–Zilber conjecturehttps://projecteuclid.org/euclid.ant/1541732440<strong>Dragos Ghioca</strong>, <strong>Khoa Dang Nguyen</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 7, 1749--1771.</p><p><strong>Abstract:</strong><br/>
Let [math] be polynomials of degree [math] such that no [math] is conjugate to [math] or to [math] , where [math] is the Chebyshev polynomial of degree [math] . We let [math] be their coordinatewise action on [math] , i.e., [math] is given by [math] . We prove a dynamical version of the Pink–Zilber conjecture for subvarieties [math] of [math] with respect to the dynamical system [math] , if [math] .
</p>projecteuclid.org/euclid.ant/1541732440_20181108220048Thu, 08 Nov 2018 22:00 ESTHomogeneous length functions on groupshttps://projecteuclid.org/euclid.ant/1541732441<strong>Tobias Fritz</strong>, <strong>Siddhartha Gadgil</strong>, <strong>Apoorva Khare</strong>, <strong>Pace Nielsen</strong>, <strong>Lior Silberman</strong>, <strong>Terence Tao</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 7, 1773--1786.</p><p><strong>Abstract:</strong><br/>
A pseudolength function defined on an arbitrary group [math] is a map [math] obeying [math] , the symmetry property [math] , and the triangle inequality [math] for all [math] . We consider pseudolength functions which saturate the triangle inequality whenever [math] , or equivalently those that are homogeneous in the sense that [math] for all [math] . We show that this implies that [math] for all [math] . This leads to a classification of such pseudolength functions as pullbacks from embeddings into a Banach space. We also obtain a quantitative version of our main result which allows for defects in the triangle inequality or the homogeneity property.
</p>projecteuclid.org/euclid.ant/1541732441_20181108220048Thu, 08 Nov 2018 22:00 ESTWhen are permutation invariants Cohen–Macaulay over all fields?https://projecteuclid.org/euclid.ant/1541732442<strong>Ben Blum-Smith</strong>, <strong>Sophie Marques</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 7, 1787--1821.</p><p><strong>Abstract:</strong><br/>
We prove that the polynomial invariants of a permutation group are Cohen–Macaulay for any choice of coefficient field if and only if the group is generated by transpositions, double transpositions, and 3-cycles. This unites and generalizes several previously known results. The “if” direction of the argument uses Stanley–Reisner theory and a recent result of Christian Lange in orbifold theory. The “only if” direction uses a local-global result based on a theorem of Raynaud to reduce the problem to an analysis of inertia groups, and a combinatorial argument to identify inertia groups that obstruct Cohen–Macaulayness.
</p>projecteuclid.org/euclid.ant/1541732442_20181108220048Thu, 08 Nov 2018 22:00 ESTOn the relative Galois module structure of rings of integers in tame extensionshttps://projecteuclid.org/euclid.ant/1545361461<strong>Adebisi Agboola</strong>, <strong>Leon R. McCulloh</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 8, 1823--1886.</p><p><strong>Abstract:</strong><br/>
Let [math] be a number field with ring of integers [math] and let [math] be a finite group. We describe an approach to the study of the set of realisable classes in the locally free class group [math] of [math] that involves applying the work of McCulloh in the context of relative algebraic [math] theory. For a large class of soluble groups [math] , including all groups of odd order, we show (subject to certain mild conditions) that the set of realisable classes is a subgroup of [math] . This may be viewed as being a partial analogue in the setting of Galois module theory of a classical theorem of Shafarevich on the inverse Galois problem for soluble groups.
</p>projecteuclid.org/euclid.ant/1545361461_20181220220432Thu, 20 Dec 2018 22:04 ESTCategorical representations and KLR algebrashttps://projecteuclid.org/euclid.ant/1545361464<strong>Ruslan Maksimau</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 8, 1887--1921.</p><p><strong>Abstract:</strong><br/>
We prove that the KLR algebra associated with the cyclic quiver of length [math] is a subquotient of the KLR algebra associated with the cyclic quiver of length [math] . We also give a geometric interpretation of this fact. This result has an important application in the theory of categorical representations. We prove that a category with an action of [math] contains a subcategory with an action of [math] . We also give generalizations of these results to more general quivers and Lie types.
</p>projecteuclid.org/euclid.ant/1545361464_20181220220432Thu, 20 Dec 2018 22:04 ESTOn nonprimitive Weierstrass pointshttps://projecteuclid.org/euclid.ant/1545361465<strong>Nathan Pflueger</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 8, 1923--1947.</p><p><strong>Abstract:</strong><br/>
We give an upper bound for the codimension in [math] of the variety [math] of marked curves [math] with a given Weierstrass semigroup. The bound is a combinatorial quantity which we call the effective weight of the semigroup; it is a refinement of the weight of the semigroup, and differs from the weight precisely when the semigroup is not primitive. We prove that whenever the effective weight is less than [math] , the variety [math] is nonempty and has a component of the predicted codimension. These results extend previous results of Eisenbud, Harris, and Komeda to the case of nonprimitive semigroups. We also survey other cases where the codimension of [math] is known, as evidence that the effective weight estimate is correct in wider circumstances.
</p>projecteuclid.org/euclid.ant/1545361465_20181220220432Thu, 20 Dec 2018 22:04 ESTBounded generation of $\mathrm{SL}_2$ over rings of $S$-integers with infinitely many unitshttps://projecteuclid.org/euclid.ant/1545361466<strong>Aleksander V. Morgan</strong>, <strong>Andrei S. Rapinchuk</strong>, <strong>Balasubramanian Sury</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 8, 1949--1974.</p><p><strong>Abstract:</strong><br/>
Let [math] be the ring of [math] -integers in a number field [math] . We prove that if the group of units [math] is infinite then every matrix in [math] is a product of at most 9 elementary matrices. This essentially completes a long line of research in this direction. As a consequence, we obtain a new proof of the fact that [math] is boundedly generated as an abstract group that uses only standard results from algebraic number theory.
</p>projecteuclid.org/euclid.ant/1545361466_20181220220432Thu, 20 Dec 2018 22:04 ESTTensor triangular geometry of filtered moduleshttps://projecteuclid.org/euclid.ant/1545361467<strong>Martin Gallauer</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 8, 1975--2003.</p><p><strong>Abstract:</strong><br/>
We compute the tensor triangular spectrum of perfect complexes of filtered modules over a commutative ring and deduce a classification of the thick tensor ideals. We give two proofs: one by reducing to perfect complexes of graded modules which have already been studied in the literature by Dell’Ambrogio and Stevenson (2013, 2014) and one more direct for which we develop some useful tools.
</p>projecteuclid.org/euclid.ant/1545361467_20181220220432Thu, 20 Dec 2018 22:04 ESTThe Euclidean distance degree of smooth complex projective varietieshttps://projecteuclid.org/euclid.ant/1545361468<strong>Paolo Aluffi</strong>, <strong>Corey Harris</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 8, 2005--2032.</p><p><strong>Abstract:</strong><br/>
We obtain several formulas for the Euclidean distance degree (ED degree) of an arbitrary nonsingular variety in projective space: in terms of Chern and Segre classes, Milnor classes, Chern–Schwartz–MacPherson classes, and an extremely simple formula equating the Euclidean distance degree of [math] with the Euler characteristic of an open subset of [math] .
</p>projecteuclid.org/euclid.ant/1545361468_20181220220432Thu, 20 Dec 2018 22:04 ESTMicrolocal lifts and quantum unique ergodicity on $GL_2(\mathbb{Q}_p)$https://projecteuclid.org/euclid.ant/1546657274<strong>Paul D. Nelson</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 9, 2033--2064.</p><p><strong>Abstract:</strong><br/>
We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of [math] for automorphic forms belonging to the principal series. We interpret this conclusion in terms of the equidistribution of eigenfunctions on covers of a fixed regular graph or along nested sequences of regular graphs.
Our results are the first of their kind on any [math] -adic arithmetic quotient. They may be understood as analogues of Lindenstrauss’s theorem on the equidistribution of Maass forms on a compact arithmetic surface. The new ingredients here include the introduction of a representation-theoretic notion of “ [math] -adic microlocal lifts” with favorable properties, such as diagonal invariance of limit measures; the proof of positive entropy of limit measures in a [math] -adic aspect, following the method of Bourgain–Lindenstrauss; and some analysis of local Rankin–Selberg integrals involving the microlocal lifts introduced here as well as classical newvectors. An important input is a measure-classification result of Einsiedler–Lindenstrauss.
</p>projecteuclid.org/euclid.ant/1546657274_20190104220133Fri, 04 Jan 2019 22:01 ESTHeights on squares of modular curveshttps://projecteuclid.org/euclid.ant/1546657275<strong>Pierre Parent</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 9, 2065--2122.</p><p><strong>Abstract:</strong><br/>
We develop a strategy for bounding from above the height of rational points of modular curves with values in number fields, by functions which are polynomial in the curve’s level. Our main technical tools come from effective Arakelov descriptions of modular curves and jacobians. We then fulfill this program in the following particular case:
If [math] is a not-too-small prime number, let [math] be the classical modular curve of level [math] over [math] . Assume Brumer’s conjecture on the dimension of winding quotients of [math] . We prove that there is a function [math] (depending only on [math] ) such that, for any quadratic number field [math] , the [math] -height of points in [math] which are not lifts of elements of [math] is less or equal to [math] .
</p>projecteuclid.org/euclid.ant/1546657275_20190104220133Fri, 04 Jan 2019 22:01 ESTA formula for the Jacobian of a genus one curve of arbitrary degreehttps://projecteuclid.org/euclid.ant/1546657276<strong>Tom Fisher</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 9, 2123--2150.</p><p><strong>Abstract:</strong><br/>
We extend the formulae of classical invariant theory for the Jacobian of a genus one curve of degree [math] to curves of arbitrary degree. To do this, we associate to each genus one normal curve of degree [math] , an [math] alternating matrix of quadratic forms in [math] variables, that represents the invariant differential. We then exhibit the invariants we need as homogeneous polynomials of degrees [math] and [math] in the coefficients of the entries of this matrix.
</p>projecteuclid.org/euclid.ant/1546657276_20190104220133Fri, 04 Jan 2019 22:01 ESTRandom flag complexes and asymptotic syzygieshttps://projecteuclid.org/euclid.ant/1546657277<strong>Daniel Erman</strong>, <strong>Jay Yang</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 9, 2151--2166.</p><p><strong>Abstract:</strong><br/>
We use the probabilistic method to construct examples of conjectured phenomena about asymptotic syzygies. In particular, we use Stanley–Reisner ideals of random flag complexes to construct new examples of Ein and Lazarsfeld’s nonvanishing for asymptotic syzygies and of Ein, Erman, and Lazarsfeld’s conjecture on how asymptotic Betti numbers behave like binomial coefficients.
</p>projecteuclid.org/euclid.ant/1546657277_20190104220133Fri, 04 Jan 2019 22:01 ESTGrothendieck rings for Lie superalgebras and the Duflo–Serganova functorhttps://projecteuclid.org/euclid.ant/1546657278<strong>Crystal Hoyt</strong>, <strong>Shifra Reif</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 9, 2167--2184.</p><p><strong>Abstract:</strong><br/>
We show that the Duflo–Serganova functor on the category of finite-dimensional modules over a finite-dimensional contragredient Lie superalgebra induces a ring homomorphism on a natural quotient of the Grothendieck ring, which is isomorphic to the ring of supercharacters. We realize this homomorphism as a certain evaluation of functions related to the supersymmetry property. We use this realization to describe the kernel and image of the homomorphism induced by the Duflo–Serganova functor.
</p>projecteuclid.org/euclid.ant/1546657278_20190104220133Fri, 04 Jan 2019 22:01 ESTDynamics on abelian varieties in positive characteristichttps://projecteuclid.org/euclid.ant/1546657279<strong>Jakub Byszewski</strong>, <strong>Gunther Cornelissen</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 9, 2185--2235.</p><p><strong>Abstract:</strong><br/>
We study periodic points and orbit length distribution for endomorphisms of abelian varieties in characteristic [math] . We study rationality, algebraicity and the natural boundary property for the dynamical zeta function (the latter using a general result on power series proven by Royals and Ward in the appendix), as well as analogues of the prime number theorem, also for tame dynamics, ignoring orbits whose order is divisible by [math] . The behavior is governed by whether or not the action on the local [math] -torsion group scheme is nilpotent.
</p>projecteuclid.org/euclid.ant/1546657279_20190104220133Fri, 04 Jan 2019 22:01 ESTHigher weight on GL(3), II: The cusp formshttps://projecteuclid.org/euclid.ant/1550113223<strong>Jack Buttcane</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 10, 2237--2294.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to collect, extend, and make explicit the results of Gel’fand, Graev and Piatetski-Shapiro and Miyazaki for the [math] cusp forms which are nontrivial on [math] . We give new descriptions of the spaces of cusp forms of minimal [math] -type and from the Fourier–Whittaker expansions of such forms give a complete and completely explicit spectral expansion for [math] , accounting for multiplicities, in the style of Duke, Friedlander and Iwaniec’s paper. We do this at a level of uniformity suitable for Poincaré series which are not necessarily [math] -finite. We directly compute the Jacquet integral for the Whittaker functions at the minimal [math] -type, improving Miyazaki’s computation. These results will form the basis of the nonspherical spectral Kuznetsov formulas and the arithmetic/geometric Kuznetsov formulas on [math] . The primary tool will be the study of the differential operators coming from the Lie algebra on vector-valued cusp forms.
</p>projecteuclid.org/euclid.ant/1550113223_20190213220031Wed, 13 Feb 2019 22:00 ESTStark systems over Gorenstein local ringshttps://projecteuclid.org/euclid.ant/1550113224<strong>Ryotaro Sakamoto</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 10, 2295--2326.</p><p><strong>Abstract:</strong><br/>
In this paper, we define a Stark system over a complete Gorenstein local ring with a finite residue field. Under some standard assumptions, we show that the module of Stark systems is free of rank 1 and that these systems control all the higher Fitting ideals of the Pontryagin dual of the dual Selmer group. This is a generalization of the theory, developed by B. Mazur and K. Rubin, on Stark (or Kolyvagin) systems over principal ideal local rings. Applying our result to a certain Selmer structure over the cyclotomic Iwasawa algebra, we propose a new method for controlling Selmer groups using Euler systems.
</p>projecteuclid.org/euclid.ant/1550113224_20190213220031Wed, 13 Feb 2019 22:00 ESTJordan blocks of cuspidal representations of symplectic groupshttps://projecteuclid.org/euclid.ant/1550113225<strong>Corinne Blondel</strong>, <strong>Guy Henniart</strong>, <strong>Shaun Stevens</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 10, 2327--2386.</p><p><strong>Abstract:</strong><br/>
Let [math] be a symplectic group over a nonarchimedean local field of characteristic zero and odd residual characteristic. Given an irreducible cuspidal representation of [math] , we determine its Langlands parameter (equivalently, its Jordan blocks in the language of Mœglin) in terms of the local data from which the representation is explicitly constructed, up to a possible unramified twist in each block of the parameter. We deduce a ramification theorem for [math] , giving a bijection between the set of endoparameters for [math] and the set of restrictions to wild inertia of discrete Langlands parameters for [math] , compatible with the local Langlands correspondence. The main tool consists in analyzing the Hecke algebra of a good cover, in the sense of Bushnell–Kutzko, for parabolic induction from a cuspidal representation of [math] , seen as a maximal Levi subgroup of a bigger symplectic group, in order to determine reducibility points; a criterion of Mœglin then relates this to Langlands parameters.
</p>projecteuclid.org/euclid.ant/1550113225_20190213220031Wed, 13 Feb 2019 22:00 ESTRealizing 2-groups as Galois groups following Shafarevich and Serrehttps://projecteuclid.org/euclid.ant/1550113226<strong>Peter Schmid</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 10, 2387--2401.</p><p><strong>Abstract:</strong><br/>
Let [math] be a finite [math] -group for some prime [math] , say of order [math] . For odd [math] the inverse problem of Galois theory for [math] has been solved through the (classical) work of Scholz and Reichardt, and Serre has shown that their method leads to fields of realization where at most [math] rational primes are (tamely) ramified. The approach by Shafarevich, for arbitrary [math] , has turned out to be quite delicate in the case [math] . In this paper we treat this exceptional case in the spirit of Serre’s result, bounding the number of ramified primes at least by an integral polynomial in the rank of [math] , the polynomial depending on the [math] -class of [math] .
</p>projecteuclid.org/euclid.ant/1550113226_20190213220031Wed, 13 Feb 2019 22:00 ESTHeights of hypersurfaces in toric varietieshttps://projecteuclid.org/euclid.ant/1550113227<strong>Roberto Gualdi</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 10, 2403--2443.</p><p><strong>Abstract:</strong><br/>
For a cycle of codimension 1 in a toric variety, its degree with respect to a nef toric divisor can be understood in terms of the mixed volume of the polytopes associated to the divisor and to the cycle. We prove here that an analogous combinatorial formula holds in the arithmetic setting: the global height of a 1-codimensional cycle with respect to a toric divisor equipped with a semipositive toric metric can be expressed in terms of mixed integrals of the [math] -adic roof functions associated to the metric and the Legendre–Fenchel dual of the [math] -adic Ronkin function of the Laurent polynomial of the cycle.
</p>projecteuclid.org/euclid.ant/1550113227_20190213220031Wed, 13 Feb 2019 22:00 ESTDegree and the Brauer–Manin obstructionhttps://projecteuclid.org/euclid.ant/1550113228<strong>Brendan Creutz</strong>, <strong>Bianca Viray</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 10, 2445--2470.</p><p><strong>Abstract:</strong><br/>
Let [math] be a smooth projective variety of degree [math] over a number field [math] and suppose that [math] is a counterexample to the Hasse principle explained by the Brauer–Manin obstruction. We consider the question of whether the obstruction is given by the [math] -primary subgroup of the Brauer group, which would have both theoretic and algorithmic implications. We prove that this question has a positive answer in the case of torsors under abelian varieties, Kummer surfaces and (conditional on finiteness of Tate–Shafarevich groups) bielliptic surfaces. In the case of Kummer surfaces we show, more specifically, that the obstruction is already given by the [math] -primary torsion, and indeed that this holds for higher-dimensional Kummer varieties as well. We construct a conic bundle over an elliptic curve that shows that, in general, the answer is no.
</p>projecteuclid.org/euclid.ant/1550113228_20190213220031Wed, 13 Feb 2019 22:00 ESTBounds for traces of Hecke operators and applications to modular and elliptic curves over a finite fieldhttps://projecteuclid.org/euclid.ant/1550113229<strong>Ian Petrow</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 10, 2471--2498.</p><p><strong>Abstract:</strong><br/>
We give an upper bound for the trace of a Hecke operator acting on the space of holomorphic cusp forms with respect to certain congruence subgroups. Such an estimate has applications to the analytic theory of elliptic curves over a finite field, going beyond the Riemann hypothesis over finite fields. As the main tool to prove our bound on traces of Hecke operators, we develop a Petersson formula for newforms for general nebentype characters.
</p>projecteuclid.org/euclid.ant/1550113229_20190213220031Wed, 13 Feb 2019 22:00 EST2-parts of real class sizeshttps://projecteuclid.org/euclid.ant/1550113230<strong>Hung P. Tong-Viet</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 12, Number 10, 2499--2514.</p><p><strong>Abstract:</strong><br/>
We investigate the structure of finite groups whose noncentral real class sizes have the same [math] -part. In particular, we prove that such groups are solvable and have [math] -length one. As a consequence, we show that a finite group is solvable if it has two real class sizes. This confirms a conjecture due to G. Navarro, L. Sanus and P. Tiep.
</p>projecteuclid.org/euclid.ant/1550113230_20190213220031Wed, 13 Feb 2019 22:00 EST