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 EDTBounded 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 ESTHigh moments of the Estermann functionhttps://projecteuclid.org/euclid.ant/1553565643<strong>Sandro Bettin</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 2, 251--300.</p><p><strong>Abstract:</strong><br/>
For [math] the Estermann function is defined as [math] if [math] and by meromorphic continuation otherwise. For [math] prime, we compute the moments of [math] at the central point [math] , when averaging over [math] .
As a consequence we deduce the asymptotic for the iterated moment of Dirichlet [math] -functions [math] , obtaining a power saving error term.
Also, we compute the moments of certain functions defined in terms of continued fractions. For example, writing [math] where [math] is the continued fraction expansion of [math] we prove that for [math] and [math] primes one has [math] as [math] .
</p>projecteuclid.org/euclid.ant/1553565643_20190325220054Mon, 25 Mar 2019 22:00 EDTLe théorème de Fermat sur certains corps de nombres totalement réelshttps://projecteuclid.org/euclid.ant/1553565644<strong>Alain Kraus</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 2, 301--332.</p><p><strong>Abstract:</strong><br/>
Let [math] be a totally real number field. For all prime number [math] , let us denote by [math] the Fermat curve of equation [math] . Under the assumption that [math] is totally ramified in [math] , we establish some results about the set [math] of points of [math] rational over [math] . We obtain a criterion so that the asymptotic Fermat’s last theorem is true over [math] , criterion related to the set of Hilbert modular cuspidal newforms over [math] , of parallel weight [math] and of level the prime ideal above [math] . It is often simply testable numerically, particularly if the narrow class number of [math] is [math] . Furthermore, using the modular method, we prove Fermat’s last theorem effectively, over some number fields whose degrees over [math] are [math] and [math] .
</p>projecteuclid.org/euclid.ant/1553565644_20190325220054Mon, 25 Mar 2019 22:00 EDT$G$-valued local deformation rings and global liftshttps://projecteuclid.org/euclid.ant/1553565645<strong>Rebecca Bellovin</strong>, <strong>Toby Gee</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 2, 333--378.</p><p><strong>Abstract:</strong><br/>
We study [math] -valued Galois deformation rings with prescribed properties, where [math] is an arbitrary (not necessarily connected) reductive group over an extension of [math] for some prime [math] . In particular, for the Galois groups of [math] -adic local fields (with [math] possibly equal to [math] ) we prove that these rings are generically regular, compute their dimensions, and show that functorial operations on Galois representations give rise to well-defined maps between the sets of irreducible components of the corresponding deformation rings. We use these local results to prove lower bounds on the dimension of global deformation rings with prescribed local properties. Applying our results to unitary groups, we improve results in the literature on the existence of lifts of mod [math] Galois representations, and on the weight part of Serre’s conjecture.
</p>projecteuclid.org/euclid.ant/1553565645_20190325220054Mon, 25 Mar 2019 22:00 EDTFunctorial factorization of birational maps for qe schemes in characteristic 0https://projecteuclid.org/euclid.ant/1553565646<strong>Dan Abramovich</strong>, <strong>Michael Temkin</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 2, 379--424.</p><p><strong>Abstract:</strong><br/>
We prove functorial weak factorization of projective birational morphisms of regular quasiexcellent schemes in characteristic 0 broadly based on the existing line of proof for varieties. From this general functorial statement we deduce factorization results for algebraic stacks, formal schemes, complex analytic germs, Berkovich analytic and rigid analytic spaces, answering a present need in nonarchimedean geometry. Techniques developed for this purpose include a method for functorial factorization of toric maps, variation of GIT quotients relative to general noetherian qe schemes, and a GAGA theorem for Stein compacts.
</p>projecteuclid.org/euclid.ant/1553565646_20190325220054Mon, 25 Mar 2019 22:00 EDTEffective generation and twisted weak positivity of direct imageshttps://projecteuclid.org/euclid.ant/1553565647<strong>Yajnaseni Dutta</strong>, <strong>Takumi Murayama</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 2, 425--454.</p><p><strong>Abstract:</strong><br/>
We study pushforwards of log pluricanonical bundles on projective log canonical pairs [math] over the complex numbers, partially answering a Fujita-type conjecture due to Popa and Schnell in the log canonical setting. We show two effective global generation results. First, when [math] surjects onto a projective variety, we show a quadratic bound for generic generation for twists by big and nef line bundles. Second, when [math] is fibered over a smooth projective variety, we show a linear bound for twists by ample line bundles. These results additionally give effective nonvanishing statements. We also prove an effective weak positivity statement for log pluricanonical bundles in this setting, which may be of independent interest. In each context we indicate over which loci positivity holds. Finally, using the description of such loci, we show an effective vanishing theorem for pushforwards of certain log-sheaves under smooth morphisms.
</p>projecteuclid.org/euclid.ant/1553565647_20190325220054Mon, 25 Mar 2019 22:00 EDTLovász–Saks–Schrijver ideals and coordinate sections of determinantal varietieshttps://projecteuclid.org/euclid.ant/1553565648<strong>Aldo Conca</strong>, <strong>Volkmar Welker</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 2, 455--484.</p><p><strong>Abstract:</strong><br/>
Motivated by questions in algebra and combinatorics we study two ideals associated to a simple graph [math] :
the Lovász-Saks-Schrijver ideal defining the
[math] -dimensional
orthogonal representations of the graph complementary to
[math] ,
and
the determinantal ideal of the
[math] -minors
of a generic symmetric matrix with
[math]
in positions prescribed by the graph
[math] .
In characteristic [math] these two ideals turn out to be closely related and algebraic properties such as being radical, prime or a complete intersection transfer from the Lovász–Saks–Schrijver ideal to the determinantal ideal. For Lovász–Saks–Schrijver ideals we link these properties to combinatorial properties of [math] and show that they always hold for [math] large enough. For specific classes of graphs, such a forests, we can give a complete picture and classify the radical, prime and complete intersection Lovász–Saks–Schrijver ideals.
</p>projecteuclid.org/euclid.ant/1553565648_20190325220054Mon, 25 Mar 2019 22:00 EDTOn rational singularities and counting points of schemes over finite ringshttps://projecteuclid.org/euclid.ant/1553565649<strong>Itay Glazer</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 2, 485--500.</p><p><strong>Abstract:</strong><br/>
We study the connection between the singularities of a finite type [math] -scheme [math] and the asymptotic point count of [math] over various finite rings. In particular, if the generic fiber [math] is a local complete intersection, we show that the boundedness of [math] in [math] and [math] is in fact equivalent to the condition that [math] is reduced and has rational singularities. This paper completes a recent result of Aizenbud and Avni.
</p>projecteuclid.org/euclid.ant/1553565649_20190325220054Mon, 25 Mar 2019 22:00 EDTThe Maillot–Rössler current and the polylogarithm on abelian schemeshttps://projecteuclid.org/euclid.ant/1553565650<strong>Guido Kings</strong>, <strong>Danny Scarponi</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 2, 501--511.</p><p><strong>Abstract:</strong><br/>
We give a structural proof of the fact that the realization of the degree-zero part of the polylogarithm on abelian schemes in analytic Deligne cohomology can be described in terms of the Bismut–Köhler higher analytic torsion form of the Poincaré bundle. Furthermore, we provide a new axiomatic characterization of the arithmetic Chern character of the Poincaré bundle using only invariance properties under isogenies. For this we obtain a decomposition result for the arithmetic Chow group of independent interest.
</p>projecteuclid.org/euclid.ant/1553565650_20190325220054Mon, 25 Mar 2019 22:00 EDTEssential dimension of inseparable field extensionshttps://projecteuclid.org/euclid.ant/1553565651<strong>Zinovy Reichstein</strong>, <strong>Abhishek Kumar Shukla</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 2, 513--530.</p><p><strong>Abstract:</strong><br/>
Let [math] be a base field, [math] be a field containing [math] , and [math] be a field extension of degree [math] . The essential dimension [math] over [math] is a numerical invariant measuring “the complexity” of [math] . Of particular interest is
τ
(
n
)
=
max
{
ed
(
L
∕
K
)
∣
L
∕
K
is a separable extension of degree
n
}
,
also known as the essential dimension of the symmetric group [math] . The exact value of [math] is known only for [math] . In this paper we assume that [math] is a field of characteristic [math] and study the essential dimension of inseparable extensions [math] . Here the degree [math] is replaced by a pair [math] which accounts for the size of the separable and the purely inseparable parts of [math] , respectively, and [math] is replaced by
τ
(
n
,
e
)
=
max
{
ed
(
L
∕
K
)
∣
L
∕
K
is a field extension of type
(
n
,
e
)
}
.
The symmetric group [math] is replaced by a certain group scheme [math] over [math] . This group scheme is neither finite nor smooth; nevertheless, computing its essential dimension turns out to be easier than computing the essential dimension of [math] . Our main result is a simple formula for [math] .
</p>projecteuclid.org/euclid.ant/1553565651_20190325220054Mon, 25 Mar 2019 22:00 EDTOrdinary algebraic curves with many automorphisms in positive characteristichttps://projecteuclid.org/euclid.ant/1553652019<strong>Gábor Korchmáros</strong>, <strong>Maria Montanucci</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 1, 1--18.</p><p><strong>Abstract:</strong><br/>
Let [math] be an ordinary (projective, geometrically irreducible, nonsingular) algebraic curve of genus [math] defined over an algebraically closed field [math] of odd characteristic [math] . Let [math] be the group of all automorphisms of [math] which fix [math] elementwise. For any solvable subgroup [math] of [math] we prove that [math] . There are known curves attaining this bound up to the constant [math] . For [math] odd, our result improves the classical Nakajima bound [math] and, for solvable groups [math] , the Gunby–Smith–Yuan bound [math] where [math] for some positive constant [math] .
</p>projecteuclid.org/euclid.ant/1553652019_20190326220041Tue, 26 Mar 2019 22:00 EDTVariance of arithmetic sums and $L$-functions in $\mathbb{F}_q[t]$https://projecteuclid.org/euclid.ant/1553652021<strong>Chris Hall</strong>, <strong>Jonathan P. Keating</strong>, <strong>Edva Roditty-Gershon</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 1, 19--92.</p><p><strong>Abstract:</strong><br/>
We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain [math] -functions of degree 2 and higher in [math] , in the limit as [math] . This is achieved by establishing appropriate equidistribution results for the associated Frobenius conjugacy classes. The variances are thus related to matrix integrals, which may be evaluated. Our results differ significantly from those that hold in the case of degree-1 [math] -functions (i.e., situations considered previously using this approach). They correspond to expressions found recently in the number field setting assuming a generalization of the pair correlation conjecture. Our calculations apply, for example, to elliptic curves defined over [math] .
</p>projecteuclid.org/euclid.ant/1553652021_20190326220041Tue, 26 Mar 2019 22:00 EDTExtended eigenvarieties for overconvergent cohomologyhttps://projecteuclid.org/euclid.ant/1553652022<strong>Christian Johansson</strong>, <strong>James Newton</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 1, 93--158.</p><p><strong>Abstract:</strong><br/>
Recently, Andreatta, Iovita and Pilloni constructed spaces of overconvergent modular forms in characteristic [math] , together with a natural extension of the Coleman–Mazur eigencurve over a compactified (adic) weight space. Similar ideas have also been used by Liu, Wan and Xiao to study the boundary of the eigencurve. This all goes back to an idea of Coleman.
In this article, we construct natural extensions of eigenvarieties for arbitrary reductive groups [math] over a number field which are split at all places above [math] . If [math] is [math] , then we obtain a new construction of the extended eigencurve of Andreatta–Iovita–Pilloni. If [math] is an inner form of [math] associated to a definite quaternion algebra, our work gives a new perspective on some of the results of Liu–Wan–Xiao.
We build our extended eigenvarieties following Hansen’s construction using overconvergent cohomology. One key ingredient is a definition of locally analytic distribution modules which permits coefficients of characteristic [math] (and mixed characteristic). When [math] is [math] over a totally real or CM number field, we also construct a family of Galois representations over the reduced extended eigenvariety.
</p>projecteuclid.org/euclid.ant/1553652022_20190326220041Tue, 26 Mar 2019 22:00 EDTA tubular variant of Runge's method in all dimensions, with applications to integral points on Siegel modular varietieshttps://projecteuclid.org/euclid.ant/1553652025<strong>Samuel Le Fourn</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 1, 159--209.</p><p><strong>Abstract:</strong><br/>
Runge’s method is a tool to figure out integral points on algebraic curves effectively in terms of height. This method has been generalized to varieties of any dimension, but unfortunately the conditions needed to apply it are often too restrictive. We provide a further generalization intended to be more flexible while still effective, and exemplify its applicability by giving finiteness results for integral points on some Siegel modular varieties. As a special case, we obtain an explicit finiteness result for integral points on the Siegel modular variety [math] .
</p>projecteuclid.org/euclid.ant/1553652025_20190326220041Tue, 26 Mar 2019 22:00 EDTAlgebraic cycles on genus-2 modular fourfoldshttps://projecteuclid.org/euclid.ant/1553652028<strong>Donu Arapura</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 1, 211--225.</p><p><strong>Abstract:</strong><br/>
This paper studies universal families of stable genus-2 curves with level structure. Among other things, it is shown that the [math] -part is spanned by divisor classes, and that there are no cycles of type [math] in the third cohomology of the first direct image of [math] under projection to the moduli space of curves. Using this, it shown that the Hodge and Tate conjectures hold for these varieties.
</p>projecteuclid.org/euclid.ant/1553652028_20190326220041Tue, 26 Mar 2019 22:00 EDTAverage nonvanishing of Dirichlet $L$-functions at the central pointhttps://projecteuclid.org/euclid.ant/1553652029<strong>Kyle Pratt</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 1, 227--249.</p><p><strong>Abstract:</strong><br/>
The generalized Riemann hypothesis implies that at least 50% of the central values [math] are nonvanishing as [math] ranges over primitive characters modulo [math] . We show that one may unconditionally go beyond GRH, in the sense that if one averages over primitive characters modulo [math] and averages [math] over an interval, then at least 50.073% of the central values are nonvanishing. The proof utilizes the mollification method with a three-piece mollifier, and relies on estimates for sums of Kloosterman sums due to Deshouillers and Iwaniec.
</p>projecteuclid.org/euclid.ant/1553652029_20190326220041Tue, 26 Mar 2019 22:00 EDTFundamental gerbeshttps://projecteuclid.org/euclid.ant/1554775222<strong>Niels Borne</strong>, <strong>Angelo Vistoli</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 3, 531--576.</p><p><strong>Abstract:</strong><br/>
For a class of affine algebraic groups [math] over a field [math] , we define the notion of [math] -fundamental gerbe of a fibered category, generalizing what we did for finite group schemes in a 2015 paper.
We give necessary and sufficient conditions on [math] implying that a fibered category [math] over [math] satisfying mild hypotheses admits a Nori [math] -fundamental gerbe. We also give a tannakian interpretation of the gerbe that results by taking as [math] the class of virtually unipotent group schemes, under a properness condition on [math] .
Finally, we prove a general duality result, generalizing the duality between group schemes of multiplicative type and Galois modules, that yields a construction of the multiplicative gerbe of multiplicative type which is independent of the previous theory, and requires weaker hypotheses. This gives a conceptual interpretation of the universal torsor of Colliot-Thélène and Sansuc.
</p>projecteuclid.org/euclid.ant/1554775222_20190408220040Mon, 08 Apr 2019 22:00 EDTA new proof of the Waldspurger formula, Ihttps://projecteuclid.org/euclid.ant/1554775223<strong>Rahul Krishna</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 3, 577--642.</p><p><strong>Abstract:</strong><br/>
We provide the first steps towards a new relative trace formula proof of the celebrated formula of Waldspurger relating the square of a toric period integral on [math] to the central value of an [math] -function.
</p>projecteuclid.org/euclid.ant/1554775223_20190408220040Mon, 08 Apr 2019 22:00 EDTAlgebraic independence for values of integral curveshttps://projecteuclid.org/euclid.ant/1554775224<strong>Tiago J. Fonseca</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 3, 643--694.</p><p><strong>Abstract:</strong><br/>
We prove a transcendence theorem concerning values of holomorphic maps from a disk to a quasiprojective variety over [math] that are integral curves of some algebraic vector field (defined over [math] ). These maps are required to satisfy some integrality property, besides a growth condition and a strong form of Zariski-density that are natural for integral curves of algebraic vector fields.
This result generalizes a theorem of Nesterenko concerning algebraic independence of values of the Eisenstein series [math] , [math] , [math] . The main technical improvement in our approach is the replacement of a rather restrictive hypothesis of polynomial growth on Taylor coefficients by a geometric notion of moderate growth formulated in terms of value distribution theory.
</p>projecteuclid.org/euclid.ant/1554775224_20190408220040Mon, 08 Apr 2019 22:00 EDTGenerically split octonion algebras and $\mathbb{A}^1$-homotopy theoryhttps://projecteuclid.org/euclid.ant/1554775225<strong>Aravind Asok</strong>, <strong>Marc Hoyois</strong>, <strong>Matthias Wendt</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 3, 695--747.</p><p><strong>Abstract:</strong><br/>
We study generically split octonion algebras over schemes using techniques of [math] -homotopy theory. By combining affine representability results with techniques of obstruction theory, we establish classification results over smooth affine schemes of small dimension. In particular, for smooth affine schemes over algebraically closed fields, we show that generically split octonion algebras may be classified by characteristic classes including the second Chern class and another “mod [math] ” invariant. We review Zorn’s “vector matrix” construction of octonion algebras, generalized to rings by various authors, and show that generically split octonion algebras are always obtained from this construction over smooth affine schemes of low dimension. Finally, generalizing P. Gille’s analysis of octonion algebras with trivial norm form, we observe that generically split octonion algebras with trivial associated spinor bundle are automatically split in low dimensions.
</p>projecteuclid.org/euclid.ant/1554775225_20190408220040Mon, 08 Apr 2019 22:00 EDTArtin's criteria for algebraicity revisitedhttps://projecteuclid.org/euclid.ant/1558144821<strong>Jack Hall</strong>, <strong>David Rydh</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 4, 749--796.</p><p><strong>Abstract:</strong><br/>
Using notions of homogeneity we give new proofs of M. Artin’s algebraicity criteria for functors and groupoids. Our methods give a more general result, unifying Artin’s two theorems and clarifying their differences.
</p>projecteuclid.org/euclid.ant/1558144821_20190517220029Fri, 17 May 2019 22:00 EDTDifferential characters of Drinfeld modules and de Rham cohomologyhttps://projecteuclid.org/euclid.ant/1558144822<strong>James Borger</strong>, <strong>Arnab Saha</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 4, 797--837.</p><p><strong>Abstract:</strong><br/>
We introduce differential characters of Drinfeld modules. These are function-field analogues of Buium’s [math] -adic differential characters of elliptic curves and of Manin’s differential characters of elliptic curves in differential algebra, both of which have had notable Diophantine applications. We determine the structure of the group of differential characters. This shows the existence of a family of interesting differential modular functions on the moduli of Drinfeld modules. It also leads to a canonical [math] -crystal equipped with a map to the de Rham cohomology of the Drinfeld module. This [math] -crystal is of a differential-algebraic nature and the relation to the classical cohomological realizations is presently not clear.
</p>projecteuclid.org/euclid.ant/1558144822_20190517220029Fri, 17 May 2019 22:00 EDTQuadratic twists of abelian varieties and disparity in Selmer rankshttps://projecteuclid.org/euclid.ant/1558144823<strong>Adam Morgan</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 4, 839--899.</p><p><strong>Abstract:</strong><br/>
We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarized abelian variety over a number field. Specifically, we determine the proportion of twists having odd (respectively even) 2-Selmer rank. This generalizes work of Klagsbrun–Mazur–Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich–Tate group (if finite) may have order twice a square. In particular, the statistics for parities of [math] -Selmer ranks and [math] -infinity Selmer ranks need no longer agree and we describe both.
</p>projecteuclid.org/euclid.ant/1558144823_20190517220029Fri, 17 May 2019 22:00 EDTIwasawa theory for Rankin-Selberg products of $p$-nonordinary eigenformshttps://projecteuclid.org/euclid.ant/1558144824<strong>Kâzım Büyükboduk</strong>, <strong>Antonio Lei</strong>, <strong>David Loeffler</strong>, <strong>Guhan Venkat</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 4, 901--941.</p><p><strong>Abstract:</strong><br/>
Let [math] and [math] be two modular forms which are nonordinary at [math] . The theory of Beilinson–Flach elements gives rise to four rank-one nonintegral Euler systems for the Rankin–Selberg convolution [math] , one for each choice of [math] -stabilisations of [math] and [math] . We prove (modulo a hypothesis on nonvanishing of [math] -adic [math] -functions) that the [math] -parts of these four objects arise as the images under appropriate projection maps of a single class in the wedge square of Iwasawa cohomology, confirming a conjecture of Lei–Loeffler–Zerbes.
Furthermore, we define an explicit logarithmic matrix using the theory of Wach modules, and show that this describes the growth of the Euler systems and [math] -adic [math] -functions associated to [math] in the cyclotomic tower. This allows us to formulate “signed” Iwasawa main conjectures for [math] in the spirit of Kobayashi’s [math] -Iwasawa theory for supersingular elliptic curves; and we prove one inclusion in these conjectures under our running hypotheses.
</p>projecteuclid.org/euclid.ant/1558144824_20190517220029Fri, 17 May 2019 22:00 EDTCycle integrals of modular functions, Markov geodesics and a conjecture of Kanekohttps://projecteuclid.org/euclid.ant/1558144825<strong>Paloma Bengoechea</strong>, <strong>Özlem Imamoglu</strong>. <p><strong>Source: </strong>Algebra & Number Theory, Volume 13, Number 4, 943--962.</p><p><strong>Abstract:</strong><br/>
In this paper we study the values of modular functions at the Markov quadratics which are defined in terms of their cycle integrals along the associated closed geodesics. These numbers are shown to satisfy two properties that were conjectured by Kaneko. More precisely we show that the values of a modular function [math] , along any branch [math] of the Markov tree, converge to the value of [math] at the Markov number which is the predecessor of the tip of [math] . We also prove an interlacing property for these values.
</p>projecteuclid.org/euclid.ant/1558144825_20190517220029Fri, 17 May 2019 22:00 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 EDT