Search

We consider the arithmetic background of integral representations of finite groups over -adic and algebraic number rings. Some infinite series of integral pairwise inequivalent absolutely irreducible representations of finite -groups with the extra congruence conditions are constructed, and some applications are given. Certain problems concerning integral irreducible two-dimensional representations over number rings are discussed. [ABSTRACT FROM AUTHOR]

Let D be a division ring and N be a subnormal subgroup of D*. In this paper we prove that if M is a nilpotent maximal subgroup of N, then M′ is abelian. If, furthermore every element of M is algebraic over Z(D) and M′ ⊈ F* or M/Z(M) or M′ is finitely generated, then M is abelian. The second main result of this paper concerns the subgroups of matrix groups; assume D is a noncommutative division ring, n is a natural number, N is a subnormal subgroup of GLn(D), and M is a maximal subgroup of N. We show that if M is locally finite over Z(D)*, then M is either absolutely irreducible or abelian. [ABSTRACT FROM AUTHOR]

An alternative proof of the necessary conditions on A , B ∈ q 2 * for f (X) = X + A X 1 + q 2 (q − 1) / 4 + B X 1 + 3 q 2 (q − 1) / 4 to be a permutation polynomial in q 2 , q even, is given. This proof involves standard arguments from algebraic geometry over finite fields and fast symbolic computations. [ABSTRACT FROM AUTHOR]

Let (𝕋 f , 𝔪 f) denote the mod p local Hecke algebra attached to a normalized Hecke eigenform f , which is a commutative algebra over some finite field 𝔽 q of characteristic p and with residue field 𝔽 q . By a result of Carayol we know that, if the residual Galois representation ρ ¯ f : G ℚ → GL 2 (𝔽 q) is absolutely irreducible, then one can attach to this algebra a Galois representation ρ f : G ℚ → GL 2 (𝕋 f) that is a lift of ρ ¯ f . We will show how one can determine the image of ρ f under the assumptions that (i) the image of the residual representation contains SL 2 (𝔽 q) , (ii) 𝔪 f 2 = 0 and (iii) the coefficient ring is generated by the traces. As an application we will see that the methods that we use allow to deduce the existence of certain p -elementary abelian extensions of big non-solvable number fields. [ABSTRACT FROM AUTHOR]

Copyright of International Journal of Number Theory is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let D be a division ring with centre F. Assume that M is a maximal subgroup of GLn(D) (n≥1) such that Z(M) is algebraic over F. Group identities on M and polynomial identities on the F-linear hull F[M] are investigated. It is shown that if F[M] is a PI-algebra, then [D:F]<∞. When D is non-commutative and F is infinite, it is also proved that if M satisfies a group identity and F[M] is algebraic over F, then we have either M=K* where K is a field and [D:F]<∞, or M is absolutely irreducible. For a finite dimensional division algebra D, assume that N is a subnormal subgroup of GLn(D) and M is a maximal subgroup of N. If M satisfies a group identity, it is shown that M is abelian-by-finite. [ABSTRACT FROM AUTHOR]

Let Q(x, y) be a quadratic form with discriminant D ≠ 0. We obtain non-trivial upper bound estimates for the number of solutions of the congruence Q(x, y) ≡ λ (mod p), where p is a prime and x, y lie in certain intervals of length M, under the assumption that Q(x, y) - λ is an absolutely irreducible polynomial modulo p. In particular, we prove that the number of solutions to this congruence is Mo(1) when M ≪ p1/4. These estimates generalize a previous result by Cilleruelo and Garaev on the particular congruence xy ≡ λ(mod p). [ABSTRACT FROM AUTHOR]

In this paper, we show that the conjectural mod p local Langlands correspondence can be realized in the mod p cohomology of the Lubin–Tate towers. The proof utilizes a well-known conjecture of Buzzard–Diamond–Jarvis [8, Conjecture 4.9], a study of completed cohomology of the ordinary and supersingular locus of the Shimura curves for a totally real field F and of mod l (≠ p) local Langlands correspondence as given by Emerton–Helm [20]. In the case of modular curves, a similar theorem was obtained by Chojecki [13]. [ABSTRACT FROM AUTHOR]

In this paper, we determine the finite classical simple groups of dimension n = 3, 5 which are (2, 3)-generated (the cases n = 2, 4 are known). If n = 3, they are 3(q), q ≠ 4, and 3(q2), q2 ≠ 9, 25. If n = 5 they are 5(q), for all q, and 5(q2), q2 ≥ 9. Also, the soluble group 3(4) is not (2, 3)-generated. We give explicit (2, 3)-generators of the linear preimages, in the special linear groups, of the (2, 3)-generated simple groups. [ABSTRACT FROM AUTHOR]

We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be interpreted in some monster model with an additional predicate. Third, we prove the "Weak Independence Theorem" for pseudo-algebraically closed (PAC) substructures of an ambient structure with no finite cover property (nfcp) and the property B (3). Fourth, we describe Kim-dividing in these PAC substructures and show several results related to the SOPn hierarchy. Fifth, we characterize the algebraic closure in PAC structures. [ABSTRACT FROM AUTHOR]

The paper aims to introduce the cyclotomic q -Schur superalgebra via the permutation supermodules of the cyclotomic Hecke algebra and investigate its structure. In particular, we show that the cyclotomic q -Schur superalgebra is a cellular superalgebra and establish the double centralizer property between the cyclotomic Hecke algebra and the cyclotomic q -Schur superalgebra. [ABSTRACT FROM AUTHOR]

The weights for a finite group G with respect to a prime number p were introduced by Jon Alperin, in order to formulate his celebrated conjecture. In 1992, Everett Dade formulated a refinement of Alperin's conjecture involving ordinary irreducible characters - with their defect - and, in 2000, Geoffrey Robinson proved that the new conjecture holds for p-solvable groups. But this refinement is formulated in terms of a vanishing alternating sum, without giving any possible refinement for the weights. In this note we show that, in the case of the p-solvable finite groups, the method developed in a previous paper can be suitably refined to provide, up to the choice of a polarization ω, a natural bijection - namely compatible with the action of the group of outer automorphisms of G - between the sets of absolutely irreducible characters of G and of G-conjugacy classes of suitable inductive weights, preserving blocks and defects. [ABSTRACT FROM AUTHOR]

The members of the class C6 in Aschbacher's structure theorem for subgroups of the general linear group GL(d,q) are the normalizers of certain absolutely irreducible, symplectic-type r-groups, where r is a prime, d a power of r and q ≡ 1 (mod r). For a prime r > 2 and d = r, we present a constructive one-sided Monte Carlo algorithm to recognize whether or not a given subgroup G of GL(r,q) contains a normal extra-special r-group of order r3 and exponent r. In the former case the algorithm returns a homomorphism from G into SL(2,r) with kernel being the extra-special r-group times the center. [ABSTRACT FROM AUTHOR]

We describe an algorithm for explicitly listing the irreducible monomial subgroups of GL(n,q), given a suitable list of finite irreducible monomial subgroups of ${\rm GL}(n,{\mathbb C})$, where n is 4 or a prime, and q is a prime power. Particular attention is paid to the case n=4, and the algorithm is illustrated for n=4 and q=5. Certain primitive permutation groups can be constructed from a list of irreducible monomial subgroups of GL(n,q). The paper's final section shows that the computation of automorphisms of such permutation groups reduces mainly to computation of irreducible monomial subgroups of GL(n,q), q prime. [ABSTRACT FROM AUTHOR]

We give examples of (i) a simple theory with a formula (with parameters) which does not fork over ∅ but has μ -measure 0 for every automorphism invariant Keisler measure μ and (ii) a definable group G in a simple theory such that G is not definably amenable, i.e. there is no translation invariant Keisler measure on G. We also discuss paradoxical decompositions both in the setting of discrete groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a new model theoretical tree property (TP), called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and k -ATP, and provide a criterion for theories to have not ATP (being NATP). Using these combinatorial observations, we find algebraic examples of ATP and NATP, including pure groups, pure fields and valued fields. More precisely, we prove Mekler's construction for groups, Chatzidakis' style criterion for pseudo-algebraically closed (PAC) fields, and the AKE-style principle for valued fields preserving NATP. We give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to use the improvement of Runge's theorem on diophantine equations due to Schinzel to prove that if has at least three different roots then the equation in the title has a finite number of solutions. [ABSTRACT FROM AUTHOR]

We study realization fields and integrality of characters of finite subgroups of GL n (C) and related lattices with a focus on the integrality of characters of finite groups G. We are interested in the arithmetic aspects of the integral realizability of representations of finite groups, order generated by the character values, the number of minimal realization splitting fields, and in particular, consider the conditions of realizability in the terms of Hilbert symbols and quaternion algebras and some orders generated by character values over the rings of rational and algebraic integers. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show how the modular method together with different techniques can be used to prove non-existence of primitive non-trivial solutions of the equation x 2 + d y 6 = z p for square-free values 1 ≤ d ≤ 2 0. The key ingredients are: the approach presented in [A. Pacetti and L. V. Torcomian, ℚ -curves, Hecke characters and some Diophantine equations, Math. Comp. 91(338) (2022) 2817–2865] (in particular its recipe for the space of modular forms to be computed) together with the use of the symplectic method (as developed in [E. Halberstadt and A. Kraus, Courbes de Fermat: Résultats et problèmes, J. Reine Angew. Math. 548 (2002) 167–234], although we give a variant over ramified extensions needed in our applications) to discard solutions and the use of a second Frey curve, aiming to prove large image of residual Galois representations. [ABSTRACT FROM AUTHOR]

A proof is given of several conjectures from a recent paper of Nakaya concerning the supersingular polynomial s s p (N ∗) (X) for the Fricke group Γ 0 ∗ (N) , for N ∈ { 2 , 3 , 5 , 7 }. One of these conjectures gives a formula for the square of s s p (N ∗) (X) (mod p) in terms of a certain resultant, and the other relates the primes p for which s s p (N ∗) (X) splits into linear factors (mod p) to the orders of certain sporadic simple groups. [ABSTRACT FROM AUTHOR]

This paper studies multizeta values in positive characteristic, which are function field analogs of the Euler–Riemann multizeta values. The objective of this paper is two-fold. We first define mixed Carlitz motives, which is a counterpart to mixed Tate motives in characteristic zero. After that, we identify special mixed Carlitz motives, and use them to derive algebraic independence properties of colored multizeta values. Our results generalize those in the literature on algebraic relations between multizeta values. [ABSTRACT FROM AUTHOR]

In this paper we give explicit -generators of the finite unitary groups , for all . They fit into a uniform sequence of likely -generators for all . [ABSTRACT FROM AUTHOR]

This note is devoted to the study of families of quaternionic modular forms arising from orders defined by Pizer. In this situation, the Hecke eigenspaces are 2-dimensional contrary to the classical case of Eichler orders. The main result is a Control Theorem in the spirit of Hida, interpolating these 2-dimensional Hecke eigenspaces. We restrict our attention to a definite rational quaternion algebra ramified at a single odd prime ℓ. [ABSTRACT FROM AUTHOR]

Symmetries in the external world constrain the evolution of neuronal circuits that allow organisms to sense the environment and act within it. Many small "modular" circuits can be viewed as approximate discretizations of the relevant symmetries, relating their forms to the functions they perform. The recent development of a formal theory of dynamics and bifurcations of networks of coupled differential equations permits the analysis of some aspects of network behavior without invoking specific model equations or numerical simulations. We review basic features of this theory, compare it to equivariant dynamics, and examine the subtle effects of symmetry when combined with network structure. We illustrate the relation between form and function through examples drawn from neurobiology, including locomotion, peristalsis, visual perception, balance, hearing, location detection, decision-making, and the connectome of the nematode Caenorhabditis elegans. [ABSTRACT FROM AUTHOR]

Let p ≥ 5 be a prime, and let f be a cuspidal eigenform of weight at least 2 and level coprime to p of finite slope α. Let ρ ̄ f denote the mod p Galois representation associated with f and ω the mod p cyclotomic character. Under an assumption on the weight of f , we prove that there exists a cuspidal eigenform g of weight at least 2 and level coprime to p of slope α + 1 such that ρ ̄ f ⊗ ω ≃ ρ ̄ g , up to semisimplification. The proof uses Hida–Coleman families and the theta operator acting on overconvergent forms. The structure of the reductions of the local Galois representations associated to cusp forms with slopes in the interval [ 0 , 1) were determined by Deligne, Buzzard and Gee and for slopes in [ 1 , 2) by Bhattacharya, Ganguli, Ghate, Rai and Rozensztajn. We show that these reductions, in spite of their somewhat complicated behavior, are compatible with the displayed equation above. Moreover, the displayed equation above allows us to predict the shape of the reductions of a class of Galois representations attached to eigenforms of slope larger than 2. Finally, the methods of this paper allow us to obtain upper bounds on the radii of certain Coleman families. [ABSTRACT FROM AUTHOR]

Let  p be a prime number and r a positive even integer less than  p − 1. In this paper, we find the strongly divisible modules corresponding to the Galois stable lattices in each 2-dimensional semi-stable non-crystalline representation of Gal ( ℚ ¯ p / ℚ p) with Hodge–Tate weights (0 , r) whose mod- p reductions are corresponding to nontrivial extensions of two distinct characters. We use these results to construct the irreducible components of the semi-stable deformation rings in Hodge–Tate weights (0 , r) of the non-split reducible residual representations of Gal ( ℚ ¯ p / ℚ p). [ABSTRACT FROM AUTHOR]

In this paper, as an analog of the number field case, for an elliptic curve E defined over the algebraic numbers and for any subfield F of algebraic numbers, we say that E has the Northcott property over F if there are at most finitely many F -rational points on E of uniformly bounded height, and we say that E has the property (P) over F if for any infinite subset S of F -rational points on E , f (S) = S for an F -endomorphism f of E implies that f is an automorphism. We establish some criteria for both properties and provide typical examples. We also show that the Northcott property implies the property (P), but the converse is not true. [ABSTRACT FROM AUTHOR]

We consider the finite set of isogeny classes of g -dimensional abelian varieties defined over the finite field q with endomorphism algebra being a field. We prove that the class within this set whose varieties have the maximal number of rational points is unique, for any prime even power q big enough and verifying mild conditions. We describe its Weil polynomial and we prove that the class is ordinary and cyclic outside the primes dividing an integer that only depends on g. In dimension 3 , we prove that the class is ordinary and cyclic and give explicitly its Weil polynomial, for any prime even power q. [ABSTRACT FROM AUTHOR]

Let X be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if A is a normal subgroup of a finite group G then the image of an X-maximal subgroup H of G in G/A is not, in general, X-maximal in G/A. We say that the reduction X-theorem holds for a finite group A if, for every finite group G that is an extension of A (i.e. contains A as a normal subgroup), the number of conjugacy classes of X-maximal subgroups in G and G/A is the same. The reduction X-theorem for A implies that HA/A is X-maximal in G/A for every extension G of A and every X-maximal subgroup H of G. In this paper, we prove that the reduction X-theorem holds for A if and only if all X-maximal subgroups of A are conjugate in A and classify the finite groups with this property in terms of composition factors. [ABSTRACT FROM AUTHOR]

Let  G be a finite group. We explore the model-theoretic properties of the class of differential fields of characteristic zero in  m commuting derivations equipped with a  G -action by differential field automorphisms. In the language of  G -differential rings (i.e. the language of rings with added symbols for derivations and automorphisms), we prove that this class has a model-companion — denoted  G − DCF 0 , m . We then deploy the model-theoretic tools developed in the first author's paper [D. M. Hoffmann, Model theoretic dynamics in a Galois fashion, Ann. Pure Appl. Logic 170(7) (2019) 755–804] to show that any model of  G − DCF 0 , m is supersimple (but unstable when  G is nontrivial), a PAC-differential field (and hence differentially large in the sense of the second author and Tressl [Differentially large fields, preprint (2020), arXiv:2005.00888, available at https://arxiv.org/abs/2005.00888 ]), and admits elimination of imaginaries after adding a tuple of parameters. We also address model-completeness and supersimplicity of theories of bounded PAC-differential fields (extending the results of Chatzidakis and Pillay [Generic structures and simple theories, Ann. Pure Appl. Logic 95 (1998) 71–92] on bounded PAC-fields). [ABSTRACT FROM AUTHOR]

The global deformation theory of residually reducible Galois representations with fixed auxiliary conditions is studied. We show that ρ ̄ : Gal (ℚ ̄ / ℚ) → GL 2 ( ̄ p) lifts to a Hida line for which the weights range over a congruence class modulo- p 2 . The advantage of the purely Galois theoretic approach is that it allows us to construct p -adic families of Galois representations lifting the actual representation ρ ̄ , and not just the semisimplification. [ABSTRACT FROM AUTHOR]

In this paper, for a geometrically integral projective scheme, we will give an upper bound of the product of the norms of its non-geometrically integral reductions over an arbitrary number field. For this aim, we take the adelic viewpoint on this subject. Résumé (Contrôle des réductions non-géométriquement intègres) Dans cet article, pour un schéma projectif géométriquement intègre, on donnera une majoration du produit des norms de ses réductions non-géométriquement intègre sur un corps de nombres arbitraire. Pour le but, on prend le point de vue adélique autour de ce sujet. [ABSTRACT FROM AUTHOR]

In this paper, we construct planar polynomials of the type f A , B (x) = x (x q 2 + A x q + B x) ∈ q 3 [ x ] , with A , B ∈ q . In particular, we completely classify the pairs (A , B) ∈ q 2 such that f A , B (x) is planar using connections with algebraic curves over finite fields. [ABSTRACT FROM AUTHOR]

Let M be a Krull monoid. Then every element of M may be written as a finite product of irreducible elements. If for every a ∈ M, each two factorizations of a have the same number of irreducible elements, then M is called half-factorial. Using a property of element exponentiation, we provide a new characterization of half-factoriality, valid for all Krull monoids whose class group has torsion-free rank at most one. [ABSTRACT FROM AUTHOR]

For n = 2,3 and finite field 피 of characteristic greater than n, we provide a complete and irredundant list of soluble irreducible subgroups of GL(n,피). The insoluble irreducible subgroups of GL(2,피) are similarly determined. Each group is given explicitly by a generating set of matrices. The lists are available electronically. [ABSTRACT FROM AUTHOR]

Let p be a prime and let ℂ be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of GL (p , ℂ) up to conjugacy. That is, we give a complete and irredundant list of GL (p , ℂ) -conjugacy class representatives as generating sets of monomial matrices. Copious structural information about non-solvable finite irreducible monomial subgroups of GL (p , ℂ) is also proved, enabling a classification of all such groups bar one family. We explain the obstacles in that exceptional case. For p ≤ 3 , we classify all finite irreducible subgroups of GL (p , ℂ). Our classifications are available publicly in MAGMA. [ABSTRACT FROM AUTHOR]

In the context of stationary bifurcation problems admitting a symmetry, this paper is focused on the key notion of Fixed Subspace (FS), and provides a review of some applications aimed at detecting bifurcating solutions in various situations. We start recalling, in its commonly used simplified version, the old Equivariant Bifurcation Lemma (EBL), where the FS is one-dimensional; then we provide a first generalization in a typical case of non-semisimple critical eigenvalues, where the presence of the symmetry produces a non-trivial situation. Next, we consider the case of FSs of dimension > 1 in very different contexts. First, relying on the topological index theory and in particular on the Krasnosel'skii theorem, we provide a largely applicable statement of an extension of the EBL. Second, we propose a completely different and new application which combines symmetry properties with the notion of stability of bifurcating solutions. We also provide some simple examples, constructed ad hoc to illustrate the various situations. [ABSTRACT FROM AUTHOR]

This article reviews recent progress in signal frequency up-conversion and down-conversion, both theory and experiments with network implementations. The fundamental idea is to exploit the inherent symmetry of networks to produce collective behavior in which certain oscillators tend to oscillate at different frequencies. This concept is significantly different from other techniques, e.g. master-slave systems, in the sense that the collective behavior arises naturally from the mutual interactions of the individual units, and without any external forcing. In this manuscript, we present a comprehensive review of the basic ideas, methods, and experiments of the symmetry-based phenomenon of frequency conversion. In addition, we present a review of a device implementation of a broad spectrum analyzer, which motivated the development of systematic methods to up- and down-convert frequencies of oscillations. This device is made up of large parallel arrays of analog nonlinear oscillators with the ability to receive complex signals containing multiple frequencies and instantaneously lock-on or respond to a received signal in a few oscillation cycles. [ABSTRACT FROM AUTHOR]

In this paper we prove that for every integer d ≥ 1 , there exists an explicit constant B d such that the following holds. Let K be a number field of degree d , let q > max { d − 1 , 5 } be any rational prime that is totally inert in K and E any elliptic curve defined over K such that E has potentially multiplicative reduction at the prime above q. Then for every rational prime p > B d , E has an irreducible mod p Galois representation. This result has Diophantine applications within the "modular method". We present one such application in the form of an Asymptotic version of Fermat's Last Theorem that has not been covered in the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider roots of multivariate polynomials over a finite grid. When given information on the leading monomial with respect to a fixed monomial ordering, the footprint bound [Footprints or generalized Bezout's theorem, IEEE Trans. Inform. Theory46(2) (2000) 635–641, On (or in) Dick Blahut's 'footprint', Codes , Curves Signals (1998) 3–9] provides us with an upper bound on the number of roots, and this bound is sharp in that it can always be attained by trivial polynomials being a constant times a product of an appropriate combination of terms consisting of a variable minus a constant. In contrast to the one variable case, there are multivariate polynomials attaining the footprint bound being not of the above form. This even includes irreducible polynomials. The purpose of the paper is to determine a large class of polynomials for which only the mentioned trivial polynomials can attain the bound, implying that to search for other polynomials with the maximal number of roots one must look outside this class. [ABSTRACT FROM AUTHOR]

Given a non-commutative finite-dimensional F -central division ring D , N a subnormal subgroup of GL n (D) and M a non-abelian maximal subgroup of N , then either M contains a non-cyclic free subgroup or there exists a non-central maximal normal abelian subgroup A of M such that K = F [ A ] is a subfield of M n (D) , K / F is Galois and Gal (K / F) ≅ M / (K ∗ ∩ N) , also Gal (K / F) is a finite simple group with F [ M ] = M n (D). [ABSTRACT FROM AUTHOR]

We consider the proportion of genus one curves over ℚ of the form z 2 = f (x , y) where f (x , y) ∈ ℤ [ x , y ] is a binary quartic form (or more generally of the form z 2 + h (x , y) z = f (x , y) where also h (x , y) ∈ ℤ [ x , y ] is a binary quadratic form) that have points everywhere locally. We show that the proportion of these curves that are locally soluble, computed as a product of local densities, is approximately 75.96%. We prove that the local density at a prime p is given by a fixed degree- 9 rational function of p for all odd p (and for the generalized equation, the same rational function gives the local density at every prime). An additional analysis is carried out to estimate rigorously the local density at the real place. [ABSTRACT FROM AUTHOR]

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields. [ABSTRACT FROM AUTHOR]

We prove that the equation (x − y) 4 + x 4 + (x + y) 4 = z n has no integer solutions x , y , z with gcd (x , y) = 1 for all integers n > 1. We mainly use a modular approach with two Frey ℚ -curves defined over the field ℚ (3 0). [ABSTRACT FROM AUTHOR]

In this paper, we extend the equivariant Hopf bifurcation theory for semilinear functional differential equations in general Banach spaces and then apply it to reaction–diffusion models with delay effect and homogeneous Dirichlet boundary condition on a general open domain with a smooth boundary. In the process we derive the criteria for the existence and directions of branches of bifurcating periodic solutions, avoiding the process of center manifold reduction. [ABSTRACT FROM AUTHOR]

Let f be a modular form with complex multiplication. If f has critical slope, then Coleman's classicality theorem implies that there is a p -adic overconvergent generalized Hecke eigenform with the same Hecke eigenvalues as f. We give a formula for the Fourier coefficients of this generalized Hecke eigenform. We also investigate the dimension of the generalized Hecke eigenspace of p -adic overconvergent forms containing f. [ABSTRACT FROM AUTHOR]

Recent work of Freitas and Siksek showed that an asymptotic version of Fermat's Last Theorem (FLT) holds for many totally real fields. This result was extended by Deconinck to the generalized Fermat equation of the form A x p + B y p + C z p = 0 , where A , B , C are odd integers belonging to a totally real field. Later Şengün and Siksek showed that the asymptotic FLT holds over number fields assuming two standard modularity conjectures. In this work, combining their techniques, we show that the generalized Fermat's Last Theorem (GFLT) holds over number fields asymptotically assuming the standard conjectures. We also give three results which show the existence of families of number fields on which asymptotic versions of FLT or GFLT hold. In particular, we prove that the asymptotic GFLT holds for a set of imaginary quadratic number fields of density 5/6. [ABSTRACT FROM AUTHOR]

This paper was motivated by a recent paper by Krumm and Pollack ([Twists of hyperelliptic curves by integers in progressions modulo p , preprint (2018); https://arXiv.org/abs/1807.00972]) investigating modulo- p behavior of quadratic twists with rational points of a given hyperelliptic curve, conditional on the abc-conjecture. We extend those results to twisted Galois covers with arbitrary Galois groups. The main point of this generalization is to interpret those results as statements about the sets of specializations of a given Galois cover under restrictions on the discriminant. In particular, we make a connection with existing heuristics about the distribution of discriminants of Galois extensions such as the Malle conjecture: our results show in a precise sense the non-existence of "local obstructions" to such heuristics, in many cases essentially only under the assumption that G occurs as the Galois group of a Galois cover defined over ℚ. This complements and generalizes a similar result in the direction of the Malle conjecture by Dèbes ([On the Malle conjecture and the self-twisted cover, Israel J. Math.218(1) (2017) 101–131]). [ABSTRACT FROM AUTHOR]