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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]

Weil-type zeta functions defined by the numbers of absolutely irreducible $ \mathrm{SL}_2$ knot group over finite fields are computed explicitly. They are expressed in terms of the congruence zeta functions of reductions of a certain elliptic curve defined over the rational number field. Then the Hasse-Weil type zeta function of the figure $ 8$ knot and a certain family of elliptic curves. [ABSTRACT FROM AUTHOR]

It is known that for finite simple groups it is possible for a faithful absolutely irreducible module to have 1-cohomology of dimension at least 3. However, until now no explicit examples have been found. We present two explicit examples where the dimension is exactly 3. It remains an open question as to whether the dimension can be bigger than 3. [ABSTRACT FROM AUTHOR]

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]

We refer to the hypotheses that for an absolutely irreducible representation of a compact Lie group there exists at least one subgroup with an odd dimensional fixed point space as the Ize conjecture (IC). If the IC is true, then it follows that loss of stability through an absolutely irreducible representation of a compact Lie group leads to bifurcation of steady states. Lauterbach and Matthews have shown that the (IC) is in general not true and have constructed three infinite families of finite subgroups of $$\mathop {\mathbf{SO}(4)}$$ which act absolutely irreducibly on $$\mathbb {R}^4$$ and have no odd dimensional fixed point space. They also have shown that in spite of this failure of the (IC) the nontrivial isotropy types are generically symmetry breaking at least for the groups in two of these three families. In this paper we show a similar bifurcation result for the third family defined by Lauterbach and Matthews. We go on and construct a family of groups acting absolutely irreducibly on $$\mathbb {R}^8$$ which have only even dimensional fixed point spaces. Then we discuss the steady state bifurcations in this case. Key ingredients are an abstract group theoretic construction and a kind of inductive step reducing the issue of bifurcations to a problem in $$\mathbb {R}^4$$ . We end this paper with a discussion on how to extend the results by Lauterbach and Matthews to larger sets of groups which act on $$\mathbb {R}^{4}$$ and $$\mathbb {R}^8$$ . In this context we point out, that the inductive step, which is important for our arguments, does not work in general and this gives rise to interesting new questions. [ABSTRACT FROM AUTHOR]

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $\bar{\unicode[STIX]{x1D70C}}$ be a continuous, absolutely irreducible representation of its absolute Galois group with values in a finite field of characteristic $p$. We prove that the Galois representations that become crystalline of a fixed regular weight after an abelian extension are Zariski-dense in the generic fiber of the universal deformation ring of $\bar{\unicode[STIX]{x1D70C}}$. In fact we deduce this from a similar density result for the space of trianguline representations. This uses an embedding of eigenvarieties for unitary groups into the spaces of trianguline representations as well as the corresponding density claim for eigenvarieties as a global input. [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]

The article explores a proof for automorphy lifting theorems for l-adic Galois representations over CM or an imaginary extension of a real number field. Topics covered include the need for residual representation to be absolutely irreducible, the different approach that can be taken with automorphic forms on unitary groups and the theorem that proves an automorphy result for irreducible three-dimensional Galois representations over a quadratic imaginary field.

The number of rank two ...ℓ-local systems, or (...ℓ-smooth sheaves, on (X - {u}) ... 픽, where X is a smooth projective absolutely irreducible curve over 픽q, 픽 an algebraic closure of Fq and u is a closed point of X, with principal unipotent monodromy at u, and fixed by Gal(픽/ 픽q), is computed. It is expressed as the trace of the Frobenius on the virtual (...ℓ-smooth sheaf found in the author's work with Deligne on the moduli stack of curves with étale divisors of degree M ≥ 1. This completes the work with Deligne in rank two. This number is the same as that of representations of the fundamental group π1((X - {u}) ... 픽) invariant under the Frobenius Frq with principal unipotent monodromy at u, or cuspidal representations of GL(2) over the function field F = 픽q(X) of X over 픽q with Steinberg component twisted by an unramified character at u and unramified elsewhere, trivial at the fixed idèle α of degree 1. This number is computed in Theorem 4.1 using the trace formula evaluated at fu ∏v≠u χKv, with an Iwahori component fu = χIu/|Iu|, hence also the pseudo-coefficient ... of the Steinberg representation twisted by any unramified character, at u. Theorem 2.1 records the trace formula for GL(2) over the function field F. The proof of the trace formula of Theorem 2.1 recently appeared elsewhere. Theorem 3.1 computes, following Drinfeld, the number of ...ℓ-local systems, or ...ℓ-smoofh sheaves, on ... 픽, fixed by Frq, namely (...ℓ-representations of the absolute fundamental group π(X ... 픽) invariant under the Frobenius, by counting the nowhere ramified cuspidal representations of GL(2) trivial at a fixed idèle α of degree 1. This number is expressed as the trace of the Frobenius of a virtual ...ℓ-smooth sheaf on a moduli stack. This number is obtained on evaluating the trace formula at the characteristic function ∏v χKv of the maximal compact subgroup, with volume normalized by |Kv| = 1. Section 5, based on a letter of P. Deligne to the author dated August 8, 2012, computes the number of such objects with any unipotent monodromy, principal or trivial, in our rank two case. Surprisingly, this number depends only on X and deg(S), and not on the degrees of the points in S1. [ABSTRACT FROM AUTHOR]

In the paper ‘Bruhat–Tits theory from Berkovich's point of view. I. Realizations and compactifications of buildings’, we investigated various realizations of the Bruhat–Tits building $\mathcal{B}(\mathrm{G},k)$ of a connected and reductive linear algebraic group G over a non-Archimedean field k in the framework of Berkovich's non-Archimedean analytic geometry. We studied in detail the compactifications of the building which naturally arise from this point of view. In the present paper, we give a representation theoretic flavour to these compactifications, following Satake's original constructions for Riemannian symmetric spaces.We first prove that Berkovich compactifications of a building coincide with the compactifications, previously introduced by the third named author and obtained by a gluing procedure. Then we show how to recover them from an absolutely irreducible linear representation of G by embedding $\mathcal{B}(\mathrm{G},k)$ in the building of the general linear group of the representation space, compactified in a suitable way. Existence of such an embedding is a special case of Landvogt's general results on functoriality of buildings, but we also give another natural construction of an equivariant embedding, which relies decisively on Berkovich geometry. [ABSTRACT FROM PUBLISHER]

Abstract: A theorem of N. Katz (1990) , p. 45, states that an irreducible differential operator L over a suitable differential field k, which has an isotypical decomposition over the algebraic closure of k, is a tensor product of an absolutely irreducible operator M over k and an irreducible operator N over k having a finite differential Galois group. Using the existence of the tensor decomposition , an algorithm is given in É. Compoint and J.-A. Weil (2004) , which computes an absolutely irreducible factor F of L over a finite extension of k. Here, an algorithmic approach to finding M and N is given, based on the knowledge of F. This involves a subtle descent problem for differential operators which can be solved for explicit differential fields k which are -fields. [Copyright &y& Elsevier]

First cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts, including Wiles''s famous paper (1995). Internal to group theory, $1$-cohomology plays a role in the general theory of maximal subgroups of finite groups, as developed by Aschbacher and Scott (1985).One can pass to the case where the group acts faithfully and the underlying module is absolutely irreducible. In this case, R. Guralnick (1986) conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups. This paper provides the first general positive results on this conjecture, proving that the generic 1-cohomology $H^1_{operatorname {gen}}(G,L) :=underset {qto infty }{lim } H^1(G(q),L)$ (see Cline, Parshall, Scott, and van der Kallen) (1977) of a finite group $G(q)$ of Lie type, with absolutely irreducible coefficients $L$ (in the defining characteristic of $G$), is bounded by a constant depending only on the root system. In all cases, we are able to improve this result to a bound on $H^1(G(q),L)$ itself, still depending only on the root system. The generic $H^1$ result, and related results for $operatorname {Ext}^1$, emerge here as a consequence of a general study, of interest in its own right, of the homological properties of certain rational modules $Delta ^{text {rm red}}(lambda ), nabla _{text {rm red}}(lambda )$, indexed by dominant weights $lambda $, for a reductive group $G$. The modules $Delta ^{text {rm red}}(lambda )$ and $nabla _{text {rm red}}(lambda )$ arise naturally from irreducible representations of the quantum enveloping algebra $U_zeta $ (of the same type as $G$) at a $p$th root of unity, where $p>0$ is the characteristic of the defining field for $G$. Finally, we apply our $operatorname {Ext}^1$-bounds, results of Bendel, Nakano, and Pillen (2006), as well as results of Sin (1993), (1992), (1994) on the Ree and Suzuki groups to obtain the (non-generic) bounds on $H^1(G(q),L)$. [ABSTRACT FROM AUTHOR]

We prove that the restriction of any absolutely irreducible representation of Steinberg's triality groups 3 D4( q) in characteristic coprime to q to any proper subgroup is reducible. [ABSTRACT FROM AUTHOR]

We provide a detailed structural description of the nilpotent primitive subgroups of GL(n, IF), where IF is a finite field. [ABSTRACT FROM AUTHOR]

Finding minimal fields of definition for representations is one of the most important unsolved problems of computational representation theory. While good methods exist for representations over finite fields, it is still an open question in the case of number fields. In this paper we give a practical method for finding minimal defining subfields for absolutely irreducible representations. We illustrate the new algorithm by determining a minimal field for each absolutely irreducible representation of Sz(8). [Copyright &y& Elsevier]

This is a final step in solving the problem of recognition of the simple groups L3(pk) by element orders. It is proven that when L3(pk) acts on an elementary abelian p-group, there always appears an element of new order. A model is proposed for constructing the absolutely irreducible p-modular representations of L3(pk) in polynomial spaces. [ABSTRACT FROM AUTHOR]

We study a certain family of simple fusion systems over finite 3-groups, ones that involve Todd modules of the Mathieu groups 2 ⁢ M 12 , M 11 , and A 6 = O 2 ⁢ (M 10) over F 3 , and show that they are all isomorphic to the 3-fusion systems of almost simple groups. As one consequence, we give new 3-local characterizations of Conway's sporadic simple groups. [ABSTRACT FROM AUTHOR]

Suppose ρ 1 and ρ 2 are two pure ℓ -adic degree n representations of the absolute Galois group of a number field K of weights k 1 and k 2 respectively, having equal normalized Frobenius traces T r ( ρ 1 ( σ v ) ) / N v k 1 / 2 and T r ( ρ 2 ( σ v ) ) / N v k 2 / 2 at a set of primes v of K with positive upper density. Assume further that the algebraic monodromy group of ρ 1 is connected and ρ 1 is absolutely irreducible. We prove that ρ 1 and ρ 2 are twists of each other by a power of the ℓ -adic cyclotomic character times a character of finite order. As a corollary, we deduce a theorem of Murty and Pujahari proving a refinement of the strong multiplicity one theorem for normalized eigenvalues of newforms. [ABSTRACT FROM AUTHOR]

We give a new local proof of the Breuil-Mézard conjecture in the case of a reducible representation of the absolute Galois group of Qp, p>2, that has scalar semi-simplification, via a formalism of Paškūnas. [ABSTRACT FROM AUTHOR]

We prove quantitative versions of Borel and Harish-Chandra's theorems on reduction theory for arithmetic groups. Firstly, we obtain polynomial bounds on the lengths of reduced integral vectors in any rational representation of a reductive group. Secondly, we obtain polynomial bounds in the construction of fundamental sets for arithmetic subgroups of reductive groups, as the latter vary in a real conjugacy class of subgroups of a fixed reductive group. Our results allow us to apply the Pila–Zannier strategy to the Zilber–Pink conjecture for the moduli space of principally polarised abelian surfaces. Building on our previous paper, we prove this conjecture under a Galois orbits hypothesis. Finally, we establish the Galois orbits hypothesis for points corresponding to abelian surfaces with quaternionic multiplication, under certain geometric conditions. [ABSTRACT FROM AUTHOR]

Reduction algorithms are an important tool for understanding structural properties of groups. They play an important role in algorithms designed to investigate matrix groups over a finite field. One such algorithm was designed by Brooksbank et al. for members of the class C6 in Aschbacher's theorem, namely groups N that are normalizers in GL(d; q) of certain absolutely irreducible symplectic-typeγ-groups R, whereγis a prime and d = rn with n > 2. However, the analysis of this algorithm has only been completed when d = r2 and when d = rn and n > 2, in the latter case under the condition that G=RZ(G)≃= N=RZ(N) . We prove that the algorithm runs successfully for some groups in the case of d = r3 without any assumption. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the finite simple groups $\text{PSp}_{6}(q)$, ${\rm\Omega}_{7}(q)$ and $\text{PSU}_{7}(q^{2})$ are $(2,3)$-generated for all $q$. In particular, this result completes the classification of the $(2,3)$-generated finite classical simple groups up to dimension 7. [ABSTRACT FROM AUTHOR]

The symplectic groupSp(6, 2) has a 14−dimensional absolutely irreducible module over. Hence a split extension group of the formḠ= 214:Sp(6,2) does exist. In this paper we first determine the conjugacy classes ofḠusing the coset analysis technique. The structures of inertia factor groups were determined. The inertia factor groups areSp(6, 2), (21+4× 22):(S3×S3),S3×S6,PSL(2, 8), (((22×Q8):3):2):2,S3×A5,and 2×S4×S3.We then determine the Fischer matrices and apply the Clifford-Fischer theory to compute the ordinary character table of.The Fischer matrices ofare all integer valued, with size ranging from 4 to 16. The full character table ofis a 186×186 complex valued matrix. [ABSTRACT FROM PUBLISHER]

We show that the symplectic groups PSp6(q) are Hurwitz for allq = pm ≥ 5, withpan odd prime. The result cannot be improved since, forqeven andq = 3, it is known that PSp6(q) is not Hurwitz. In particular,n = 6 turns out to be the smallest degree for which a family of classical simple groups of degreen, over 𝔽p m, contains Hurwitz groups for infinitely many values ofm. This fact, for a given (possibly large)p, also follows from [9] and [10]. [ABSTRACT FROM AUTHOR]

Majorana representations have been introduced by Ivanov [ Cambridge Tracts in Mathematics , Cambridge University Press, Cambridge, 2009] in order to provide an axiomatic framework for studying the actions on the Griess algebra of the Monster and of its subgroups generated by Fischer involutions. A crucial step in this programme is to obtain an explicit description of the Majorana representations of A_{12} (by Franchi, Ivanov, and Mainardis [J. Algebraic Combin. 44 (2016), pp. 265-292], the largest alternating group admitting a Majorana representation) for this might eventually lead to a new and independent construction of the Monster group (see A.A Ivanov [ Group theory and computation , Indian Stat. Inst. Ser., Springer, Singapore, 2018, Section 4, page 115]). In this paper we prove that A_{12} has two possible Majorana sets, one of which is the set \mathcal X_b of involutions of cycle type 2^2, the other is the union of \mathcal X_b with the set \mathcal X_s of involutions of cycle type 2^6. The latter case (the saturated case) is most interesting, since the Majorana set is precisely the set of involutions of A_{12} that fall into the class of Fischer involutions when A_{12} is embedded in the Monster. We prove that A_{12} has a unique saturated Majorana representation and we determine its degree and decomposition into irreducibles. As consequences we get that the Harada-Norton group has, up to equivalence, a unique Majorana representation and every Majorana algebra, affording either a Majorana representation of the Harada-Norton group or a saturated Majorana representation of A_{12}, satisfies the Straight Flush Conjecture (see A. A. Ivanov [ Contemp. Math. , Amer. Math. Soc., Providence, RI, 2017, pp. 11-17] and A. A. Ivanov [ Group theory and computation , Indian Stat. Inst. Ser., Springer, Singapore, 2018, pp. 107-118]). As a by-product we also determine the degree and the decomposition into irreducibles of the Majorana representation induced on A_8, the four point stabilizer subgroup of A_{12}. We finally state a conjecture about Majorana representations of the alternating groups A_n, 8\leq n\leq 12. [ABSTRACT FROM AUTHOR]

We study in this paper the extent to which one can detect fusion in certain finite groups Γ from information about the universal deformation rings R (Γ , V ) of absolutely irreducible F p Γ -modules V. The Γ we consider are extensions of either abelian or dihedral groups G of order prime to p by an elementary abelian p-group of rank 2. [ABSTRACT FROM AUTHOR]

Abstract: We classify all pairs , where is a proper subgroup of , even, and is an -modular representation of for which is absolutely irreducible as a representation of . This problem is motivated by the Aschbacher–Scott program on classifying maximal subgroups of finite classical groups. [Copyright &y& Elsevier]

Abstract: Let be a continuous, 2-dimensional and absolutely irreducible mod p representation of the absolute Galois group of a number field . In this work, we study the existence of lifts of to , for which the restrictions to the decomposition groups above p are abelian. The tools and philosophy come from the Taylor–Ramakrishna method. As an application we produce finitely ramified extensions over with Galois group , for some p. These extensions are unramified at p. [Copyright &y& Elsevier]

Abstract: We will classify the absolutely irreducible subgroups which are not realizable modulo scalars over any proper subfield of , which are “nearly simple”, and which contain elements of prime power order greater than . [Copyright &y& Elsevier]

Abstract: Let φ be a Drinfeld A-module of arbitrary rank and arbitrary characteristic over a finitely generated field K, and set . Let . We show that for almost all primes of A the image of the group ring in is the commutant of E. In the special case it follows that the representation of on the -torsion points of φ is absolutely irreducible for almost all . [Copyright &y& Elsevier]

Abstract  We propose a simple and efficient deterministic extractor for an ordinary elliptic curve E, defined over $$\mathbb{F}_{2^n}$$ , where n = 2ℓ and ℓ is a positive integer. Our extractor, for a given point P on E, outputs the first $${\mathbb{F}}_{2^\ell}$$ -coefficient of the abscissa of the point P. We also propose a deterministic extractor for the main subgroup G of E, where E has minimal 2-torsion. We show that if a point P is chosen uniformly at random in G, the bits extracted from the point P are indistinguishable from a uniformly random bit-string of length ℓ. [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 f be a newform of weight k ≥ 3 with Fourier coefficients in a number field K. We show that the universal deformation ring of the mod λ Galois representation associated to f is unobstructed, and thus isomorphic to a power series ring in three variables over the Witt vectors, for all but finitely many primes λ of K. We give an explicit bound on such λ for the 6 known cusp forms of level 1, trivial character, and rational Fourier coefficients. We also prove a somewhat weaker result for weight 2. [ABSTRACT FROM AUTHOR]

Abstract: Let f be a newform of weight , level N with coefficients in a number field K, and A the adjoint motive of the motive M associated to f. We carefully discuss the construction of the realisations of M and A, as well as natural integral structures in these realisations. We then use the method of Taylor and Wiles to verify the λ-part of the Tamagawa number conjecture of Bloch and Kato for and . Here λ is any prime of K not dividing , and so that the mod λ representation associated to f is absolutely irreducible when restricted to the Galois group over where . The method also establishes modularity of all lifts of the mod λ representation which are crystalline of Hodge–Tate type . [Copyright &y& Elsevier]

In Section 2 of this paper, the maximum number of real elements possible in a covering group of C2(n) is determined, and a description of those covering groups in which this maximum is attained is given. Among these “maximally real” examples is that covering group G of C2(n) which is generated by n involutions. For this particular group, the Schur indices of real-valued irreducible characters of each degree are investigated in Section 4. The main result of this section is a set of recurrence relations describing the number of absolutely irreducible characters of G of a given degree of each of three types (non-real-valued and real-valued of index 1 or 2 over R) in terms of related numbers for the corresponding group on n−1 generators. [Copyright &y& Elsevier]

Hecke algebras for the complex reflection groups G(r,p,n) (where p&ges;1 and p∣r) were introduced in the work of Ariki [J. Algebra 177 (1995) 164–185], Broue´ and Malle [Aste´risque 212 (1993) 119–189]. In this paper we consider modular representation theory for these algebras in the case where r=p. We assume that the field K contains a primitive pth root of unity &epsiv;. Our method is to study the restrictions of Specht modules Sλ for Hecke algebras Hp,n of type G(p,1,n) with parameters (q;1,&epsiv;,…,&epsiv;p−1). Suppose that f(q,&epsiv;)≠0 in K (see 4.7 for definition of f(q,&epsiv;)). For any multipartition λ=(λ(1),…,λ(p)) of n, we prove that Sλ↓Hp,p,n≅Sλ[k]↓Hp,p,n for any 1&les;k&les;p−1, where λ[k]=(λ(k+1),λ(k+2),…,λ(p),λ(1),λ(2),…,λ(k)); and if k is the smallest positive integer such that λ=λ[k] (hence k∣p), we explicitly decompose Sλ↓Hp,p,n into a direct sum of p/k smaller Hp,p,n-submodules with the same dimensions. As a result, when f(q,&epsiv;)≠0 in K, we show that Hp,p,n is split over K and get a complete classification of all the absolutely irreducible Hp,p,n-modules. This generalizes earlier work of [C. Pallikaros, J. Algebra 169 (1994) 20–48] and [J. Hu, Manuscripta Math. 108 (2002) 409–430] on Hecke algebras of type Dn (which are included as a special case of our main results). [Copyright &y& Elsevier]

The purpose of this note is to give a description of class 2 representations of the absolute Galois group of a field K of characteristic 0 which satisfy a certain condition of Kummer type. This description is based on Galois cohomology and on the theory of projective representations of finite abelian groups. [ABSTRACT FROM AUTHOR]

Let p be a prime. This paper classifies finite connected reductive groups G in characteristic p with the property that all complex character values of G belong to an unramified above p extension of the field of rational numbers. The main application of these results is to the problem of describing the irreducible complex (or p-adic) representations of G that remain absolutely irreducible under the Brauer reduction modulo p. An efficient approach to solve this problem for p>3 has been developed in our paper [Proc. London Math. Soc. (3) 84 (2002) 439]. Together with [Proc. London Math. Soc. (3) 84 (2002) 439], Theorem 1.9 of the paper solves this problem for many finite connected reductive groups in characteristic p>3. Additionally, we show that all complex representations of any finite connected reductive group with no composition factor of type E7(2f), E8(2f), and E8(5f) can be realized over a quadratic extension of an unramified (above p) extension of Q. [Copyright &y& Elsevier]

Let R be an algebra over a field K, and let G be a finite group of units in R. Suppose that either char K = 0 and G is nonabelian, or K is a nonabsolute field of characteristic π > 0 and G/[This character is not converted in Asci text) π(G) is nonabelian. Then we show that there are two cyclic subgroups X and Y of G of prime power order, and two special units u[SUBX] &epsis; KX ⊆ R and u[SUBY] &epsis; KY ⊆ R, such that 〈u[SUBX], u[SUBY]〉 contains a nonabelian free group. Indeed, we obtain a rather precise description of these units, generalizing an earlier result where R = K[G] was the group algebra of G over K. [ABSTRACT FROM AUTHOR]

Dedicated to the memory of Professor A. I. KostrikinThe main problem under discussion is to determine, for quasi-simple groups of Lie type G, irreducible representations φ of G that remain irreducible under reduction modulo the natural prime p. The method is new. It works only for p >3 and for representations φ that can be realized over an unramified extension of Qp, the field of p -adic numbers. Under these assumptions, the main result says that the trivial and the Steinberg representations of G are the only representations in question provided G is not of type A1. This is not true for G=SL(2, p). The paper contains a result of independent interest on infinitesimally irrreducible representations ρ of G over an algebraically closed field of characteristic p. Assuming that G is not of rank 1 and G≠ G2(5), it is proved that either the Jordan normal form of a root element contains a block of size d with 1

We present a new approach to unramified descent ("descente étale") in Bruhat-Tits theory of reductive groups over a discretely valued field k with Henselian valuation ring which appears to be conceptually simpler, and more geometric, than the original approach of Bruhat and Tits. We are able to derive the main results of the theory over k from the theory over the maximal unramified extension K of k. Even in the most interesting case for number theory and representation theory, where k is a locally compact nonarchimedean field, the geometric approach described in this paper appears to be considerably simpler than the original approach. [ABSTRACT FROM AUTHOR]

Let E be a modular elliptic curve over a totally real cubic field. We have a cubic-triple product motive over Q constructed from E through multiplicative induction; it is of rank 8. We show that, under certain assumptions on E, the nonvanishing of the central critical value of the L-function attached to the motive implies that the dimension of the associated Bloch-Kato Selmer group is 0. [ABSTRACT FROM AUTHOR]

We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute EndK(A) when A is the Jacobian of a nice genus-2 curve over a number field K. We use this algorithm to confirm that the description of the structure of the geometric endomorphism ring of Jac(C) given in the LMFDB (L-functions and modular forms database) is correct for all the genus-2 curves C currently listed in it. We also discuss the determination of the field of definition of the endomorphisms in some special cases. [ABSTRACT FROM AUTHOR]

Let G be a finite group. A proper subgroup H of G is said to be weakly monomial if the order of H satisfies | H | 2 > | G |. In this paper, we determine all the weakly monomial maximal subgroups of finite simple groups. [ABSTRACT FROM AUTHOR]

Let p and l be distinct primes, and let p be an orthogonal or symplectic representation of the absolute Galois group of an l-adic field over a finite field of characteristic p. We define and study a liftable deformation condition of lifts of p 'ramified no worse than p', generalizing the minimally ramified deformation condition for GLn studied in Clozel et al. [ Automorphy for some l-adic lifts of automorphic mod l Galois representations , Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181; MR 2470687 (2010j:11082)]. The key insight is to restrict to deformations where an associated unipotent element does not change type when deforming. This requires an understanding of nilpotent orbits and centralizers of nilpotent elements in the relative situation, not just over fields. [ABSTRACT FROM AUTHOR]