Smith theory and Hecke operators

The cohomology of arithmetic groups is made up of two pieces, the cuspidal and noncuspidal parts. Within the cuspidal cohomology is a subspace— the f -cuspidal cohomology—spanned by the classes that generate representations of the associated finite Lie group which are cuspidal in the sense of finite Lie group theory. Few concrete examples of f -cuspidal cohomology have been computed geometrically, outside the cases of rational rank 1, or where the symmetric space has a Hermitian structure. This paper presents new computations of the f -cuspidal cohomology of principal congruence subgroups Γ ( p ) \Gamma (p) of GL ( 3 , Z ) {\text {GL}}(3,\mathbb {Z}) of prime level p . We show that the f -cuspidal cohomology of Γ ( p ) \Gamma (p) vanishes for all p ⩽ 19 p \leqslant 19 with p ≠ 11 p \ne 11 , but that it is nonzero for p = 11 p = 11 . We give a precise description of the f -cuspidal cohomology for Γ ( 11 ) \Gamma (11) in terms of the f -cuspidal representations of the finite Lie group GL ( 3 , Z / 11 ) {\text {GL}}(3,\mathbb {Z}/11) . We obtained the result, ultimately, by proving that a certain large complex matrix M is rank-deficient. Computation with the SVD algorithm gave strong evidence that M was rank-deficient; but to prove it, we mixed ideas from numerical analysis with exact computation in algebraic number fields and finite fields.

Cohomology of arithmetic groups, automorphic forms and L-functions

We now apply the cohomology theory of the previous chapter to the study of the Brauer group. However, we shall have to use a slightly modified construction which takes into account the fact that the absolute Galois group of a field is determined by its finite quotients. This is the cohomology theory of profinite groups, which we develop first. As a fruit of our labours, we identify the Brauer group of a field with a second, this time commutative, cohomology group of the absolute Galois group. As applications, we obtain easy proofs of some basic facts concerning the Brauer group, and give an important characterization of the index of a central simple algebra. Last but not least, one of the main objects of study in this book makes its appearance: the Galois symbol. The cohomology theory of profinite groups was introduced in the late 1950s by John Tate, motivated by sheaf-theoretic considerations of Alexander Grothendieck. His original aim was to find the appropriate formalism for developing class field theory. Tate himself never published his work, which thus became accessible to the larger mathematical community through the famous account of Serre [4], which also contains many original contributions. It was Brauer himself who described the Brauer group as a second cohomology group, using his language of factor systems. We owe to Serre the insight that descent theory can be used to give a more conceptual proof. The Galois symbol was defined by Tate in connection with the algebraic theory of power residue symbols, a topic extensively studied in the 1960s by Bass, Milnor, Moore, Serre and others. Profinite groups and Galois groups It can be no surprise that the main application of the cohomological techniques of the previous chapter will be in the case when G is the Galois group of a finite Galois extension. However, it will be convenient to consider the case of infinite Galois extensions as well, and first and foremost that of the extension k s |k , where k s is a separable closure of k .

On the cohomology of Coxeter groups and their finite parabolic subgroups. II by T. Akita Linear source modules and trivial source modules by R. Boltje Resolutions de foncteurs de Mackey by S. Bouc Homotopy uniqueness of $BPU(3)$ by C. Broto and A. Viruel Lowering operators for $GL(n)$ and quantum $GL(n)$ by J. Brundan Homomorphisms in higher complexity quotient categories by J. F. Carlson and W. W. Wheeler On the homotopy theory of $p$-completed classifying spaces by F. R. Cohen and R. Levi $H^*V$ is of bounded type over $\mathcal {A} (p)$ by M. D. Crossley Classifying spaces of central group extensions by J. Dietz and T. Ratliff Sharp homology decompositions for classifying spaces of finite groups by W. G. Dwyer On the geometry and cohomology of the simple groups $G_2(q)$ and $^3D_4(q)$ by P. Fong and R. J. Milgram Transfer and Chern classes for extraspecial $p$-groups by D. J. Green and P. A. Minh The infinite order of the symplectic classes in the cohomology of the stable mapping class group by R. Grieder A variant of the proof of the Landweber Stong conjecture by H.-W. Henn Unstable modules over the Steenrod algebra and cohomology of groups by H.-W. Henn The category of Mackey functors for a compact Lie group by L. G. Lewis, Jr. Comparison of extensions of modules for algebraic groups and quantum groups by Z. Lin The defect of the completion of a Green functor by F. Luca Simple modules of Ariki-Koike algebras by A. Mathas On the geometry and cohomology of the simple groups $G_2(q)$ and $^3D_4(q)$: II by R. J. Milgram On the relation between simple groups and homotopy theory by R. J. Milgram Varieties for $G_rT$-modules by D. K. Nakano On defect groups of the Mackey algebras for finite groups by F. Oda Applications of stable classifying space theory to group cohomology by S. Priddy The spectra $k$ and $kO$ are not Thom spectra by Yu. B. Rudyak The code of a regular generalized quadrangle of even order by N. S. N. Sastry and P. Sin Quivers of blocks with normal defect group by M. Schaps, D. Shapira, and O. Shlomo Cohomology of a homogeneous space $E_6/T^1\cdot SU(6)$ by T. Watanabe Problem session, Seattle 1996 by D. Benson.

We define Hecke operators on vector-valued modular forms of the type that appear as characters of rational conformal field theories (RCFTs). These operators extend the previously studied Galois symmetry of the modular representation and fusion algebra of RCFTs to a relation between RCFT characters. We apply our results to derive a number of relations between characters of known RCFTs with different central charges and also explore the relation between Hecke operators and RCFT characters as solutions to modular linear differential equations. We show that Hecke operators can be used to construct an infinite set of possible characters for RCFTs with two independent characters and increasing central charge. These characters have multiplicity one for the vacuum representation, positive integer coefficients in their q expansions, and are associated to a two-dimensional representation of the modular group which leads to non-negative integer fusion coefficients as determined by the Verlinde formula.

We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of ℚp, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localization at the primes above p is an extension of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a conjecture on the existence of a "zeta element", whose image under the regulator map is a p-adic L-function. We show that this conjecture implies the known properties of the 2-variable p-adic L-functions constructed by Perrin-Riou and Kim.

Let X X be a complete smooth variety defined over a number field K K and let i i be an integer. The absolute Galois group G a l K \mathrm {Gal}_K of K K acts on the i i th étale cohomology group H e ´ t i ( X K ¯ , Q ℓ ) H^i_{\mathrm {\acute {e}t}}(X_{\bar K},\mathbb {Q}_\ell ) for all primes ℓ \ell , producing a system of ℓ \ell -adic representations { Φ ℓ } ℓ \{\Phi _\ell \}_\ell . The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of Φ ℓ \Phi _\ell admits a reductive Q \mathbb {Q} -form that is independent of ℓ \ell if X X is projective. Denote by Γ ℓ \Gamma _\ell and G ℓ \mathbf {G}_\ell respectively the monodromy group and the algebraic monodromy group of Φ ℓ s s \Phi _\ell ^{\mathrm {ss}} , the semisimplification of Φ ℓ \Phi _\ell . Assuming that G ℓ 0 \mathbf {G}_{\ell _0} satisfies some group theoretic conditions for some prime ℓ 0 \ell _0 , we construct a connected quasi-split Q \mathbb {Q} -reductive group G Q \mathbf {G}_{\mathbb {Q}} which is a common Q \mathbb {Q} -form of G ℓ ∘ \mathbf {G}_\ell ^\circ for all sufficiently large ℓ \ell . Let G Q s c \mathbf {G}_{\mathbb {Q}}^{\mathrm {sc}} be the universal cover of the derived group of G Q \mathbf {G}_{\mathbb {Q}} . As an application, we prove that the monodromy group Γ ℓ \Gamma _\ell is big in the sense that Γ ℓ s c ≅ G Q s c ( Z ℓ ) \Gamma _\ell ^{\mathrm {sc}}\cong \mathbf {G}_{\mathbb {Q}}^{\mathrm {sc}}(\mathbb {Z}_\ell ) for all sufficiently large ℓ \ell .

The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups H. We provide a unified study of the action of both cover Galois group H and the absolute Galois group $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.

We prove that the sign of the functional equation of the L-function of an orthogonal motive of even weight is always positive, if we admit the conjecture of the product formula for the epsilon factor of Deligne [D3] (5.2). More precisely, we define the sign independently of any conjecture by using the product formula as the definition and by working with one chosen "good" prime f and we prove its positivity. The positivity has been established for the Artin motives [Fr-Q][D2], for the symmetric square of a modular elliptic curve [C-Sc] and in positive characteristic by Serre [Fr-Q]. As in the proof in [D2], we consider the second Stiefel-Whitney class in zBr(K) of an orthogonal representation of the absolute Galois group G~: of an algebraic number field K. The positivity of the sign is then a consequence of the reciprocity law of the square residue symbol. Fixing a "good" prime f, we work with an f-adic representation and compare the local epsilon factor and the local Stiefel-Whitney class. For a prime vXr it is a theorem of Deligne [D2]. For a prime vie , we compute the Stiefel-Whitney class assuming the representation is cristalline (Theorem 2) using a generalization of a theorem of Fr6hlich [Fr] to f-adic representation of Hodge-Tate type (Theorem 1). By combining them with an elementary computation for the Hodge structures at r iot , we conclude that the sign is positive (Theorem 3). The crucial step in the proof of Theorem 2, given in section 1, is the determination by Fontaine and Lafaille [Fo-L] of the action of the tame inertia on the semisimplification of reduction modulo p of a cristalline representation. The theory of orthogonal representations of Galois group including a proof of Fr6hlich's theorem is reviewed in a preliminary section 0.

ABSTRACTA result of Segal states that every complex irreducible representation of a finitely generated nilpotent group G is monomial if and only if G is abelian-by-finite. A conjecture of Parshin, recently proved affirmatively by Beloshapka and Gorchinskii (2016), characterizes the monomial irreducible representations of finitely generated nilpotent groups. This article gives a slightly shorter proof of the conjecture using ideas of Kutzko and Brown. We also give a characterization of the finite-dimensional irreducible representations of two-step nilpotent groups and describe these completely for two-step groups whose center has rank one.

The Well theorem [I] on local rigidity is one of the fundamental results of the theory of discrete subgroups of Lie groups. The theorem asserts that for any connected semisimple Lie group G without compact components whose Lie algebra does not have any sl(2: R) factors, the orbit of any uniform lattice F c G (with respect to the adjoint action adG) is open in Hom(F, G). The assertion follows from the fact that the cohomology group HI(F, Ad) is trivial (for the definitions, see e.g. [2]). The above result of Weil can often be generalized ([3-6], etc.). Garland and Raghunathan [7] proved the "disappearance theorem," according to which for any lattice F in a simple connected Lie group G of real rank 1 that is not locally isomorphic with SL2, the cohomology group HI(F, Ad) is equal to zero. Thurston [8] proved that the corresponding "disappearance theorem" is no longer valid for nonuniform lattices in the case G----SL~(C). Namely, if F is a lattice in SL2(C) that has no finite-order elements and n is the number of conjugacy classes of the maximal parabolic subgroups of F, then the complex dimension of Horn(F, SL2(C))/adSL2(C) at any point corresponding to an irreducible representation p is not less than n. Alternative proofs of this fact were presented in [9, i0]. However, each of the proofs rests upon some algebraic (or geometric) properties of the group SLy(C) , the representations in which were considered in these articles. ]in particular~ Thurston's proof was based on the fact that for any a, b~SL~(C) and for any word w(a, b) = i, the word w(a -l, b -I) is also equal to i.

A characterization of number fields by p-closed extensions

In the theory of representations of a finite group by linear transformations the closely related notions of imprimitivity and of play a prominent role. In [18] the author has generalized these notions to the case in which the group is a separable locally compact topological group and the linear transformations are unitary transformations in Hilbert space. It turns out (and this is the principle theorem of [18]) that the classical theorem of Frobenius according to which every imprimitive representation of a finite group is in a certain canonical fashion by a representation of a subgroup may be reformulated so as to remain true under the more general circumstances indicated above. This connection between representations of groups and representations of their subgroups has many interesting and useful properties in the finite case and it naturally occurs to one to study the extent to which these properties persist in general. The present paper is the first of a projected series in which it is planned to investigate this question in a systematic manner. Our principal results, formulated as Theorems 12.1, 12.2 and 13.1, are closely related; each being essentially a corollary of its predecessor. The first asserts that if L is a representation of the closed subgroup G1 of (M and UL is the corresponding induced representation of (M then the restriction of UL to the closed subgroup G2 is a sum over the G1: G2 double cosets of certain induced representations o G 2 . The second gives a similar decomposition of the Kronecker product UL f? UM where L and M are representations of G1 and G2 respectively. The third provides a usable formula for computing the strong intertwining numbers of the induced representations UL and UM of (M. The in question are ordinary discrete sums only when there are at most countably many non trivial double cosets. In general direct integrals as defined by von Neumann in [26] must be used and we must restrict ourselves to the case in which the relevant double coset decomposition of (M is measurable. As we have shown in detail elsewhere [20] these theorems for finite groups imply certain classical results; in particular the Frobenius reciprocity theorem and the Shoda criteria for the irreducibility and unitary equivalence of monomial representations. Our theorems yield generalizations of these results but these generalizations may be regarded as satisfactory only insofar as they deal with representations whose irreducible constituents are discrete and finite dimensional. More far reaching generalizations must be sought in other directions. We

The theory of differential characters is developed completely from a de Rham-Federer viewpoint. Characters are defined as equivalence classes of special currents, called sparks, which appear naturally in the theory of singular connections. As in de Rham-Federer cohomology, many different spaces of currents are shown to yield the character groups. The fundamental exact sequences are derived, and a multiplication of de Rham-Federer characters is defined using new transversality/intersection results for flat and rectifiable currents. An equivalence of de Rham-Federer characters with the classical Cheeger-Simons characters is given, as in de Rham cohomology, via integration. This discussion rounds out the approach to characters introduced by Gillet-Soulé and Harris. The groups of differential characters have a natural topology and smooth Pontrjagin duals (introduced here) which fit into fundamental exact sequences. A principal result is the formulation and proof of duality for characters on oriented manifolds. It is shown that the pairing ( a, b ) [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] a * b ([ X ]) defined by multiplication and evaluation on the fundamental cycle, gives an isomorphism of the group of differential characters of degree k with the dual to characters in degree n - k -1 where n = dim( X ). Thom homomorphisms, which refine the usual ones in de Rham theory and integral cohmology, are established for differential characters. Gysin maps for characters are similarly established. New examples, namely Morse sparks and Hodge sparks, are introduced and examined in detail.

We describe a method for counting the number of extensions of $\\mathbb{Q}_p$\nwith a given Galois group $G$, founded upon the description of the absolute\nGalois group of $\\mathbb{Q}_p$ due to Jannsen and Wingberg. Because this\ndescription is only known for odd $p$, our results do not apply to\n$\\mathbb{Q}_2$. We report on the results of counting such extensions for $G$ of\norder up to $2000$ (except those divisible by $512$), for $p=3,5,7,11,13$. In\nparticular, we highlight a relatively short list of minimal $G$ that do not\narise as Galois groups. Motivated by this list, we prove two theorems about the\ninverse Galois problem for $\\mathbb{Q}_p$: one giving a necessary condition for\n$G$ to be realizable over $\\mathbb{Q}_p$ and the other giving a sufficient\ncondition.\n