Victor Snaith

The University of Sheffield | Sheffield · School of Mathematics and Statistics

In this, the eighth article in my Derived Langlands series, I describe the construction of a 2-variable L-function for two representations of general linear groups of a p-adic local field. Due to extenuating health circumstances, many of the proofs and ideas herein are suggestive rather than the finished definitive versions.

This is a sequel to the book "Derived Langlands" (World Scientific 2018) which studies an embedding of the category of admissible representations of a locally p-adic group into a derived category. This essay introduces the hyperHecke algebra - an elaboration of the usual Hecke algebra. Its bar resolution is the basis of the embedding and its centre...

Notes on how Shintani base change for finite general linear groups gives an algebra map between associated PSH algebras

Let K be a p-adic local field where p is an odd prime and let A be the unique quaternion division algebra whose centre is K. By means of Stiefel–Whitney classes, we define an exponential homomorphism ϒK from the orthogonal representations of A*/K* to fourth roots of unity. We then evaluate this homomorphism in terms of the local root numbers of two...

We give a new proof of the Glauberman correspondence using the Explicit Brauer Induction technique developed by Boltje–Snaith–Symonds.

In the local, unramified case the determinantal functions associated to the group-ring of a finite group satisfy Galois descent. This note examines the obstructions to Galois determinantal descent in the ramified case. Comment: 12 pages

Let $p$ be a prime. We calculate the connective unitary K-theory of the smash product of two copies of the classifying space for the cyclic group of order $p$, using a K\"{u}nneth formula short exact sequence. As a corollary, using the Bott exact sequence and the mod $2$ Hurewicz homomorphism we calculate the connective orthogonal K-theory of the s...

As an application of the upper triangular technology method of (V.P. Snaith: {\em Stable homotopy -- around the Arf-Kervaire invariant}; Birkh\"{a}user Progress on Math. Series vol. 273 (April 2009)) it is shown that there do not exist stable homotopy classes of $ {\mathbb RP}^{\infty} \wedge {\mathbb RP}^{\infty}$ in dimension $2^{s+1}-2$ with $s...

This work surveys classical and recent advances around the existence of exotic differentiable structures on spheres and its connection to stable homotopy theory.

In our earlier article we described a power series formula for the Borel regulator evaluated on the odd-dimensional homology of the general linear group of a number field and, concentrating on dimension three for simplicity, described a computer algorithm which calculates the value to any chosen degree of accuracy. In this sequel we give an algorit...

We present an infinite series formula based on the Karoubi-Hamida integral, for the universal Borel class evaluated on H_{2n+1}(GL(\mathbb{C})). For a cyclotomic field F we define a canonical set of elements in K_3(F) and present a novel approach (based on a free differential calculus) to constructing them. Indeed, we are able to explicitly constru...

We derive a power series formula for the $p$-adic regulator on the higher dimensional algebraic K-groups of number fields. This formula is designed to be well suited to computer calculations and to reduction modulo powers of $p$. In addition we describe a series of regulator questions concerning higher dimensional K-theoretic analogues of conjectur...

The objective of this chapter is to determine the conjugacy class of the map 1 ∧ψ3 in the upper triangular group U∞ℤ2 in the sense of Chapter 3, Theorem 3.1.2. §1 recapitulates the background and states the main result (Theorem 5.1.2). §2 contains the central calculations in which the effect of 1 ∧ ψ3 is estimated with respect to the ℤ2-module basi...

The objective of this chapter is to prove the conjecture of [30] in its original form, as stated in Chapter 1, Theorem 1.8.10. This result is reiterated in this chapter as Theorem 7.2.2. The conjecture states that an element of \( \pi _{2^{n + 1} - 2} (\sum ^\infty \mathbb{R}\mathbb{P}^\infty ) \) corresponds under the Kahn-Priddy map to the class...

The object of this chapter is to establish the basic result which relates the upper triangular group to operations in connective K-theory. This result will identify a certain group of operations with the infinite upper triangular group with entries in the 2-adic integers. This identification will be canonical up to inner automorphisms. The 2-adic i...

The objective of this chapter is to sketch the historical and technical stable homotopy background which we shall need in the course of this book. § 1 deals with the history of the calculations of stable homotopy groups of spheres (the so-called “stable stems”). §2 describes the framed manifold approach of Pontrjagin and Thom. § 3 introduces the cl...

The objective of this chapter is to present the cohomological calculations (in MU*, KU* and -BP*) which will be needed in this and later chapters for the study of maps of the form $$ g:\sum ^\infty S^{2^{k + 1} - 2} \to \sum ^\infty \mathbb{R}\mathbb{P}^{2^{k + 1} - 2} $$ and related Whitehead product maps. In [30] it is shown that if g* is non-zer...

The fractional Galois ideal of [Victor P. Snaith, Stark's conjecture and new Stickelberger phenomena, Canad. J. Math. 58 (2) (2006) 419--448] is a conjectural improvement on the higher Stickelberger ideals defined at negative integers, and is expected to provide non-trivial annihilators for higher K-groups of rings of integers of number fields. In...

We show that the motivic spectrum representing algebraic $K$-theory is a localization of the suspension spectrum of $\mathbb{P}^\infty$, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspension spectrum of $BGL$. In particular, working over $\mathbb{C}$ and passing to spaces of $\mathbb{C...

Without Abstract

Without Abstract

We introduce a new conjecture concerning the construction of elements in the annihilator ideal associated to a Galois action on the higher-dimensional algebraic K-groups of rings of integers in number fields. Our conjecture is motivic in the sense that it involves the (transcendental) Borel regulator as well as being related to l-adic étale cohomol...

In the 2-local stable homotopy category the group of left-bu-module automorphisms of bu ∧ bo which induce the identity on mod 2 homology is isomorphic to the group of infinite, invertible upper triangular matrices with entries in the 2-adic integers. We identify the conjugacy class of the matrix corresponding to 1∧ ψ3, where ψ3 is the Adams operati...

When G is abelian and l is a prime we show how elements of the relative K-group K0(Zl[G], Ql give rise to annihilator/Fitting ideal relations of certain associated Z[G]-modules. Examples of this phenomenon are ubiquitous. Particularly, we give examples in which G is the Galois group of an extension of global fields and the resulting annihilator/Fit...

In the 2-local stable homotopy category the group of left-bu-module automorphisms of bu\wedge bo which induce the identity on mod 2 homology is isomorphic to the group of infinite upper triangular matrices with entries in the 2-adic integers. We identify the conjugacy class of the matrix corresponding to 1\wedge\psi^3, where \psi^3 is the Adams ope...

Explicit Brauer Induction formulae with certain natural behaviour have been developed for complex representations, for example by work of Boltje, Snaith and Symonds. In this paper we present induction formulae for symplectic and orthogonal representations of finite groups. The problems are motivated by number theoretical and topological questions....

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We lift the Euler characteristic of a nearly perfect complex to a relative algebraic K-group by passing to its l-adic Euler characteristics.

Inspired by the work of Bloch and Kato in [2], David Burns constructed several ‘equivariant Tamagawa invariants’ associated to motives of number fields. These invariants lie in relative K-groups of group-rings of Galois groups, and in [3] Burns gave several conjectures (see Conjecture 3.1) about their values. In this paper I shall verify Burns' con...

Let L/K be a finite Galois extension of local fields in positive characteristic with group G. The weight-two motivic complex defines an element of . We show, after inverting the prime 2, that cup-product with this 2-extension induces an isomorphism on Tate cohomology. In fact we show that this isomorphism coincides with cup-product by the K2/K3 loc...

We lift the Euler characteristic of a nearly perfect complex to a relative algebraic K-group by passing to its l-adic Euler characteristics.

We verify one of Burn's equivariant Tamagawa number conjectures for some families of quaternion fields.

In this paper BP-theory is used to give a proof that there exists a stable homotopy element in π2n+1-2S (RP∞) with non-zero Hurewicz image in ju-theory if and only if there exists an element of π2n+1-2S(S0) that is represented by a framed manifold of Arf invariant one.

In this chapter we shall examine invariants of the Galois module structure on the higher algebraic K-groups of local fields. These will be constructed by the method of Example 2.1.8(i) as the Euler characteristic of suitable 2-extensions, called the local fundamental classes lying in \( {\rm E}xt_{{\rm Z}\left[ {G\left( {L/K} \right)} \right]}^2\le...

In this chapter we shall be concerned with the fundamental classes of Theorem 3.1.17 and Example 3.1.21 associated to K 2 and K 3 of a local field in characteristic p > 0. We are going to calculate the Euler characteristic, in the sense of Example 2.1.8, of the 2-extension described in Theorem 3.1.22. For the calculations of this section we shall m...

In this chapter we shall recall the salient facts about low-dimensional K-groups and localisation sequence for the integral group-ring, Z [G], of a finite group G. This material is recounted just as well, if not better, in many other places. Nonetheless, repeating it here will serve both to remind the reader and also to establish our conventions, p...

The aim of this chapter is to introduce by example the study of the manner in which the Galois structure on various motivic objects from arithmetic influence the special values of associated L-functions. In these examples our motivic object will be K 2 of rings of algebraic integers in a Galois extension, whose Galois structure influences the value...

In this chapter we shall relate the higher-dimensional K-theory Galois module structure invariants, Ωn(N/K,3), to relations between annihilator ideals of K-groups and of étale cohomology groups. The constructions are very simple and work particularly well when the K-groups are finite. In §7.1 we show how to obtain a chain of annihilator ideal relat...

In this chapter we shall examine invariants of the Galois module structure on the higher-dimensional algebraic K-groups of rings of algebraic integers in number fields. Special cases of these invariants were given in §§1.3.4-1.3.13. The material of this chapter originated in ([132] Chapter VII) where generalizations of the Chin-burg invariant, Ω0(L...

In this chapter I want to introduce a conjecture (Conjecture 6.3.4) whose cryptic form would be: “The Wiles Unit is a determinant” The “Wiles Unit” unit is the p-adic unit-valued function on Galois representations given (approximately) by the ratio of the p-adic L-function to the Iwasawa polynomial. The values of this function are p-adic units by t...

Throughout number theory and arithmetic-algebraic geometry one encounters objects endowed with a natural action by a Galois group. In particular this applies to algebraic K-groups and étale cohomology groups. This volume is concerned with the construction of algebraic invariants from such Galois actions.

An alternative proof is given of a result, originally due to Guido Mislin, giving necessary and sufficient conditions for the inclusion of a subgroup to induce an isomorphism in mod p cohomology.

The Second Chinburg Conjecture relates the Galois module structure of rings of integers in number fields to the values of the Artin root number on the symplectic representations of the Galois group. We establish the Second Chinburg Conjecture for all quaternion fields.

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HEIGHTS OF POLYNOMIALS AND ENTROPY IN ALGEBRAIC DYNAMICS (Universitext) By GRAHAM EVEREST and THOMAS WARD: 212 pp., £35.00, ISBN 1 85233 125 9 (Springer, 1999). - - Volume 32 Issue 4 - VICTOR P. SNAITH

Weprovethattwo, apparentlydifferent, class-group valued Galois modulestructureinvariantsasso- ciated to the algebraic K-groups of rings of algebraic integers coincide. This comparison result is particularly important in making explicit calculations.

We define a generalization of the Euler characteristic of a perfect complex of modules for the group ring of a finite group. This is combined with work of Lichtenbaum and Saito to define an equivariant Euler characteristic for G on regular projective surfaces over Z having a free action of a finite group. In positive characteristic we relate the Eu...

In this paper BP-theory is used to give a proof that there exists a stable homotopy element in S 2n+1 2(RP 1) with non-zero Hurewicz image in ju-theory if and only if there exists an element of S 2n+1 2(S 0 )w hich is represented by a framed manifold of Arf invariant one.

We define a generalization of the Euler characteristic of a perfect complex of modules for the group ring of a finite group. This is combined with work of Lichtenbaum and Saito to define an equivariant Euler characteristic for G on regular projective surfaces over Z having a free action of a finite group. In positive characteristic we relate the Eu...

N/Kbe a Galois extension of number fields with finite Galois group G.We describe a new approach for constructing invariants of the G-module structure of the K groups of the ring of integers of N in the Grothendieck group of finitely generated projective Z[G]modules. In various cases we can relate these classes, and their function field counterparts...

L et Gbe a finite group. To a set of subgroups of order two we asso- ciate a mod 2 Hecke algebra and construct a homomorphism, , from its units to the class-group of Z(G). We show that this homomorphism takes values in the subgroup, D(Z(G)). Alternative constructions of Chinburg invariants arising from the Galois mod- ule structure of higher-dimens...

Let G be a finite group. To a set of subgroups of order two we associate a mod 2 Hecke algebra and construct a homomorphism, ?, from its units to the class-group of Z[G]. We show that this homomorphism takes values in the subgroup, D(Z[G]). Alternative constructions of Chinburg invariants arising from the Galois module structure of higher-dimension...

We study the second Chinburg invariant of a Galois extension of number fields. The Chinburg invariant lies in the class-group of the integral group-ring of the Galois group of the extension. A procedure is given whereby to evaluate the invariant in the case of the real cyclotomic case of regular prime power conductor and their subextensions of p-po...

Explicit Brauer Induction is an important technique in algebra, discovered by the author in 1986. It solves an old problem, giving a canonical formula for Brauer's induction theorem. In this 1994 book it is derived algebraically, following a method of R. Boltje - thereby making the technique, previously topological, accessible to algebraists. Once...

Using the technique of Explicit Brauer Induction an integer-valued conductor homomorphism is constructed for Galois representations of complete, discrete valuation fields. In the special case in which the residue field extension is separable the new conductor coincides with the classical Swan conductor. In the one-dimensional case the new conductor...

Algebraicisation of explicit Brauer induction / Robert Boltje, Victor Snaith, and Peter Symonds. - In: Journal of algebra. 148. 1992. S. 504-527

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Let G be a finite group. The class group of the group ring of G is a quotient of the idlic homomorphisms of R(G) by the global functions and the determinantal functions. Using Explicit Brauer Induction, we prove Adams operations, applied to determinantal functions, satisfy some congruences which were conjectured by M. J. Taylor. As an application,...

Let K be a number field and let O K denote the integers of K. The locally free class groups, Cl( O K [ G ]), furnish a fundamental collection of invariants of a finite group, G. In this paper I will construct some new, non-trivial homomorphisms, called restricted determinants , which map the N G H -invariant idèlic units of O k ([H ab ] to Cl( O K...

THE Segal conjecture has been subjected to a number of usefulgeneralizations over the years [1,3,9,14,19]. We here give a still furthergeneralization which includes those in all of the cited papers. Weconsider an extension1->AT->G->F->1,where G is a compact Lie group, K is a (closed) normal subgroup, andthe quotient G/K = F is finite. There is a cl...

Explicit Brauer Induction is a canonical form for Brauer’s induction theorem. It is designed for use in the construction of invariants of representations from invariants of one-dimensional characters. This paper gives a number of further applications including some new ‘change of field’ maps between representation rings, the behaviour of the canoni...

Let KO and KU respectively denote the real and complex periodic K -theory spectra [ 1 , Part III]. Let KSC denote the spectrum representing self-conjugate K -theory [ 2 , G]. Thus we have a fibring 1.1 where T is induced by complex conjugation on the unitary group. The following result is due to R. Wood [ 1 , p. 206] and, I believe, to D. W. Anders...

Write F for the finite field, , having 2m elements. Let W2(F) denote the Witt vectors of length two over F (for a definition, see [4] or [10], §10). Write F(q) for the truncated polynomial ring, F[t]/(tq).(Received March 14 1985)(Revised March 14 1986)

We calculate the topological K -theory of the cotangent sphere bundle of ℝP ⁿ and show the manner in which it is detected by the eta invariant.

According to [Car= P. Cartier, Lect. Notes Math. 842, 112-138 (1981; Zbl 0498.12013)] in 1923 E. Artin [A= Abh. Math. Semin. Univ. Hamb. 3, 89-108 (1923)] introduced L-functions into the theory of Galois representations in order to determine the multiplicative relations between Dedekind zeta functions of number fields [see A, pp. 96/97]. I will des...

Let K be a field of characteristic different from two. Let L be a finite separable extension of K. If is the separable closure of K, we have a continuous homomorphism π : Ga( /K) → ∑ n (n - [L : K]). We give a very short proof of Serre's formula which evaluates the Hasse-Witt invariant of a symmetric bilinear form, transferred from L, in terms of t...