title: Book review: Decentering Empire: Britain, India and the Transcolonial World

authors: Beattie, James John

subjects: book review

abstract: This article reviews the book: “Decentering Empire: Britain, India and the Transcolonial World”, by Durba Ghosh and Dane Kennedy.

text: Book review: Decentering Empire: Britain, India and the Transcolonial World This article reviews the book: “Decentering Empire: Britain, India and the Transcolonial World”, by Durba Ghosh and Dane Kennedy.

year: 2007

title: L-invariants and congruences for Galois representations of dimension 3, 4, and 8

authors: Gilmore, Hamish Julian

abstract: The arithmetic of Galois representations plays a central role in modern number theory. In this thesis we consider representations arising from tensor products of the two-dimensional representations attached to modular forms by Deligne. In particular, we shall study the Iwasawa theory of the adjoint representation, as well as certain double and triple products of Deligne's representations.

In the first half we will undertake a computational study of L-invariants attached to symmetric squares of modular forms. Let ƒ be a primitive modular form of weight 𝘬 and level 𝘕, and 𝘱 ∤ 𝘕 a prime greater than two for which the attached representation is ordinary. The 𝘱-adic 𝘓-function Sym²ƒ always vanishes at s=1, even though the complex 𝘓-function does not have a zero there. The L-invariant itself appears on the right-hand side of the formula
	
	\frac{\text{\rm d}\;}{\text{\rm d}s}\mathbf{L}_p(\text{\rm Sym}^2f,s)
	\Bigg|_{s=k-1}=
	\mathcal{L}_p(\text{\rm Sym}^2f) \times 
	 (1-\alpha_p^{-2}p^{k-2})(1-\alpha_p^{-2}p^{k-1}) \times
	\frac{L_{\infty}(\text{\rm Sym}^2f,k-1)}{\pi^{k-1}\langle f,f\rangle_N}
	
where X²-aₚ(ƒ)𝘟 + p = (𝘟-αₚ)(𝘟-βₚ) with αₚ ∈ Zˣₚ.

Now let 𝘌 be an elliptic curve over Q with associated modular form ƒᴇ, and 𝘱 ≠ 2 a prime of good ordinary reduction. We devise a method to calculate Lₚ(Sym²ƒᴇ) effectively, then show it is non-trivial for almost all pairs of elliptic curves 𝘌 of conductor 𝘕ᴇ ≤ 300, with 4|𝘕ᴇ, and ordinary primes 𝘱 < 17. Hence, in these cases at least, the order of the zero in Lₚ(Sym²ƒᴇ, 𝘴) at 𝘴 = 1 is exactly one. We also generalise this method to compute symmetric square L-invariants for modular forms of weight 𝘬 > 2.

In the second half we will establish congruences between 𝘱-adic 𝘓-functions. In the late 1990s, Vatsal showed that a congruence modulo 𝘱 ᵛ between two newforms implied a congruence between their respective 𝘱-adic 𝘓-functions. We shall prove an analogous statement for both the double product and triple product 𝘱-adic 𝘓-functions, Lₚ(f⊗g) and Lₚ (f⊗g⊗h): the former is cyclotomic in its nature, while the latter is over the weight space.

As a corollary, we derive transition formulae relating analytic  λ-invariants for pairs of congruent Galois representations for 𝘝𝒻 ⊗ 𝘝𝓰, and for 𝘝𝒻 ⊗ 𝘝𝓰⊗ 𝘝ₕ.

text: L-invariants and congruences for Galois representations of dimension 3, 4, and 8 The arithmetic of Galois representations plays a central role in modern number theory. In this thesis we consider representations arising from tensor products of the two-dimensional representations attached to modular forms by Deligne. In particular, we shall study the Iwasawa theory of the adjoint representation, as well as certain double and triple products of Deligne's representations.

In the first half we will undertake a computational study of L-invariants attached to symmetric squares of modular forms. Let ƒ be a primitive modular form of weight 𝘬 and level 𝘕, and 𝘱 ∤ 𝘕 a prime greater than two for which the attached representation is ordinary. The 𝘱-adic 𝘓-function Sym²ƒ always vanishes at s=1, even though the complex 𝘓-function does not have a zero there. The L-invariant itself appears on the right-hand side of the formula
	
	\frac{\text{\rm d}\;}{\text{\rm d}s}\mathbf{L}_p(\text{\rm Sym}^2f,s)
	\Bigg|_{s=k-1}=
	\mathcal{L}_p(\text{\rm Sym}^2f) \times 
	 (1-\alpha_p^{-2}p^{k-2})(1-\alpha_p^{-2}p^{k-1}) \times
	\frac{L_{\infty}(\text{\rm Sym}^2f,k-1)}{\pi^{k-1}\langle f,f\rangle_N}
	
where X²-aₚ(ƒ)𝘟 + p = (𝘟-αₚ)(𝘟-βₚ) with αₚ ∈ Zˣₚ.

Now let 𝘌 be an elliptic curve over Q with associated modular form ƒᴇ, and 𝘱 ≠ 2 a prime of good ordinary reduction. We devise a method to calculate Lₚ(Sym²ƒᴇ) effectively, then show it is non-trivial for almost all pairs of elliptic curves 𝘌 of conductor 𝘕ᴇ ≤ 300, with 4|𝘕ᴇ, and ordinary primes 𝘱 < 17. Hence, in these cases at least, the order of the zero in Lₚ(Sym²ƒᴇ, 𝘴) at 𝘴 = 1 is exactly one. We also generalise this method to compute symmetric square L-invariants for modular forms of weight 𝘬 > 2.

In the second half we will establish congruences between 𝘱-adic 𝘓-functions. In the late 1990s, Vatsal showed that a congruence modulo 𝘱 ᵛ between two newforms implied a congruence between their respective 𝘱-adic 𝘓-functions. We shall prove an analogous statement for both the double product and triple product 𝘱-adic 𝘓-functions, Lₚ(f⊗g) and Lₚ (f⊗g⊗h): the former is cyclotomic in its nature, while the latter is over the weight space.

As a corollary, we derive transition formulae relating analytic  λ-invariants for pairs of congruent Galois representations for 𝘝𝒻 ⊗ 𝘝𝓰, and for 𝘝𝒻 ⊗ 𝘝𝓰⊗ 𝘝ₕ.

year: 2020

title: Where to from here: Charting a way forward for language and education policy in Aotearoa/New Zealand

authors: May, Stephen

subjects: New Zealand

abstract: The principle of cultural and linguistic homogeneity, upon which the organization of modern nation-states is predicated, is becoming increasingly hard to defend and maintain.  At the demographic level, nation-states comprise (indeed, have always comprised) a variety of different cultural and linguistic groups.  These include, alongside majority populations, indigenous peoples and other national minorities, as well as migrant communities.  This long-standing demographic diversity has also increased markedly in modern times, particularly with patterns of increased migration and the forced relocation of refugees.  These demographic changes have, in turn, placed increasing pressure on the public policies of nation-states which have historically been inimical or formal recognition of cultural and linguistic diversity.  As a result, nation-states are having to address more seriously the 'politics of multiculturalism' - that is, the degree to which the languages and cultures of so called 'minority' groups can be (or should be) accorded recognition in the public domain.  This also necessarily involves addressing directly issues of bilingualism and multilingualism and their implications for language and education policy and practice.

This paper explores these broader debates in specific relation to Aotearoa/New Zealand which itself has seen a marked increase in migration, and attendant cultural and linguistic diversity, over the last ten years.  Debates on multiculturalism, and their implications for public policy, are contentious enough in themselves, but are further complicated in Aotearoa/New Zealand by prior bicultural commitments to Maori.  Can multicultural/multilingual commitments be extended without compromising, or undermining biculturalism?  What are the specific implications of this potential dialectic between biculturalism and multiculturalism for the further development of language and education policy in Aotearoa/New Zealand?  The paper will attempt to develop a set of general principles as a basis for moving these debates forward, drawing on discussions of language rights, and language planning and policy, as well as recent developments in international law.  Specific implications for language education will also be discussed.

text: Where to from here: Charting a way forward for language and education policy in Aotearoa/New Zealand The principle of cultural and linguistic homogeneity, upon which the organization of modern nation-states is predicated, is becoming increasingly hard to defend and maintain.  At the demographic level, nation-states comprise (indeed, have always comprised) a variety of different cultural and linguistic groups.  These include, alongside majority populations, indigenous peoples and other national minorities, as well as migrant communities.  This long-standing demographic diversity has also increased markedly in modern times, particularly with patterns of increased migration and the forced relocation of refugees.  These demographic changes have, in turn, placed increasing pressure on the public policies of nation-states which have historically been inimical or formal recognition of cultural and linguistic diversity.  As a result, nation-states are having to address more seriously the 'politics of multiculturalism' - that is, the degree to which the languages and cultures of so called 'minority' groups can be (or should be) accorded recognition in the public domain.  This also necessarily involves addressing directly issues of bilingualism and multilingualism and their implications for language and education policy and practice.

This paper explores these broader debates in specific relation to Aotearoa/New Zealand which itself has seen a marked increase in migration, and attendant cultural and linguistic diversity, over the last ten years.  Debates on multiculturalism, and their implications for public policy, are contentious enough in themselves, but are further complicated in Aotearoa/New Zealand by prior bicultural commitments to Maori.  Can multicultural/multilingual commitments be extended without compromising, or undermining biculturalism?  What are the specific implications of this potential dialectic between biculturalism and multiculturalism for the further development of language and education policy in Aotearoa/New Zealand?  The paper will attempt to develop a set of general principles as a basis for moving these debates forward, drawing on discussions of language rights, and language planning and policy, as well as recent developments in international law.  Specific implications for language education will also be discussed.

year: 2002-01-01