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: Fibre Degrading Enzymes from Butyrivibrio proteoclasticus

authors: Till, Marisa

abstract: The rumen harbours a large and diverse microbial population that is responsible for the breakdown of plant material into smaller compounds, which can then be utilised by the animal. Butyrivibrio proteoclasticus is an anaerobic, Gram-positive bacterium originally isolated from the rumen of New Zealand cows. The entire genome of B. proteoclasticus has been sequenced. This revealed that a large proportion of the genome is devoted to polysaccharide degradation and reassembly1. Prior to the start of the research described in this thesis, 44 of the genes from the B. proteoclasticus genome annotated as being involved in fibre degradation had been cloned and expression of many has been tested. Two of these enzymes were expressed, purified and had had their 3D structures determined. Further characterisation of these two enzymes is presented here, with site directed mutagenesis used to probe the proposed mechanism for each. The results support the proposed catalytic mechanisms of both enzymes. Kinetic parameters were measured with model substrates for both enzymes and this allowed comparison with similar enzymes produced by other organisms. A further nine fibre degrading enzymes from B. proteoclasticus were chosen for structural and/or functional investigation. Preliminary structural investigation gave crystallisation conditions for one enzyme, Xsa43E. The structure of Xsa43E is presented along with functional analysis of the enzyme. Four residues were found to be important for catalysis, three previously identified catalytic acidic residues common to all GH43 enzymes and a fourth residue, a histidine that is important for pKa modulation of the catalytic acid. The large central ion present in the structure of Xsa43E is identified as calcium and shown to be important both for activity and structural stability.

text: Fibre Degrading Enzymes from Butyrivibrio proteoclasticus The rumen harbours a large and diverse microbial population that is responsible for the breakdown of plant material into smaller compounds, which can then be utilised by the animal. Butyrivibrio proteoclasticus is an anaerobic, Gram-positive bacterium originally isolated from the rumen of New Zealand cows. The entire genome of B. proteoclasticus has been sequenced. This revealed that a large proportion of the genome is devoted to polysaccharide degradation and reassembly1. Prior to the start of the research described in this thesis, 44 of the genes from the B. proteoclasticus genome annotated as being involved in fibre degradation had been cloned and expression of many has been tested. Two of these enzymes were expressed, purified and had had their 3D structures determined. Further characterisation of these two enzymes is presented here, with site directed mutagenesis used to probe the proposed mechanism for each. The results support the proposed catalytic mechanisms of both enzymes. Kinetic parameters were measured with model substrates for both enzymes and this allowed comparison with similar enzymes produced by other organisms. A further nine fibre degrading enzymes from B. proteoclasticus were chosen for structural and/or functional investigation. Preliminary structural investigation gave crystallisation conditions for one enzyme, Xsa43E. The structure of Xsa43E is presented along with functional analysis of the enzyme. Four residues were found to be important for catalysis, three previously identified catalytic acidic residues common to all GH43 enzymes and a fourth residue, a histidine that is important for pKa modulation of the catalytic acid. The large central ion present in the structure of Xsa43E is identified as calcium and shown to be important both for activity and structural stability.

year: 2011

title: Adult literacy and numeracy in Aotearoa New Zealand: How does current research and organisational reporting align with the Living Standards Framework?

authors: Hockings, Nicki Samantha

subjects: Literacy

abstract: Literacy and numeracy are foundational skills that are important in everyday life. Dominant western perspectives of literacy and numeracy focus on skills attainment for economic benefit and international comparison; this is a neoliberal approach that influences policy development in developed countries worldwide. However, there is growing recognition that high rates of economic productivity do not provide an accurate indicator of success for either individuals or nations. In support of this position, this thesis favours a sociomaterial approach to literacy and numeracy, prioritising people’s abilities to communicate and navigate their world effectively, while engaging in meaningful social practices that allow them to participate in family and community. This master’s research sought to understand the connections between adult literacy and numeracy learners, wellbeing outcomes, and the Living Standards Framework. The New Zealand Treasury’s Living Standards Framework (LSF) was developed to provide a view of intergenerational wellbeing in Aotearoa New Zealand; it underpins the government’s wellbeing focus for budgetary and policy decisions. A community psychology and Kaupapa Māori lens was applied to critically evaluate whether the LSF aligns with community psychology values and principles in Aotearoa New Zealand, and, most importantly, whether it reflects Māori cultural views and values. The value of the LSF in the adult literacy and numeracy education context was also examined, with particular emphasis on similarities between the LSF, literacy and numeracy learning opportunities, and learner outcomes; and whether the use of the LSF in adult education programmes and policies would complement the existing approach. The study involved a document analysis of six key LSF documents and six literacy and numeracy research reports. Each document was analysed using deductive and inductive means, which enabled the identification of themes and subthemes that reflected underlying values, approaches, and outcomes. To augment the document analysis, six interviews were held with report authors and key personnel from relevant organisations; these added value to the research by exploring the values that underpin literacy and numeracy research and reporting in Aotearoa New Zealand. The findings conclude that the 2018 LSF was based on sound international principles, but lacked a genuine perspective of wellbeing in Aotearoa New Zealand. A recently-launched 2021 LSF has undergone an ‘evolution’ and is presented in a much more holistic style, acknowledging its previous shortcomings in addressing concepts of wellbeing in Aotearoa New Zealand. Two accompanying Māori-led documents: An Indigenous Approach to the Living Standards Framework, and He Ara Waiora, both offer a holistic, te ao Māori perspective of wellbeing. The document analysis highlighted a sociomaterial, values based approach to adult literacy and numeracy education that prioritised holistic wellbeing outcomes for learners and their whānau. These findings were supported by the interview data. The final discussion and conclusion posited that the 2021 Living Standards Framework provides a welcome update and potential for genuine, Aotearoa-specific wellbeing considerations in policy-making and governance; however, it remains to be seen whether this will translate into changes in policy-making and programme delivery for the literacy and numeracy sector.

text: Adult literacy and numeracy in Aotearoa New Zealand: How does current research and organisational reporting align with the Living Standards Framework? Literacy and numeracy are foundational skills that are important in everyday life. Dominant western perspectives of literacy and numeracy focus on skills attainment for economic benefit and international comparison; this is a neoliberal approach that influences policy development in developed countries worldwide. However, there is growing recognition that high rates of economic productivity do not provide an accurate indicator of success for either individuals or nations. In support of this position, this thesis favours a sociomaterial approach to literacy and numeracy, prioritising people’s abilities to communicate and navigate their world effectively, while engaging in meaningful social practices that allow them to participate in family and community. This master’s research sought to understand the connections between adult literacy and numeracy learners, wellbeing outcomes, and the Living Standards Framework. The New Zealand Treasury’s Living Standards Framework (LSF) was developed to provide a view of intergenerational wellbeing in Aotearoa New Zealand; it underpins the government’s wellbeing focus for budgetary and policy decisions. A community psychology and Kaupapa Māori lens was applied to critically evaluate whether the LSF aligns with community psychology values and principles in Aotearoa New Zealand, and, most importantly, whether it reflects Māori cultural views and values. The value of the LSF in the adult literacy and numeracy education context was also examined, with particular emphasis on similarities between the LSF, literacy and numeracy learning opportunities, and learner outcomes; and whether the use of the LSF in adult education programmes and policies would complement the existing approach. The study involved a document analysis of six key LSF documents and six literacy and numeracy research reports. Each document was analysed using deductive and inductive means, which enabled the identification of themes and subthemes that reflected underlying values, approaches, and outcomes. To augment the document analysis, six interviews were held with report authors and key personnel from relevant organisations; these added value to the research by exploring the values that underpin literacy and numeracy research and reporting in Aotearoa New Zealand. The findings conclude that the 2018 LSF was based on sound international principles, but lacked a genuine perspective of wellbeing in Aotearoa New Zealand. A recently-launched 2021 LSF has undergone an ‘evolution’ and is presented in a much more holistic style, acknowledging its previous shortcomings in addressing concepts of wellbeing in Aotearoa New Zealand. Two accompanying Māori-led documents: An Indigenous Approach to the Living Standards Framework, and He Ara Waiora, both offer a holistic, te ao Māori perspective of wellbeing. The document analysis highlighted a sociomaterial, values based approach to adult literacy and numeracy education that prioritised holistic wellbeing outcomes for learners and their whānau. These findings were supported by the interview data. The final discussion and conclusion posited that the 2021 Living Standards Framework provides a welcome update and potential for genuine, Aotearoa-specific wellbeing considerations in policy-making and governance; however, it remains to be seen whether this will translate into changes in policy-making and programme delivery for the literacy and numeracy sector.

year: 2022