title: Iwasawa theory for tensor products of Hilbert modular forms

authors: Corpuz, Raiza

abstract: The main conjecture of Iwasawa theory bridges two seemingly disjoint areas of mathematics: arithmetic and analysis. In particular, it provides a deep connection between the p-adic L-function which interpolates critical values of the complex L-series, and the Selmer group which is an important object used to control the growth of arithmetic data. In this thesis, we explore some questions that arise organically from the Iwasawa Main Conjecture, applied to Hilbert modular forms and their tensor products.
Greenberg and Vatsal developed an approach to study the main conjecture for a large class of elliptic curves simultaneously. They showed that if a given pair of elliptic curves share the same residual Galois representation, then the main conjecture holds for one if and only if it does for the other. The first part of this thesis investigates whether the ideas of Greenberg and Vatsal work for elliptic curves twisted by a CM-Hecke character. The second part of this thesis then extends the method to treat non-ordinary classical modular forms (without any twist). The former permits one to study rational elliptic curves base-changed to an arbitrary number field, whilst the latter requires techniques crafted by Pollack and Kobayashi in the early 2000s.
Finally, the third part of this thesis concerns Euler systems and their applications to the arithmetic of motives arising from modular forms—these objects are indispensable tools which can be used to prove half of the Iwasawa main conjecture. However they often give rise to additional “junk” error terms, as well as causing the p-adic L-function to be stripped away of certain bad Euler factors. For modular forms and their tensor products we devise a method to dispose of the error terms and to replenish the missing factors, allowing one to genuinely obtain half of the main conjecture free from any discrepancies.

text: Iwasawa theory for tensor products of Hilbert modular forms The main conjecture of Iwasawa theory bridges two seemingly disjoint areas of mathematics: arithmetic and analysis. In particular, it provides a deep connection between the p-adic L-function which interpolates critical values of the complex L-series, and the Selmer group which is an important object used to control the growth of arithmetic data. In this thesis, we explore some questions that arise organically from the Iwasawa Main Conjecture, applied to Hilbert modular forms and their tensor products.
Greenberg and Vatsal developed an approach to study the main conjecture for a large class of elliptic curves simultaneously. They showed that if a given pair of elliptic curves share the same residual Galois representation, then the main conjecture holds for one if and only if it does for the other. The first part of this thesis investigates whether the ideas of Greenberg and Vatsal work for elliptic curves twisted by a CM-Hecke character. The second part of this thesis then extends the method to treat non-ordinary classical modular forms (without any twist). The former permits one to study rational elliptic curves base-changed to an arbitrary number field, whilst the latter requires techniques crafted by Pollack and Kobayashi in the early 2000s.
Finally, the third part of this thesis concerns Euler systems and their applications to the arithmetic of motives arising from modular forms—these objects are indispensable tools which can be used to prove half of the Iwasawa main conjecture. However they often give rise to additional “junk” error terms, as well as causing the p-adic L-function to be stripped away of certain bad Euler factors. For modular forms and their tensor products we devise a method to dispose of the error terms and to replenish the missing factors, allowing one to genuinely obtain half of the main conjecture free from any discrepancies.

year: 2025

title: The influence of reef topography on storm-driven sand flux

authors: Bosserelle, Cyprien

abstract: Natural formations of rock and coral can support geologically controlled beaches, where the beach dynamics are significantly influenced by these structures. However, little is known about how alongshore variations in geological controls influence beach morphodynamics. Therefore, in this study we focus on the storm response of a beach (Yanchep in south Western Australia) that has strong alongshore variation in the level of geological control because of the heterogeneous calcarenite limestone reef. We used a modified version of XBeach to simulate the beach morphodynamics during a significant winter storm event. We find that the longshore variation in topography of the reef resulted in: (1) strong spatial difference in current distribution, including areas with strong currents jets; and (2) significant alongshore differences in sand flux, with larger fluxes in areas strongly geologically controlled by reefs. In particular, this resulted in enhanced beach erosion at the boundary of the reef where strong currents jet-exited the nearshore.

text: The influence of reef topography on storm-driven sand flux Natural formations of rock and coral can support geologically controlled beaches, where the beach dynamics are significantly influenced by these structures. However, little is known about how alongshore variations in geological controls influence beach morphodynamics. Therefore, in this study we focus on the storm response of a beach (Yanchep in south Western Australia) that has strong alongshore variation in the level of geological control because of the heterogeneous calcarenite limestone reef. We used a modified version of XBeach to simulate the beach morphodynamics during a significant winter storm event. We find that the longshore variation in topography of the reef resulted in: (1) strong spatial difference in current distribution, including areas with strong currents jets; and (2) significant alongshore differences in sand flux, with larger fluxes in areas strongly geologically controlled by reefs. In particular, this resulted in enhanced beach erosion at the boundary of the reef where strong currents jet-exited the nearshore.

year: 2021

title: Te hau mihi ata mātauranga Māori and science - Cross cultural dialogue

authors: Joseph, Robert

abstract: The Te Hau Mihi Ata Mātauranga Māori and Science research project (the research project) will explore ways to link mātauranga Māori and western science. The purpose of the research project is to open a new creative knowledge space to unleash innovative thinking and cross-cultural connection that is unique to New Zealand which can be a source of untapped economic, social, environmental and cultural potential. The historic marginalisation of alternative value and knowledge systems has prevented quality engagement and dialogue between these two knowledge systems in New Zealand. The research project will develop, inter alia, a new inter-cultural dialogue process between Māori educated in western-based science and experts in mātauranga Māori.

text: Te hau mihi ata mātauranga Māori and science - Cross cultural dialogue The Te Hau Mihi Ata Mātauranga Māori and Science research project (the research project) will explore ways to link mātauranga Māori and western science. The purpose of the research project is to open a new creative knowledge space to unleash innovative thinking and cross-cultural connection that is unique to New Zealand which can be a source of untapped economic, social, environmental and cultural potential. The historic marginalisation of alternative value and knowledge systems has prevented quality engagement and dialogue between these two knowledge systems in New Zealand. The research project will develop, inter alia, a new inter-cultural dialogue process between Māori educated in western-based science and experts in mātauranga Māori.

year: 2008-11