We consider the problem of explicitly computing Beilinson–Bloch heights of homologically trivial cycles on varieties defined over number fields. Recent results have established a congruence, up to... Show moreWe consider the problem of explicitly computing Beilinson–Bloch heights of homologically trivial cycles on varieties defined over number fields. Recent results have established a congruence, up to the rational span of logarithms of primes, between the height of certain limit mixed Hodge structures and certain Beilinson–Bloch heights obtained from odd-dimensional hypersurfaces with a node. This congruence suggests a new method to compute Beilinson–Bloch heights. Here we explain how to compute the relevant limit mixed Hodge structures in practice, then apply our computational method to a nodal quartic curve and a nodal cubic threefold. In both cases we explain the nature of the primes occurring in the congruence. Show less
This thesis deals with properties of Jacobians of genus two curves that cover elliptic curves. If E is an elliptic curve and C is a curve of genus two that covers it n-to-1 so that the covering... Show moreThis thesis deals with properties of Jacobians of genus two curves that cover elliptic curves. If E is an elliptic curve and C is a curve of genus two that covers it n-to-1 so that the covering does not factor through an isogeny, then C also covers another elliptic curve n-to-1 in such a way and the Jacobian of C is isogenous to the product of the two elliptic curves. The Jacobian is said to be (n,n)-split and the elliptic curves are said to be glued along their n-torsion. The first chapter deals with the geometric aspects of this setup. We describe two approaches to constructing (n,n)-split Jacobians and we investigate which curves can appear in the setup. The second chapter deals with the arithmetic aspects, focusing on height functions and the Lang-Silverman conjecture in particular. We show that this conjecture holds for families of (n,n)-split Jacobians if and only if it holds for the corresponding families of elliptic curves that can be glued along their n-torsion. Show less
With focus on study of binary forms and their discriminants and resultants over function fields, we developed an analogue of the geometry of numbers and generalized Mason's ABC theorem. Then we... Show moreWith focus on study of binary forms and their discriminants and resultants over function fields, we developed an analogue of the geometry of numbers and generalized Mason's ABC theorem. Then we proved a conjecture, which is possibly first formulated by Evertse, over the rational function field and effectively bounded S-distance of algebraic functions, and improved results of root separation problem. Show less
For South Asian adults lowered BMI cut-offs for overweight and obesity are recommended as these are more in agreement with their body composition (low muscle mass, high fat mass) and the... Show moreFor South Asian adults lowered BMI cut-offs for overweight and obesity are recommended as these are more in agreement with their body composition (low muscle mass, high fat mass) and the considerably higher risk of cardiovascuar and metabolic disease. For South Asian children living in a Western country lowered BMI norms have not been established, nor other ethnic specific growth standards. The aims of this thesis are 1. to gain more insight into the normal physical growth of South Asian children 2. to develop South Asian specific height-for-age (0-21 years) and BMI-for-age (2-18 years) references, including BMI cut-offs. The studies in this thesis show that South Asian children are generally lighter and shorter than other ethnic groups. The mean birth weight is at 3012 grams around 450 grams lower than of Dutch babies and has remained unchanged for the past 35 years. In children 3-15 years thinness is highly prevalent and overweight prevalence is relatively low. Application of the developed height-for-age charts and lowered BMI cut-offs will expectedly lead to a better assessment of height and nutritional status (thinness, overweight) in this group. Consequently, thinness rates will be lower while overweight will be detected earlier, leading to earlier preventive interventions. Show less
