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Robin de Jong
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In this thesis, we construct explicit formulas for canonical local height function on low genus curves. We do this based on an abstract definition of canonical local heights. In the case of elliptic curves over number fields there is a full ‘formulaire’ by Néron and Tate. We present a new proof of the formula for the canonical local height at a non-archimedean place, if the curve has multiplicative reduction at that place. We do this by evaluating the normalized tropical Riemann theta function on the Jacobian of the reduction graph.
In the case of a Jacobian of a genus 2 curve with semistable reduction, we can again evaluate the normalized tropical Riemann theta function, to obtain formulas for the Néron correction term, at non-archimedean places. This yields seven formulas, one for every semistable reduction type of the curve. These formulas are dependent on the tropical theta characteristic, which is generally hard to find. Under certain assumptions on the tropical theta characteristic, these formulas resemble formulas for canonical local heights on the Kummer surface. This leads us to explicitly calculate the intersection multiplicity of a point on the Jacobian, with the Theta divisor, by evaluating functions on the Kummer surface. This suggests that our assumptions may be correct, and should bring us closer to understanding the tropical theta characteristic ...
In the case of a Jacobian of a genus 2 curve with semistable reduction, we can again evaluate the normalized tropical Riemann theta function, to obtain formulas for the Néron correction term, at non-archimedean places. This yields seven formulas, one for every semistable reduction type of the curve. These formulas are dependent on the tropical theta characteristic, which is generally hard to find. Under certain assumptions on the tropical theta characteristic, these formulas resemble formulas for canonical local heights on the Kummer surface. This leads us to explicitly calculate the intersection multiplicity of a point on the Jacobian, with the Theta divisor, by evaluating functions on the Kummer surface. This suggests that our assumptions may be correct, and should bring us closer to understanding the tropical theta characteristic ...
In this thesis, we construct explicit formulas for canonical local height function on low genus curves. We do this based on an abstract definition of canonical local heights. In the case of elliptic curves over number fields there is a full ‘formulaire’ by Néron and Tate. We present a new proof of the formula for the canonical local height at a non-archimedean place, if the curve has multiplicative reduction at that place. We do this by evaluating the normalized tropical Riemann theta function on the Jacobian of the reduction graph.
In the case of a Jacobian of a genus 2 curve with semistable reduction, we can again evaluate the normalized tropical Riemann theta function, to obtain formulas for the Néron correction term, at non-archimedean places. This yields seven formulas, one for every semistable reduction type of the curve. These formulas are dependent on the tropical theta characteristic, which is generally hard to find. Under certain assumptions on the tropical theta characteristic, these formulas resemble formulas for canonical local heights on the Kummer surface. This leads us to explicitly calculate the intersection multiplicity of a point on the Jacobian, with the Theta divisor, by evaluating functions on the Kummer surface. This suggests that our assumptions may be correct, and should bring us closer to understanding the tropical theta characteristic
In the case of a Jacobian of a genus 2 curve with semistable reduction, we can again evaluate the normalized tropical Riemann theta function, to obtain formulas for the Néron correction term, at non-archimedean places. This yields seven formulas, one for every semistable reduction type of the curve. These formulas are dependent on the tropical theta characteristic, which is generally hard to find. Under certain assumptions on the tropical theta characteristic, these formulas resemble formulas for canonical local heights on the Kummer surface. This leads us to explicitly calculate the intersection multiplicity of a point on the Jacobian, with the Theta divisor, by evaluating functions on the Kummer surface. This suggests that our assumptions may be correct, and should bring us closer to understanding the tropical theta characteristic
This thesis is about Van Kampen's theorem and fundamental groupoids. Van Kampen's Theorem is a classical result in algebraic topology, which proposes a way of calculating the fundamental group of a topological spaces using the fundamental groups of certain subspaces. In this thesis we will construct the fundamental group, which intuitively counts "holes" in a topological space and is mainly used to distinguish topological spaces. Van Kampen's theorem is then proven for a arbitrary large cover using covering spaces. In the proof we glue topological spaces together and an example of this gluing is given for the Klein bottle.
Van Kampen's theorem does not work for every topological space, so we take a look at a generalization of the fundamental group, the so-called fundamental groupoid. Van Kampen's theorem can be upgraded to calculate fundamental groupoid and this theorem is proven in this thesis as well. ...
Van Kampen's theorem does not work for every topological space, so we take a look at a generalization of the fundamental group, the so-called fundamental groupoid. Van Kampen's theorem can be upgraded to calculate fundamental groupoid and this theorem is proven in this thesis as well. ...
This thesis is about Van Kampen's theorem and fundamental groupoids. Van Kampen's Theorem is a classical result in algebraic topology, which proposes a way of calculating the fundamental group of a topological spaces using the fundamental groups of certain subspaces. In this thesis we will construct the fundamental group, which intuitively counts "holes" in a topological space and is mainly used to distinguish topological spaces. Van Kampen's theorem is then proven for a arbitrary large cover using covering spaces. In the proof we glue topological spaces together and an example of this gluing is given for the Klein bottle.
Van Kampen's theorem does not work for every topological space, so we take a look at a generalization of the fundamental group, the so-called fundamental groupoid. Van Kampen's theorem can be upgraded to calculate fundamental groupoid and this theorem is proven in this thesis as well.
Van Kampen's theorem does not work for every topological space, so we take a look at a generalization of the fundamental group, the so-called fundamental groupoid. Van Kampen's theorem can be upgraded to calculate fundamental groupoid and this theorem is proven in this thesis as well.
This thesis is about homological algebra and singular (co)homology.
In the first chapter the notions of complexes of abelian groups, (co)homology of these complexes and injective resolutions will be introduced. Then Ext-groups will be defined and various properties dervied. A particularly interesting group, Ext(Q,Z), will be calculated which involves the p-adic integers. Lastly we will prove the universal coefficient theorem for complexes of free abelian groups.
In the second chapter we will use the tools provided by the previous chapter to calculate the singular homology groups of topological spaces. First we will explicitely describe the zero'th and first singular homology groups for any topological space. For the spheres S^n and real projective n space P^n(R) we will calculate all singular homology and cohomology groups. For this we will use the universal coefficient and properties about Ext-groups which have been proven in chapter 1. We will also prove and use the long exact sequence of Mayer-Vietoris. This theorem proposes a way to calculate the singular homology groups of a space by using the singular homology groups of two subspaces. ...
In the first chapter the notions of complexes of abelian groups, (co)homology of these complexes and injective resolutions will be introduced. Then Ext-groups will be defined and various properties dervied. A particularly interesting group, Ext(Q,Z), will be calculated which involves the p-adic integers. Lastly we will prove the universal coefficient theorem for complexes of free abelian groups.
In the second chapter we will use the tools provided by the previous chapter to calculate the singular homology groups of topological spaces. First we will explicitely describe the zero'th and first singular homology groups for any topological space. For the spheres S^n and real projective n space P^n(R) we will calculate all singular homology and cohomology groups. For this we will use the universal coefficient and properties about Ext-groups which have been proven in chapter 1. We will also prove and use the long exact sequence of Mayer-Vietoris. This theorem proposes a way to calculate the singular homology groups of a space by using the singular homology groups of two subspaces. ...
This thesis is about homological algebra and singular (co)homology.
In the first chapter the notions of complexes of abelian groups, (co)homology of these complexes and injective resolutions will be introduced. Then Ext-groups will be defined and various properties dervied. A particularly interesting group, Ext(Q,Z), will be calculated which involves the p-adic integers. Lastly we will prove the universal coefficient theorem for complexes of free abelian groups.
In the second chapter we will use the tools provided by the previous chapter to calculate the singular homology groups of topological spaces. First we will explicitely describe the zero'th and first singular homology groups for any topological space. For the spheres S^n and real projective n space P^n(R) we will calculate all singular homology and cohomology groups. For this we will use the universal coefficient and properties about Ext-groups which have been proven in chapter 1. We will also prove and use the long exact sequence of Mayer-Vietoris. This theorem proposes a way to calculate the singular homology groups of a space by using the singular homology groups of two subspaces.
In the first chapter the notions of complexes of abelian groups, (co)homology of these complexes and injective resolutions will be introduced. Then Ext-groups will be defined and various properties dervied. A particularly interesting group, Ext(Q,Z), will be calculated which involves the p-adic integers. Lastly we will prove the universal coefficient theorem for complexes of free abelian groups.
In the second chapter we will use the tools provided by the previous chapter to calculate the singular homology groups of topological spaces. First we will explicitely describe the zero'th and first singular homology groups for any topological space. For the spheres S^n and real projective n space P^n(R) we will calculate all singular homology and cohomology groups. For this we will use the universal coefficient and properties about Ext-groups which have been proven in chapter 1. We will also prove and use the long exact sequence of Mayer-Vietoris. This theorem proposes a way to calculate the singular homology groups of a space by using the singular homology groups of two subspaces.