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Intersections of Humbert surfaces and binary quadratic forms Kani, Ernst
Description
Humbert surfaces are certain surfaces embedded in the moduli space $A_2$ of principally polarized abelian surfaces. In this talk I will explain the connection between the components of the intersection of two Humbert surfaces and classes of certain binary quadratic forms. More precisely, for each positive quadratic form $q$ in $r$ variables one can associate a closed subvariety $H(q)$ of $A_2$ (which depends only on the equivalence class of the form). If $r = 1$, then we recover the Humbert surfaces. For $r = 2$ we get curves which can be used to describe the intersection of two Humbert surfaces. (Using the reduction theory of binary quadratic forms, this can be done quite explicitly.) If q is a primitive binary quadratic form, then $H(q)$ is irreducible, but in the general case $H(q)$ is a union of the images of modular curves (modular correspondences) lying on $X(N) x X(N)$ (or on Hilbert modular surfaces). By studying conjugacy classes of matrices mod $N$, the irreducible components of $H(q)$ can be identified. Thus, one gets an explicit description of all irreducible components of the intersection of two Humbert surfaces.
Item Metadata
Title |
Intersections of Humbert surfaces and binary quadratic forms
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2017-05-30T11:01
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Description |
Humbert surfaces are certain surfaces embedded in the moduli space $A_2$ of principally polarized abelian surfaces. In this talk I will explain the connection between the components of the intersection of two Humbert surfaces and classes of certain binary quadratic forms. More precisely, for each positive quadratic form $q$ in $r$ variables one can associate a closed subvariety $H(q)$ of $A_2$ (which depends only on the equivalence class of the form). If $r = 1$, then we recover the Humbert surfaces. For $r = 2$ we get curves which can be used to describe the intersection of two Humbert surfaces. (Using the reduction theory of binary quadratic forms, this can be done quite explicitly.) If q is a primitive binary quadratic form, then $H(q)$ is irreducible, but in the general case $H(q)$ is a union of the images of modular curves (modular correspondences) lying on $X(N) x X(N)$ (or on Hilbert modular surfaces). By studying conjugacy classes of matrices mod $N$, the irreducible components of $H(q)$ can be identified. Thus, one gets an explicit description of all irreducible components of the intersection of two Humbert surfaces.
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Extent |
50 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Queen's University
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Series | |
Date Available |
2017-11-27
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0360733
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International