- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- BIRS Workshop Lecture Videos /
- Isogeny classes of rational squares of CM elliptic...
Open Collections
BIRS Workshop Lecture Videos
BIRS Workshop Lecture Videos
Isogeny classes of rational squares of CM elliptic curves Fite, Francesc
Description
Let $A$ be an abelian surface defined over $\Q$. It is well known that the $\Q$-algebra $End(A) \otimes \Q$ is either $\Q$, $\Q \times \Q$, $M_2(\Q)$, a real quadratic field $E$, the product of $\Q$ and a quadratic imaginary field of class number 1, a quartic CM field $L$, a definite division quaternion algebra $B$, or $M_2(K)$, where $K$ is a quadratic imaginary field. It is conjectured that there are only finitely many possibilities for $End(A) \otimes \Q$. While almost nothing is known about the sets of possibilities for $E$ and $B$, Murabashy and Umegaki have proved that there are precisely 19 possibilities for the quartic CM field $L$. In this talk I will report on a joint work with Xevi Guitart in which we show that there are at most 52 possibilities for the quadratic imaginary field $K$. An equivalent way to formulate the result is by saying that there are at most 52 $\bar\Q$-isogeny classes of abelian surfaces defined over $\Q$ that are isogenous to the square of an elliptic curve with CM.
Item Metadata
Title |
Isogeny classes of rational squares of CM elliptic curves
|
Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
|
Date Issued |
2017-05-31T10:59
|
Description |
Let $A$ be an abelian surface defined over $\Q$. It is well known that the $\Q$-algebra $End(A) \otimes \Q$ is either $\Q$, $\Q \times \Q$, $M_2(\Q)$, a real quadratic field $E$, the product of $\Q$ and a quadratic imaginary field of class number 1, a quartic CM field $L$, a definite division quaternion algebra $B$, or $M_2(K)$, where $K$ is a quadratic imaginary field. It is conjectured that there are only finitely many possibilities for $End(A) \otimes \Q$. While almost nothing is known about the sets of possibilities for $E$ and $B$, Murabashy and Umegaki have proved that there are precisely 19 possibilities for the quartic CM field $L$. In this talk I will report on a joint work with Xevi Guitart in which we show that there are at most 52 possibilities for the quadratic imaginary field $K$. An equivalent way to formulate the result is by saying that there are at most 52 $\bar\Q$-isogeny classes of abelian surfaces defined over $\Q$ that are isogenous to the square of an elliptic curve with CM.
|
Extent |
34 minutes
|
Subject | |
Type | |
File Format |
video/mp4
|
Language |
eng
|
Notes |
Author affiliation: Essen
|
Series | |
Date Available |
2017-11-27
|
Provider |
Vancouver : University of British Columbia Library
|
Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
|
DOI |
10.14288/1.0360753
|
URI | |
Affiliation | |
Peer Review Status |
Unreviewed
|
Scholarly Level |
Postdoctoral
|
Rights URI | |
Aggregated Source Repository |
DSpace
|
Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International