abelian variety

^7cc3c7

- Tags - #AG - Refs: - Notes from Haiyang - Learning/Abelian Varieties ac30f83986fe4dbc9a2c2f14b036550a/EllipticCurve_notes1.pdf - Learning/Abelian Varieties ac30f83986fe4dbc9a2c2f14b036550a/EllipticCurve_notes2.pdf - Learning/Abelian Varieties ac30f83986fe4dbc9a2c2f14b036550a/EllipticCurve_notes3.pdf - Notion site - https://www.jmilne.org/math/CourseNotes/AV.pdf - Links: - polarization - abelian-by-finite

See Satake compactification constructed by Faltings.

attachments/Pasted%20image%2020220502190201.png

\({\mathcal{A}_g}\) carries the Hodge bundle:

attachments/Pasted%20image%2020220501132916.png

See relative Jacobian. attachments/Pasted%20image%2020220502190103.png

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unlikely intersection

moduli stack of abelian varieties
semiabelian variety

Idea: an extension of an abelian variety by a torus.
polarization

- Tags: - #todo/untagged - Refs: - #todo/add-references - Links: - abelian variety
elliptic curve

abelian variety
Tate-Shafarevich group

For \(A\) an abelian variety or a group scheme define over a field \(k\), this is the group of 1-cocycles in \(H^1(G_K; X)\) which become boundaries at every place: \begin{align*}\sha(X_{/ {k}} ) = \cap_{v\in \mathrm{Pl}\qty{K} } \ker \qty{ H^1(G_k; X) \to H^1(G_{{ k_{\widehat{v}} }}; X)}\end{align*} Measures the extent to which the Hasse principle holds for equations with coefficients in \(k\).
Tate module

abelian variety
Neron-Tate height

A quadratic form on the Mordell-Weil group of rational points on an elliptic curve, or more generally an abelian variety.
Abel-Jacobi map

abelian variety