01:42
 There is apparently a theory of algebraic cobordism.
Padmavathi Srinivasan, UGA NT Seminar
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Reference: Padmavathi Srinivasan, UGA NT Seminar.

If \(r\) is the rank and \(g\) is the genus for \(X\) a nice curve over \({\mathbf{Q}}\) with good reduction at \(p\),
 Want to find rational points.
 If \(r<g\) (?) then ChabautyColeman applies.
 This talk: \(r=g\), allows finding a basis for \({\mathbf{Q}}_p\) valued functions on \(J({\mathbf{Q}})\). (see Jacobian).

See padic height pairing.
 Height pairing has to do with rational points on Jacobian.
 Trying to compute canonical heights.
 See global height vs local height
 See Weil height machine

Can compute heights as intersection numbers on a regular model.

See AbelJacobi map

Try to realize rational points as the zero locus of padic analytic functions.

Recent work: quadratic Chabauty used to find all rational points on the infamously cursed modular curve \(X_S(13)\). See cursed curve.

Looking for explanations through Arakelov theory instead of padic Hodge theory.

See MordellWeil rank

See NéronSeveri of the Jacobian.

Get height functions where local heights \(h_p\) can be computed by iterated Coleman integrals.

There is a canonical height machine for abelian varieties.
 Need a curvature form in \(\Omega^1(X) \otimes H^1_\mathrm{dR}(X)\).
 Get one height for each choice of idele class character

Symmetric line bundles: \(\mathcal{L}\cong [1]^* \mathcal{L}\).

Zhang defines a metric on a line bundle, which gives a way to measure the size of elements in each fiber.
 It’s a locally bounded continuous function \begin{align*}\nu: \mathcal{L}^{\times}\coloneqq{ \operatorname{Tot} }( \mathcal{L} ) \setminus\left\{{0}\right\}\to {\mathbf{R}}\end{align*} with \(\nu( \alpha \mathbf{v}) = v_p(\alpha) + \nu(\mathbf{v})\) for \(\alpha\in \mkern 1.5mu\overline{\mkern1.5mu{\mathbf{Q}}_p\mkern1.5mu}\mkern 1.5mu^{\times}, \mathbf{v} \in \mathcal{L}^{\times}\).
 Here continuous is in the (locally) analytic topology, since we’re over \({\mathbf{Q}}_p\).
 Can do all of the usual stuff carrying the additional data of the metric: tensor powers, pullbacks, etc.

General technique: when you have an integral model: take closures!
 Picking an integral model allows measuring sizes of sections.

See valuations, used to define admissible metrics.

Admissible metrics on \({\mathcal{O}}_X\) factor through the reduction graph
 See semistable model of a curve \(X_{/{ {\mathbf{Q}}_p }}\).

Adelic metric: a collection of metrics for almost every place.
 Associated height function: for \(x\in X({\mathbf{Q}})\), pick a section not vanishing at \(x\) and sum all contributions: \(x\mapsto \sum_{p \in {\operatorname{Places}}} v_p (s(x))\).

Rigidified bundles: remember a point in the fiber.

For each place, there is a canonical metric which makes certain isomorphisms into isometries in a Banach space
 Apply Banach fixedpoint theorem to the selfmap \(A \xrightarrow{[2]} A\).

Curvature form \(\mathop{\mathrm{Curv}}(\mathcal{L}_v)\) is sent to the first Chern class \(c_1(\mathcal{L}_v )\) under cup product in de Rham cohomology.

To get a canonical metric for all line bundles, it suffices to canonically metrize the Poincare bundle. Every line bundle on \(A\) is a pullback of it.

Any two metrics with the same curvature differ by \(\int \omega\) for some \(\omega\).