# 2021-05-05

## 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 Chabauty-Coleman applies.
• This talk: $$r=g$$, allows finding a basis for $${\mathbf{Q}}_p$$ valued functions on $$J({\mathbf{Q}})$$. (see Jacobian).

• 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 Abel-Jacobi map

• Try to realize rational points as the zero locus of p-adic 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 p-adic Hodge theory.

• See Mordell-Weil rank

• See Néron-Severi 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 $$\alpha\in \mkern 1.5mu\overline{\mkern-1.5mu{\mathbf{Q}}_p\mkern-1.5mu}\mkern 1.5mu^{\times}, \mathbf{v} \in \mathcal{L}^{\times}$$.
• 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 fixed-point theorem to the self-map $$A \xrightarrow{[2]} A$$.
• 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 $$\omega$$.