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 ${\mathbb{Q}}$ 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 ${\mathbb{Q}}_p$ valued functions on $J({\mathbb{Q}})$.

• See p-adic 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 regular models.

• 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 $X_S(13)$.

• 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 $\nu: \mathcal{L}^{\times}\coloneqq{ \operatorname{Tot} }( \mathcal{L} ) \setminus\left\{{0}\right\}\to {\mathbb{R}}$ with $\nu( \alpha \mathbf{v}) = v_p(\alpha) + \nu(\mathbf{v})$ for $\alpha\in \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{Q}}_p\mkern-1.5mu}\mkern 1.5mu^{\times}, \mathbf{v} \in \mathcal{L}^{\times}$.
  • Here continuous is in the locally analytic topology, since we’re over ${\mathbb{Q}}_p$.
  • Can do all of the usual stuff carrying the additional data of the metric: tensor powers, pullbacks, etc.

• When you have an integral model: take closures!

  • Picking integral models 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 $X_{/\QQp}$.

• Adelic metric: a collection of metrics for almost every place.

  • Associated height function: for $x\in X({\mathbb{Q}})$, 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 $\omega$.