• There is apparently a theory of algebraic cobordism.

Padmavathi Srinivasan, UGA NT Seminar


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).
  • 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 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 \(\omega\).