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- There is apparently a theory of algebraic cobordism.
Padmavathi Srinivasan, UGA NT Seminar
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Reference: Padmavathi Srinivasan, UGA NT Seminar.
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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).
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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
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Can compute heights as intersection numbers on a regular model.
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See Abel-Jacobi map
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Try to realize rational points as the zero locus of p-adic analytic functions.
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Recent work: quadratic Chabauty used to find all rational points on the infamously cursed modular curve \(X_S(13)\). See cursed curve.
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Looking for explanations through Arakelov theory instead of p-adic Hodge theory.
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See Mordell-Weil rank
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See Néron-Severi of the Jacobian.
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Get height functions where local heights \(h_p\) can be computed by iterated Coleman integrals.
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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
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Symmetric line bundles: \(\mathcal{L}\cong [1]^* \mathcal{L}\).
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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.
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General technique: when you have an integral model: take closures!
- Picking an integral model allows measuring sizes of sections.
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See valuations, used to define admissible metrics.
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Admissible metrics on \({\mathcal{O}}_X\) factor through the reduction graph
- See semistable model of a curve \(X_{/{ {\mathbf{Q}}_p }}\).
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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: \(x\mapsto \sum_{p \in {\operatorname{Places}}} v_p (s(x))\).
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Rigidified bundles: remember a point in the fiber.
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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 \(A \xrightarrow{[2]} A\).
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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.
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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.
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Any two metrics with the same curvature differ by \(\int \omega\) for some \(\omega\).