---
date: 2021-05-05
tags: [ web/quick-notes ]
---


# 2021-05-05

## 01:42

- There is apparently a theory of [algebraic cobordism](algebraic%20cobordism).

## Padmavathi Srinivasan, UGA NT Seminar

^22ba3a

> Reference: Padmavathi Srinivasan, UGA NT Seminar. 

- If $r$ is the rank and $g$ is the [[genus]] for $X$ a nice [[curve]] over $\QQ$ with [[good reduction]] at $p$, 
  - Want to find [[rational points]].
  - If $r<g$ (?) then [Chabauty-Coleman](Chabauty-Coleman) applies.
  - This talk: $r=g$, allows finding a basis for $\QQ_p$ valued functions on $J(\QQ)$.\ (see [Jacobian](Unsorted/Jacobian.md)).
- See p-adic [height pairing](height%20pairing).
  - Height pairing has to do with rational points on [Jacobian](Jacobian.md).
  - Trying to compute [canonical heights](canonical%20heights).
  - See [global height](global%20height) vs [local height](local%20height)
  - See [Weil height machine](Weil%20height%20machine)
- Can compute heights as intersection numbers on a [[regular model]].
- See [Abel-Jacobi map](Abel-Jacobi%20map)
- 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](modular%20curve.md) $X_S(13)$. See [[cursed curve]].
- Looking for explanations through [Arakelov theory](Arakelov%20theory) instead of [p-adic Hodge theory](p-adic%20Hodge%20theory.md).
- See [Mordell-Weil rank](Mordell-Weil%20rank.md)
- See [Néron-Severi](Néron-Severi) of the [Jacobian](Jacobian.md).
- Get height functions where local heights $h_p$ can be computed by iterated [Coleman integrals](Coleman%20integrals).
- There is a canonical height machine for abelian varieties.
  - Need a curvature form in \( \Omega^1(X) \tensor H^1_\dR(X) \).
  - Get one height for each choice of [idele class character](idele%20class%20character)
- *Symmetric line bundles*: \( \mathcal{L}\cong [1]^* \mathcal{L} \).
- Zhang defines a [metric on a line bundle](metric%20on%20a%20line%20bundle), which gives a way to measure the size of elements in each fiber.
  - It's a locally bounded continuous function 
    $$\nu: \mathcal{L}\units \da  \Tot( \mathcal{L} ) \smz \to \RR$$ with $\nu( \alpha \vector v) = v_p(\alpha) + \nu(\vector v)$ for \( \alpha\in \bar{\QQ_p}\units, \vector v \in \mathcal{L}\units   \).
  - Here continuous is in the (locally) analytic topology, since we're over $\QQ_p$.
  - 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](valuations.md), used to define **admissible metrics**.
- Admissible metrics on $\OO_X$ factor through the [reduction graph](reduction%20graph)
  - See [semistable model](semistable%20model) of a curve $X_{/\QQpadic}$.

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

- 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](Banach%20space)
  - Apply [[Banach fixed-point theorem]] to the self-map $A \mapsvia{[2]} A$.

- Curvature form $\Curv(\mathcal{L}_v)$ is sent to the first [Chern class](Chern%20class.md) $c_1(\mathcal{L}_v )$ under cup product in de Rham cohomology.

- To get a canonical metric for all line bundles, it suffices to canonically metrize the [Poincare bundle](Poincare%20bundle.md).
  Every line bundle on $A$ is a pullback of it.

- Any two metrics with the same curvature differ by $\int \omega$ for some $\omega$.