---
date: 2021-04-15
tags: [ web/quick-notes ]
---


# 2021-04-15

## Beilinson-Bloch Conjecture 

> Reference: Chao Li, "Beilinson-Bloch conjecture for unitary Shimura varieties". Priinceton/IAS NT Seminar

- What is the [Beilinson Bloch conjecture](Beilinson%20Bloch%20conjecture)?

- Beilinson-Bloch conjecture: generalizes the [Birch and Swinnerton-Dyer conjecture](Birch%20and%20Swinnerton-Dyer%20conjecture.md).
- What are higher [Chow ring](Chow%20ring.md) ?
  What do they generalize?
  - Higher Chow groups: generalize the [Mordell-Weil group](Mordell-Weil%20group) for [elliptic curve](elliptic%20curve.md)

![](attachments/image_2021-04-15-16-38-49.png)

![](attachments/image_2021-04-15-16-47-57.png)

- What is an [adele](adele)? What is an [adele](adele) point?

- I should also review what a [place](place)really is.
  Definitely what it means to be an [Unsorted/Valuations](Unsorted/Valuations.md).
  Also double-check the $v\divides \infty$ notation.

- What is an [automorphic representation](automorphic%20representation.md)?

- See [Gross-Zagier](Gross-Zagier) formula.

- What is a [modular curve](modular%20curve.md)? 
	
- What is a [Heegner divisor](Heegner%20divisor) for some [imaginary quadratic field](imaginary%20quadratic%20field) over $\QQ$ and why can one use the theory of [complex multiplication](complex%20multiplication) to get it defined over other fields?

- Gotta learn [modular form](modular%20form.md).
  They can take values in the complexification of a [Mordell-Weil group](Mordell-Weil%20group)?
  Also need to know something about [Hecke operator](Hecke%20operator).

- What is a [Shimura variety](Shimura%20variety.md)?

- What is a theta series?
  Something here called an *arithmetic theta lift*, where some pairing form generalizes Gross-Zagier (?).
  See Beilinson-Bloch height maybe?

- I should read a lot more about [Chow groups](Chow%20ring.md).

- What is [Betti cohomology](Betti%20cohomology)?

- Why is proving that something is [modular form](modular%20form.md) a *big deal*?

- Look for the Kudla Program in arithmetic geometry, and [Kudla-Rapoport conjecture](Kudla-Rapoport%20conjecture).

- Comment by Peter Sarnak: BSD was first checked numerically for CM elliptic curves! 

![](attachments/image_2021-04-15-17-20-37.png)

- What is the characteristic function of a lattice?
  What is a self-dual [lattice](lattice.md)?

![](attachments/image_2021-04-15-17-21-23.png)

- What is a Siegel [Eisenstein series](Eisenstein%20series.md)?
  Or even just an Eisenstein series.

- See [Néron-Tate height](Néron-Tate%20height) pairing?
  Seems like these BB heights can only really be computed locally, then you have to sum over places.

- What are the [Standard conjectures](Standard%20conjectures)?
  
- Main formula and big theorem:

  ![](attachments/image_2021-04-15-17-35-06.png)

  Seems that we know a lot about the LHS, the right-hand side is new.
  We don't know nondegeneracy of the RHS, for example, e.g. the pairing vanishing implying the cycle is zero.

  - Proof technique: "doubling".

- See [Tate conjecture](Tate%20conjecture.md).

- Comment from Peter Sarnak: we know very little about where $L$ functions vanish, except for $1/2$.

- Need to do [Resolution of singularities](Resolution%20of%20singularities.md) when you don't have a "regular" (integral?) model.

## 20:13

> Paper recommended by Juliette Bruce:
  <https://arxiv.org/pdf/2003.02494.pdf>

  - Jonathan Love! 
  Shows some cool consequences of the [Beilinson Bloch conjecture](Beilinson%20Bloch%20conjecture), primarily a 2-parameter family of [elliptic curve](elliptic%20curve.md)  where the image $\CH^1(E_1)_0 \tensor \CH^1(E_2)_0 \to \CH^2(E_1 \cross E_2)$ is finite.
  BB predicts this is always finite when defined over $k$ a [number field](number%20field.md).

- I should remind myself what [local fields](local%20fields) and [global fields](global%20field.md) are.