BeilinsonBloch Conjecture
Reference: Chao Li, “BeilinsonBloch conjecture for unitary Shimura varieties”. Priinceton/IAS NT Seminar

What is the Beilinson Bloch conjecture?

BeilinsonBloch conjecture: generalizes the Birch and SwinnertonDyer conjecture.

What are higher Chow ring ? What do they generalize?
 Higher Chow groups: generalize the MordellWeil group for elliptic curve

What is an adele? What is an adele point?

I should also review what a placereally is. Definitely what it means to be an Unsorted/Valuations. Also doublecheck the \(v\divides \infty\) notation.

What is an automorphic representation?

See GrossZagier formula.

What is a modular curve?

What is a Heegner divisor for some imaginary quadratic field over \({\mathbf{Q}}\) and why can one use the theory of complex multiplication to get it defined over other fields?

Gotta learn modular form. They can take values in the complexification of a MordellWeil group? Also need to know something about Hecke operator.

What is a Shimura variety?

What is a theta series? Something here called an arithmetic theta lift, where some pairing form generalizes GrossZagier (?). See BeilinsonBloch height maybe?

I should read a lot more about Chow groups.

What is Betti cohomology?

Why is proving that something is modular form a big deal?

Look for the Kudla Program in arithmetic geometry, and KudlaRapoport conjecture.

Comment by Peter Sarnak: BSD was first checked numerically for CM elliptic curves!
 What is the characteristic function of a lattice? What is a selfdual lattice?

What is a Siegel Eisenstein series? Or even just an Eisenstein series.

See NéronTate height pairing? Seems like these BB heights can only really be computed locally, then you have to sum over places.

What are the Standard conjectures?

Main formula and big theorem:
Seems that we know a lot about the LHS, the righthand 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.

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

Need to do Resolution of singularities 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, primarily a 2parameter family of elliptic curve where the image \({\operatorname{CH}}^1(E_1)_0 \otimes{\operatorname{CH}}^1(E_2)_0 \to {\operatorname{CH}}^2(E_1 \times E_2)\) is finite. BB predicts this is always finite when defined over \(k\) a number field.

I should remind myself what local fields and global fields are.