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-Bloch conjecture: generalizes the Birch and Swinnerton-Dyer conjecture.

• What are higher Chow ring ? What do they generalize?

  • Higher Chow groups: generalize the Mordell-Weil 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 Archimedean place. Also double-check the \(v\divides \infty\) notation.

• What is an automorphic representation?

• See Gross-Zagier formula.

• What is a modular curve?

• What is a Heegner divisor for some imaginary quadratic field over \({\mathbb{Q}}\) 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 Mordell-Weil 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 Gross-Zagier (?). See Beilinson-Bloch height maybe?

• I should read a lot more about [[Chow ring|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 Kudla-Rapoport 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 self-dual lattice?

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

• See Néron-Tate 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:

  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.

• 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 2-parameter 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 field|global fields]] are.