Meeting with Zagier after my talk at the MPI

William A. Stein^{∗}

10 January 2001

Introduction

I (William Stein) just spoke with Zagier for a while after giving my MPI talk. He has ideas about the parity conjecture, about the formulation of BSD, and about an analogue of the Hilbert class field in the context of visibility and abelian varieties.

NOTE: Many of the times when it says “I” below, this refers to Zagier. Other times, it refers to me! The intended audience of this short note is William Stein, and only William Stein, so this shouldn’t cause any confusion. If you aren’t William Stein, please keep this in mind.

1 Formulation of BSD

Maybe the full BSD conjecture is false! Or, rather, maybe a version of it in a closely related context, such as over a different number field or for motives is false. In parallel with many results from Iwasawa theory, one can make the following conjecture.

Conjecture 1.1 (BSD II). Let E be an elliptic curve over Q. Then

� (E) ≅ F × ∏_{all p}(Q_{p}/Z_{p})^{r_{p}},

where F is a finite group. Moreover, all of the r_{p} are equal to one another; let r_{0} be their common value. Then the analytic rank of E is equal to r_{0} + r_{alg}, and there is a formula like

L^{(r_{0}+r_{alg})}(E, 1) = \frac{Ω · Reg · ∏ c_{p} · #F}{#E(Q)_{tor}^{2}} .

One would have to do a literature search. I sort of have, and this conjecture seems consistent with everything that’s been proved towards BSD. Lang told me I should write a paper on this, but I don’t know if I will. I argued a lot with Lang about this. Coates sent me an email saying he thinks we now believe the full BSD conjecture should be true, but I [=Zagier!] think he fired it off in such a hurry that perhaps it is nonsense.

∗University of California, Berkeley, U.S.A., was@mail.math.berkeley.edu

2 The parity conjecture

I have an elementary approach to proving the parity conjecture. Away from a tedious computation at 2, I have basically proved the conjecture. Let E be an elliptic curve over Q. Then there are two numbers attached to E, the analytic rank r_{A} and the algebraic rank r_{E}. Neither of these numbers is computable! However, it is possible to compute both of them modulo 2! Let’s assume, for simplicity, that E is semistable. Then the analytic rank mod 2 is got from the sign of the functional equation, which is easy to determine from whether or not E has split reduction at each bad prime. On the other hand, assuming that � (E)[2] is a square, which would follow from finiteness of � (E) (which we don’t know!), we can compute the parity of r_{E} by computing the rank of the 2-Selmer group of E. For simplicity, assume that all of the 2-torsion of E is defined over Q. Then, using quadratic reciprocity and the standard tricks for computing the Selmer group we find that the Selmer group has the same mod-2 parity as what we get when we compute the sign of the functional equation. Viola!!! Zagier explained and re-explained the details of the previous sentence to me about five times, but I didn’t really understand because I know I can work out the details myself and want to save it as an exercise. However, the point is that his approach must work.

3 Hilbert class field for abelian varieties

My restriction-of-scalars construction is the analogue of trivializing the class group by adjoining roots of generators for an ideal. (??) This is trivial and does not lead one to the Hilbert class field. Historically, it was a hundred years from when people went from considering “ideal numbers” to proving the existence of the Hilbert class field. Is there an analogue of the Hilbert class field for abelian varieties? Is there a natural ambient abelian variety H and an embedding A ↪ H that behaves like the Hilbert class field of a number field? In particular, it should trivialize all of � (A)=class group of A.