Beginnings of the BSD Summer School

Here is my plan, to induce myself to study as well as to provide updates about my vacation, I'll be typing up some notes on the notes that I take on the conferences I attend.

There are three lecturers, Vladimir Dokchitser, Christian Wuthrich, and Tim Dokchitser. Their talks are respectively about Birch-Swinnerton-Dyer and Parity conjectures, modular symbols and BSD over abelian fields, and L-functions and root numbers. Here are some comments on the first two days of notes that I've taken.

Vlad's first talk:

Reviewed the definition and group law for elliptic curves, and the Mordell-Weil Theorem that the group of an elliptic curve is finitely generated as well as abelian; call the rank of the free part of that group the arithmetic rank. Questions you might ask are: what ranks occur for curves over Q? Over number fields? It is conjectured that there are curves of unbounded rank, however it is known (conjectured?) that half of them are rank 0, and half are rank 1. It is also known that every elliptic curve over a number field has a quadratic twist of rank 0 or 1.

The L function of an elliptic curve is the product over all primes of the inverses of polynomials:

(1-a_p*p^-s+p*p^-2s) where a_p is the number of points on the curve modulo p; with some adjustments for primes of bad reduction (this works with the norms of primes over generic number fields). For all p, the local factor at s=1 is the norm of p divided by the number of points on E mod p. L(E/k,s) has an analytic continuation to C and satisfies a functional equation. This is known over Q due to Wiles, Taylor, et. al.

We define the analytic rank of E/k to be the order of vanishing of the L function at s=1. Then the Birch and Swinnerton-Dyer conjecture (as reformulated with Tate) states:

i) the arithmetic rank is equal to the analytic rank

ii) the terms of the Taylor series are given by explicit formulas (too long to include here)

The formula involves the regulator or E/K, the terrifying Tate-Shafarevich group, Tamigawa numbers, an integral of the invariant differential for each infinite place, the discriminant of K, and the size of the torsion subgroup of E(k)

A useful fact for bounding the rank of an elliptic curve: if the n-torsion subgroup of an elliptic curve is in K, then we can look at points which are 1/n parts of points on our curve, which must lie in a field with Galois group inside (C_n)^2 and satisfy further conditions. We get a map from our free points to this Galois group which is at worst two-to-one, and this allows us to use Galois theory to bound the rank of the curve.

Using this alone, one can prove that if E/Q has good reduction at 2 or 3, the torsion group of E(Q) cannot be bigger than order 12. To do this we use the Hasse-Weil bound to note that E cannot have more than 5 points mod 2 or 7 points mod 3, and that we have an injective map from 2-torsion points to E(F_3) which must have size 2^n less than 7, so it must be 1,2, or 4. Similarly we must have an injection from 3-torsion to E(F_2) which has size power of 3 less than 5, so it must be 1 or 3. This allows us groups up to order 12. If there is any other prime torsion injecting in, the prime must be no bigger than 5, but five can inject in. However, if it does, the size of the torsion group cannot be divisible by 3 or 2, or we would have 15<5, 10<7, which is fail. So groups of order 1,2,3,4,5,6,12 are the only possibilities.