Differences between revisions 1 and 2

Jared Weinstein (UCLA) and William Stein (Univ. of Washington): Heegner Points and Kolyvagin's Euler system

Description

The celebrated Gross-Zagier theorem implies that if E/Q is an elliptic curve of analytic rank one, then E(Q) contains a subgroup of finite index generated by a single Heegner point PK which is necessarily non-torsion. (Here K is an appropriate imaginary quadratic field.) But if E has higher analytic rank, meaning that presumably E has lots of nontorsion rational points, then PK is torsion. Nonetheless, there is a remarkable theory of Kolyvagin systems which plays the role of PK. This is a family of cohomology classes τn in H1(K,E[p]), indexed by suitable square-free integers n, which satisfy a highly restrictive compatibility relationship. If just one member of the family is nonzero, then there are strong consequences; for instance, the entire p-Selmer group (which contains E(K)/pE(K) as a subgroup) will be generated by the Kolyvagin classes taun for n a product of r−1 distinct primes, where r is the rank of the p-Selmer group for E/K.

Our group will be concerned with the computation of the classes τn. By the end of the workshop, I would like to see a table of mod 3 classes tauℓ1ℓ2 for the elliptic curve 5077A of rank 3, for different values of the quadratic field K and for various pairs of primes ℓ1, ℓ2.

References

B. H. Gross, Kolyvagin’s work on modular elliptic curves, L-functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 235–256.

B. Howard, The Heegner point Kolyvagin system, Compos. Math. 140 (2004), no. 6, 1439–1472.

V. A. Kolyvagin, Euler systems, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhauser Boston, Boston, MA, 1990, pp. 435–483.

W. Stein, Heegner Points on Rank Two Elliptic Curves. http://wstein.org/papers/kolyconj2/.

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+== Description ==

The celebrated Gross-Zagier theorem implies that if  $E / Q$  is
an elliptic curve of analytic rank one, then  $E (Q)$  contains a
subgroup of finite index generated by a single Heegner point  $P_{K}$ 
which is necessarily non-torsion.   (Here  $K$  is an appropriate
imaginary quadratic field.)  But if  $E$  has higher analytic rank,
meaning that presumably  $E$  has lots of nontorsion rational points,
then  $P_{K}$  is torsion.  Nonetheless, there is a remarkable theory of
Kolyvagin systems which plays the role of  $P_{K}$ .  This is a family of
cohomology classes  $τ_{n}$  in  $H^{1} (K, E [p])$ , indexed by suitable
square-free integers  $n$ , which satisfy a highly restrictive
compatibility relationship.   If just one member of the family is
nonzero, then there are strong consequences;  for instance, the entire
 $p$ -Selmer group (which contains  $E (K) / p E (K)$  as a subgroup) will be
generated by the Kolyvagin classes  $t a u_{n}$  for  $n$  a product of  $r - 1$ 
distinct primes, where  $r$  is the rank of the  $p$ -Selmer group for
 $E / K$ .

Our group will be concerned with the computation of the classes
 $τ_{n}$ .   By the end of the workshop, I would like to see a table of
mod 3 classes  $t a u_{ℓ_{1} ℓ_{2}}$  for the elliptic curve 5077A of
rank 3, for different values of the quadratic field  $K$  and for
various pairs of primes  $ℓ_{1}$ ,  $ℓ_{2}$ .

== References ==

B. H. Gross, Kolyvagin’s work on modular elliptic curves, L-functions
and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser.,
vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 235–256.

B. Howard, The Heegner point Kolyvagin system, Compos. Math. 140
(2004), no. 6, 1439–1472.

V. A. Kolyvagin, Euler systems, The Grothendieck Festschrift, Vol. II,
Progr. Math., vol. 87, Birkhauser Boston, Boston, MA, 1990, pp.
435–483.

W. Stein,  Heegner Points on Rank Two Elliptic Curves.
http://wstein.org/papers/kolyconj2/.

Diff for "days22/weinstein"

Jared Weinstein (UCLA) and William Stein (Univ. of Washington): Heegner Points and Kolyvagin's Euler system

Description

References