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/\mathbf{Q} is an elliptic curve of analytic rank one, then E(\mathbf{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 \tau_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)/pE(K) as a subgroup) will be generated by the Kolyvagin classes \tau_n for n a product of r-1 distinct primes, where r is the rank of the p-Selmer group for E/K.

Project 1

Let E be an elliptic curve of rank 2, let p be a prime, let K/\mathbf{Q} be an imaginary quadratic field satisfying the Heegner hypothesis. Finally, let \ell be a Kolyvagin prime relative to the data E, K, p. The Kolyvagin class \tau_\ell lies in a modified Selmer group H^1_{\mathcal{F}(\ell)}(K,E[p]). (For definitions, see Howard's paper, Def. 1.2.2.)

(a) Assume that III_p(E/K)=0. Give an algorithm that finds the dimension of H^1_{\mathcal{F}(\ell)}(K,E[p]).

(b) If H^1_{\mathcal{F}(\ell)}(K,E[p]) is one-dimensional, then it must be a subspace of H^1_{\mathcal{F}}(K,E[p])=E(\mathbf{Q})\otimes\mathbf{Z}/p\mathbf{Z}. (Howard's paper, Lemma 1.5.3) Therefore \tau_\ell lives in a line inside of E(\mathbf{Q})\otimes\mathbf{Z}/p\mathbf{Z}. Which one is it?

(c) Compare the \tau_\ell from (b) (known only up to torsion) with those produced by William's function "kolyvagin_point_on_curve".

Our group will be concerned with the computation of the classes \tau_n. By the end of the workshop, I would like to see a table of mod 3 classes \tau_{\ell_1 \ell_2} for the elliptic curve 5077A of rank 3, for different values of the quadratic field K and for various pairs of primes \ell_1, \ell_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/.

W. Stein, Toward a generalization of Gross-Zagier. http://wstein.org/papers/stein-ggz/

Projects

• People: Jen Balakrishnan, Justin Walker, Robert Miller, Rebecca Bellovin, Daniel Disegni, Ian Whitehead, Donggeon Yhee, Khoa, Robert Bradshaw, Dario, Chen

Compute Kolyvagin classes mod p

• Subproject: fix trac #9302: http://trac.sagemath.org/sage_trac/ticket/9302

Using A to compute Kolyvagin classes en masse (theoretical)

Misc.

• Finding 1/2 * P: http://nt.sagenb.org/home/pub/9/

• Noam's algorithm for computing the trace: /noamtrace