|
Size: 113
Comment:
|
Size: 2682
Comment:
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 2: | Line 2: |
== 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 $E/K$. 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]] |
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.
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