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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 τn 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 τℓ1ℓ2 for the elliptic curve 5077A of rank 3, for different values of the quadratic field K and for various pairs of primes ℓ1, ℓ2.

Project 1

Let E be an elliptic curve of rank 2, let p be a prime, let K/Q be an imaginary quadratic field satisfying the Heegner hypothesis. Finally, let ℓ be a Kolyvagin prime relative to the data E, K, p. The Kolyvagin class τℓ lies in a modified Selmer group H1F(ℓ)(K,E[p]). (For definitions, see Howard's paper, Def. 1.2.2.)

(a) Assume that IIIp(E/K)=0. Give an algorithm that finds the dimension of H1F(ℓ)(K,E[p]).

(b) If H1F(ℓ)(K,E[p]) is one-dimensional, then it must be a subspace of HF1(K,E[p])=E(Q)⊗Z/pZ. (Howard's paper, Lemma 1.5.3) Therefore τℓ lives in a line inside of E(Q)⊗Z/pZ. Which one is it?

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

UPDATE: (a)-(b) has been done for the elliptic curve 389a, for discriminants up to and excluding -67, and for ℓ up to 3000. Here is the Sage code: http://standalone.sagenb.org/home/pub/9/. Part (c) has been partially checked.

Project 2

Suppose the Heegner point y1∈E(K[1]) has been computed, so that its coordinates are known as algebraic numbers. Compute the trace yK of y1.

Noam's algorithm for computing the trace: /noamtrace
Sage code for Noam's algorithm: http://www.sagenb.org/home/pub/2179/

Project 3

The Mordell-Weil group mod p is a subgroup of the mod p Selmer group. The Kolyvagin classes lie in modified Selmer groups H(ℓ):=H1F(ℓ)(K,E[p]). For small values of p (i.e. p=2!), directly compute these Selmer groups. Use the existing Selmer group functionality in SAGE (for number fields).

Go along the lines of the paper of Shaefer and Stoll: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.1056

When ρE,p is surjective, there is an injective map H1F(ℓ)(K,E[p])→A×/(A×)p, where A/K is the degree p2−1 extension obtained by adjoining one p-division point of E to K. Use Shaefor and Stoll's method to characterize the image of this injection.

Sage code (a start) for p=3: http://standalone.sagenb.org/home/pub/10/

* Sage code (more than a start) for p=3: http://standalone.sagenb.org/home/pub/13/

A Sage function that decides whether an element of A is unramified or transverse: http://sagenb.org/home/pub/2213

Project 4

Let E/Q be an elliptic curve with non-trivial 3-torsion in III(E/Q). (E.g., curve 681b.) Let K/Q be an imaginary quadratic field satisfying the Heegner hypothesis. Assume 3 doesn't divide any Tamagawa numbers of E. Can III(E/K)[3] be generated by Kolyvagin classes? Here's how you might proceed:

a) Use 3-descent on E to find two elements in A×/(A×)3 representing a basis of III(E/Q). Here A/Q is the etale algebra arising in the 3-descent computation.

b) Find primes ℓ for which dimH(ℓ)=1. Use Stein's algorithm to confirm (if possible) that τℓ/=0. Repeat until you have found two linearly independent τℓ1, τℓ2: These should span III(E/Q).

Misc

Finding the n-th roots of a point P on an elliptic curve even when they are not defined over the ground field -- this should be used in Project 3. This should probably be implemented in Sage as an option in P.division_points(n, options).

* Sage code: http://standalone.sagenb.org/home/pub/12/

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

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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  $τ_{n}$  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  $τ_{ℓ_{1} ℓ_{2}}$  for the elliptic curve 5077A of
rank 3, for different values of the quadratic field  $K$  and for
various pairs of primes  $ℓ_{1}$ ,  $ℓ_{2}$ .

== Project 1 ==

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

(a) Assume that  $I I I_{p} (E / K) = 0$ .  Give an algorithm that finds the dimension of  $H_{F (ℓ)}^{1} (K, E [p])$ .

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

(c) Compare the  $τ_{ℓ}$  from (b) (known only up to torsion) with those produced by William's function "kolyvagin_point_on_curve".

UPDATE:  (a)-(b) has been done for the elliptic curve 389a, for discriminants up to and excluding -67, and for  $ℓ$  up to  $3000$ . Here is the Sage code: http://standalone.sagenb.org/home/pub/9/.  Part (c) has been partially checked.

== Project 2 ==

Suppose the Heegner point  $y_{1} \in E (K [1])$  has been computed, so that its coordinates are known as algebraic numbers.  Compute the trace  $y_{K}$  of  $y_{1}$ .  

 * Noam's algorithm for computing the trace: [[/noamtrace]]
 * Sage code for Noam's algorithm: http://www.sagenb.org/home/pub/2179/
    

== Project 3 ==

The Mordell-Weil group mod  $p$  is a subgroup of the mod  $p$  Selmer group.  The Kolyvagin classes lie in modified Selmer groups  $H (ℓ) := H_{F (ℓ)}^{1} (K, E [p])$ .  For small values of  $p$  (i.e.  $p = 2$ !), directly compute these Selmer groups.  Use the existing Selmer group functionality in SAGE (for number fields).

Go along the lines of the paper of Shaefer and Stoll:  http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.1056

When  $ρ_{E, p}$  is surjective, there is an injective map  $H_{F (ℓ)}^{1} (K, E [p]) \to A^{\times} / (A^{\times})^{p}$ , where  $A / K$  is the degree  $p^{2} - 1$  extension obtained by adjoining one  $p$ -division point of  $E$  to  $K$ .  Use Shaefor and Stoll's method to characterize the image of this injection.  


 * Sage code (a start) for  $p = 3$ : http://standalone.sagenb.org/home/pub/10/
* Sage code (more than a start) for  $p = 3$ : http://standalone.sagenb.org/home/pub/13/

 * A Sage function that decides whether an element of A is unramified or transverse: http://sagenb.org/home/pub/2213

== Project 4 ==

Let  $E / Q$  be an elliptic curve with non-trivial 3-torsion in  $I I I (E / Q)$ .  (E.g., curve 681b.)  Let  $K / Q$  be an imaginary quadratic field satisfying the Heegner hypothesis.  Assume 3 doesn't divide any Tamagawa numbers of  $E$ .  Can  $I I I (E / K) [3]$  be generated by Kolyvagin classes?  Here's how you might proceed:

a) Use 3-descent on E to find two elements in  $A^{\times} / (A^{\times})^{3}$  representing a basis of  $I I I (E / Q)$ .  Here  $A / Q$  is the etale algebra arising in the 3-descent computation.  

b) Find primes  $ℓ$  for which  $\dim H (ℓ) = 1$ .  Use Stein's algorithm to confirm (if possible) that  $τ_{ℓ} \neq 0$ .  Repeat until you have found two linearly independent  $τ_{ℓ_{1}}$ ,  $τ_{ℓ_{2}}$ :  These should span  $I I I (E / Q)$ .







   

== Misc ==

Finding the n-th roots of a point P on an elliptic curve even when they are not defined over the ground field -- this should be used in Project 3.
This should probably be implemented in Sage as an option in P.division_points(n, options). 

* Sage code: http://standalone.sagenb.org/home/pub/12/

== 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

Diff for "days22/weinstein"