Computing Heegner Points with Sage

William Stein

University of Washington

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Relevant References

Watkins: Some remarks on Heegner point computations
Delaunay's chapter in Cohen's book: Number Theory: Tools and diophantine equations, Section 8.6.
Stein: Heegner points on rank 2 elliptic curves
Sage: schemes/elliptic_curves/heegner.py (by Stein, Bradshaw)

WARNING: Sage's Heegner points code is optimized for doing theoretical research into Kolyvagin's Euler System of Heegner Points.

It is not designed (yet!) to be used as a tool for actually efficiently computing points on curves. Mark Watkins has written code that is in Magma that is very highly optimized for finding points on curves (but not useful for research into Euler systems).

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Computing Heegner Points $y_K$

{{{id=34| E = EllipticCurve('37a'); E /// Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field }}} {{{id=28| E.plot() ///

}}} {{{id=29| E.rank() /// 1 }}} {{{id=30| E.heegner_discriminants(100) /// [-3, -4, -7, -11, -40, -47, -67, -71, -83, -84, -95] }}}

First a discriminant of class number 1

{{{id=38| QuadraticField(-3,'a').class_number() /// 1 }}} {{{id=33| P = E.heegner_point(-3); P /// Heegner point of discriminant -3 on elliptic curve of conductor 37 }}} {{{id=32| P.numerical_approx() /// (-1.00000000000000 + 2.18015221574935e-17*I : 3.24718550659409e-17 + 4.34765071166687e-17*I : 1.00000000000000) }}} {{{id=39| P.numerical_approx(prec=100) /// (-1.0000000000000000000000000000 - 9.6895704450769185435636090117e-31*I : -5.1662376187963059639095337283e-31 - 1.9379140828452461806982502823e-30*I : 1.0000000000000000000000000000) }}} {{{id=36| P.point_exact() /// (-1 : 0 : 1) }}}

Compute the image of the Heegner point on the upper half plane as a complex number, where we view $E$ as $\mathbf{C}/\Lambda$.

{{{id=40| P.map_to_complex_numbers() /// 0.204680500375773 - 1.22569469099340*I }}}

The point in the upper half plane (really on the modular curve $X_0(N)$) that maps to the above point $P$.

{{{id=44| t = P.heegner_point_on_X0N(); t /// Heegner point 1/74*sqrt(-3) - 21/74 of discriminant -3 on X_0(37) }}} {{{id=82| /// }}} {{{id=81| /// }}} {{{id=45| /// }}}

Next, an example in which the class number is $>1$.

{{{id=49| for D in E.heegner_discriminants(100): print D, QuadraticField(D,'a').class_number() /// -3 1 -4 1 -7 1 -11 1 -40 2 -47 5 -67 1 -71 7 -83 3 -84 4 -95 8 }}} {{{id=50| P = E.heegner_point(-47); P /// Heegner point of discriminant -47 on elliptic curve of conductor 37 }}} {{{id=53| P.heegner_point_on_X0N() /// Heegner point 1/74*sqrt(-47) - 29/74 of discriminant -47 on X_0(37) }}} {{{id=51| P.numerical_approx(100) /// (0.41295030345265522481209547079 - 1.2506651800425362123442536977*I : -1.2882357148097753052431772001 - 1.6283769583240139393113357931*I : 1.0000000000000000000000000000) }}} {{{id=48| y_1 = P.point_exact(100); show(y_1) ///

\newcommand{\Bold}[1]{\mathbf{#1}}\left(a : a^{4} + a - 1 : 1\right)

}}}

Note that $y_1$ is the Heegner point defined over the Hilbert Class Field. It has not been traced down to $K$.

{{{id=41| K = y_1[0].parent(); K /// Number Field in a with defining polynomial x^5 - x^4 + x^3 + x^2 - 2*x + 1 }}}

Computing $y_K$ when $h_K>1$

Not Implemented!

Project: The Heegner points package in Sage doesn't have a function that just computes $y_K = {\rm trace}_{H/K}(y_1)$. Implement such a function.

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Computing Heegner Points $y_1$

{{{id=2| E = EllipticCurve('5077a'); E /// Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 over Rational Field }}} {{{id=19| E.rank() /// 3 }}} {{{id=18| E.heegner_discriminants(100) /// [-3, -4, -7, -19, -40, -47, -55, -59, -71, -79, -84, -88] }}} {{{id=20| P = E.heegner_point(-7); P /// Heegner point of discriminant -7 on elliptic curve of conductor 5077 }}} {{{id=21| P.numerical_approx() /// (5.24763468960140e25 - 9.42689096571972e25*I : 2.63080809434194e37 + 1.12035605894541e39*I : 1.00000000000000) }}} {{{id=22| P.numerical_approx(100) /// (0 : 1.0000000000000000000000000000 : 0) }}}

Bug in released Sage -- see 9302. (It just gets the $x$-coordinate and lifts, and didn't do the right thing for point at infinity.)

{{{id=24| P.point_exact(100) /// (0 : 1 : 0) }}} {{{id=23| /// }}} {{{id=13| /// }}} {{{id=12| /// }}}

Computing Higher Heegner Points $y_n$

First our rank 1 curve 37a again:

{{{id=61| E = EllipticCurve('37a'); E /// Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field }}} {{{id=60| E.rank() /// 1 }}} {{{id=65| P = E.heegner_point(-3, 5); P /// Heegner point of discriminant -3 and conductor 5 on elliptic curve of conductor 37 }}} {{{id=64| P.numerical_approx(100) /// (1.5000000000000000000000000000 - 1.9364916731037084425896326999*I : -1.5904435122427102073514607000e-29 - 3.8729833462074168851792653998*I : 1.0000000000000000000000000000) }}} {{{id=59| P.point_exact(100) /// Traceback (most recent call last): File "", line 1, in File "_sage_input_104.py", line 10, in exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("UC5wb2ludF9leGFjdCgxMDAp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in File "/private/var/folders/FE/FEo498bGEOeewp0B4AIIP++++TM/-Tmp-/tmph90X0d/___code___.py", line 3, in exec compile(u'P.point_exact(_sage_const_100 )' + '\n', '', 'single') File "", line 1, in File "/Users/wstein/sage/build/sage/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/heegner.py", line 3360, in point_exact f = self.x_poly_exact(prec, algorithm=algorithm) File "/Users/wstein/sage/build/sage/local/lib/python2.6/site-packages/sage/misc/cachefunc.py", line 322, in __call__ return self._cachedmethod._instance_call(self._instance, *args, **kwds) File "/Users/wstein/sage/build/sage/local/lib/python2.6/site-packages/sage/misc/cachefunc.py", line 466, in _instance_call cache[key] = self._cachedfunc.f(inst, *args, **kwds) File "/Users/wstein/sage/build/sage/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/heegner.py", line 3243, in x_poly_exact n = self.ring_class_field().degree_over_K() File "/Users/wstein/sage/build/sage/local/lib/python2.6/site-packages/sage/misc/cachefunc.py", line 322, in __call__ return self._cachedmethod._instance_call(self._instance, *args, **kwds) File "/Users/wstein/sage/build/sage/local/lib/python2.6/site-packages/sage/misc/cachefunc.py", line 466, in _instance_call cache[key] = self._cachedfunc.f(inst, *args, **kwds) File "/Users/wstein/sage/build/sage/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/heegner.py", line 370, in degree_over_K return K.class_number() * self.degree_over_H() File "/Users/wstein/sage/build/sage/local/lib/python2.6/site-packages/sage/misc/cachefunc.py", line 322, in __call__ return self._cachedmethod._instance_call(self._instance, *args, **kwds) File "/Users/wstein/sage/build/sage/local/lib/python2.6/site-packages/sage/misc/cachefunc.py", line 466, in _instance_call cache[key] = self._cachedfunc.f(inst, *args, **kwds) File "/Users/wstein/sage/build/sage/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/heegner.py", line 419, in degree_over_H raise NotImplementedError NotImplementedError }}}

Project: Implement this. See sage/schemes/elliptic_curves/heegner.py, line 419: "if K.discriminant() > -5:"

The problem is just that some things aren't implemented when $K=\mathbf{Q}(i)$ or $K=\mathbf{Q}(\sqrt{-3})$.

{{{id=67| E = EllipticCurve('37a'); E P = E.heegner_point(-7, 5); P /// Heegner point of discriminant -7 and conductor 5 on elliptic curve of conductor 37 }}} {{{id=10| P.numerical_approx() /// (5.83468606702153 + 10.1265099943474*I : -0.320931045360298 - 40.0244674238506*I : 1.00000000000000) }}} {{{id=9| y_5 = P.point_exact(); y_5 /// (a : 2/369*a^5 - 11/123*a^4 + 389/369*a^3 - 144/41*a^2 + 1184/369*a - 208/123 : 1) }}} {{{id=70| factor(y_5[0].parent().discriminant()) /// -1 * 5^5 * 7^3 }}} {{{id=69| /// }}} {{{id=68| /// }}}

Computing Information About Kolyvagin Classes

{{{id=15| E = EllipticCurve('389a'); E /// Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field }}} {{{id=73| E.heegner_discriminants(20) /// [-4, -7, -11, -19] }}} {{{id=8| H = heegner_points(389); H /// Set of all Heegner points on X_0(389) }}} {{{id=72| H5 = H.reduce_mod(5); H5 /// Heegner points on X_0(389) over F_5 }}} {{{id=78| K = QuadraticField(-7,'a') /// }}} {{{id=77| p = 5 for c in primes(100): a_c = E.ap(c) print c, a_c % p, (c+1) % p, sage.schemes.elliptic_curves.heegner.is_inert(-7,c) /// 2 3 3 False 3 3 4 True 5 2 1 True 7 0 3 False 11 1 2 False 13 2 4 True 17 4 3 True 19 0 0 True 23 1 4 False 29 4 0 False 31 4 2 True 37 2 3 False 41 2 2 True 43 2 4 False 47 3 3 True 53 4 4 False 59 3 0 True 61 2 2 True 67 0 3 False 71 0 2 False 73 3 4 True 79 2 0 False 83 3 4 True 89 2 0 True 97 1 3 True }}} {{{id=76| /// }}}

Verify Kolvyagin's Conjecture for a rank 2 curve:

{{{id=88| H5 /// Heegner points on X_0(389) over F_5 }}}

The following calculation shows that if we consider the Kolyvagin class for $D=-7$

$$ [P_{17}] \in E(K_{17})/3 E(K_{17})$$

and take its image in $E(\mathbf{F}_{5^2})/3E(\mathbf{F}_{5^2})$, we get $0$.

{{{id=1| H5.kolyvagin_point_on_curve(D=-7, c=17, E=E, p=3) /// [0, 0] }}}

However, if we consider

$$ [P_{41}] \in E(K_{41})/3 E(K_{41})$$

and take its image in $E(\mathbf{F}_{5^2})/3E(\mathbf{F}_{5^2})$, we get something nonzero!

{{{id=90| H5.kolyvagin_point_on_curve(D=-7, c=41, E=E, p=3) /// [1, 0] }}}

This computation is the first ever provable verification of Kolyvagin's conjecture for a curve of rank at least 2:

Conjecture (Kolyvagin): Some equivalence class $[P_n] \in E(K_n)/p^m E(K_n)$ is nonzero.

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