CoCalc Shared Fileswww / papers / anti-cyclotomic_height_pairing / nonzero_evidence.mOpen in CoCalc with one click!
Author: William A. Stein
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K<p> := pAdicField(3);
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K`SeriesPrinting := true;
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heights := [];
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// ** 389A **: Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
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// Q(sqrt(-11))
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heights[1] := [* <"389A", -11, [* <false, false>, <-4054834521710959438*3^-3 + O(p^37), 4734502197939836345*3^-3 + O(p^37)>,
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<-5128451143030415824*3^-3 + O(p^37), 184944463937779316*3^-3 + O(p^37)>,
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<-4233587810120733751*3^-3 + O(p^37), 5449450017977127950*3^-3 + O(p^37)>
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*]> *];
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// ** 433A **: Elliptic Curve defined by y^2 + x*y = x^3 + 1 over Rational Field
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// Q(sqrt(-8))
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heights[2] := [* <"433A", -8, [*
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<-291315411869673877*3^-1 + O(p^39), false>,
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<-3822008204613305447*3^-2 + O(p^38), -3112286436193297540*3^-2 + O(p^38)>,
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<3737976811705656244*3^-2 + O(p^38), 4481636830969682309*3^-2 + O(p^38)>,
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<-4171348572200177903*3^-2 + O(p^38), 2968081566999880118*3^-2 + O(p^38)>
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*]> *];
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// ** 446D **: Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 4*x + 4 over Rational Field
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// Q(sqrt(-23))
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heights[3] := [* <"446D", -23, [*
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<4889205826410188336*3^-2 + O(p^38), false>,
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<false, false>, <false, false>, <2966449211307770182*3^-7 + O(p^33), false>
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*]> *];
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// ** 563A **: Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 15*x + 16 over Rational Field
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// Q(sqrt(-8))
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heights[4] := [**];
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// ** 571B **: Elliptic Curve defined by y^2 + y = x^3 + x^2 - 4*x + 2 over Rational Field
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// Q(sqrt(-8))
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heights[5] := [* <"571B", -8, [*
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<-1782454760770426910*3^-1 + O(p^39), 3778404476646802307 + O(p^40)>,
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<4305413687616553349*3^-2 + O(p^38), -2702125882048523978*3^-3 + O(p^37)>,
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<2450010421639998875*3^-2 + O(p^38), -1498804411813247603*3^-3 + O(p^37)>,
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<-1125839608013336218*3^-2 + O(p^38), 1655068396078319938*3^-3 + O(p^37)>
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*]> *];
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// ** 643A **: Elliptic Curve defined by y^2 + x*y = x^3 - 4*x + 3 over Rational Field
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// Q(sqrt(-8))
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heights[6] := [* <"643A", -8, [*
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<false, 1819123308739170151*3^-1 + O(p^39)>,
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<false, 5037002409645970538*3^-3 + O(p^37)>,
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<false, -3113590887348738160*3^-3 + O(p^37)>,
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<false, 2062975139883612320*3^-3 + O(p^37)>
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*]> *];
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// ** 655A **: Elliptic Curve defined by y^2 + y = x^3 - 13*x + 18 over Rational Field
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// Q(sqrt(-56))
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heights[7] := [* <"655A", -56, [*
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<false, -3132389174079978272*3 + O(p^41)>,
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<false, -1453867404683136053*3^-3 + O(p^37)>,
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<-4028903459026151389*3^-5 + O(p^35), false>,
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<false, 3035093387808238001*3^-6 + O(p^34)>
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*]> *];
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// ** 664A **: Elliptic Curve defined by y^2 = x^3 - 7*x + 10 over Rational Field
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// Q(sqrt(-47))
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heights[8] := [* <"664A", -47, [*
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<4926314537163462334*3^-1 + O(p^39), -1945641455192181074*3^-1 + O(p^39)>,
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<false, false>, <false, false>, <-3910325727473192560*3^-6 + O(p^34), -3912984832032276275*3^-6 + O(p^34)>
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*]> *];
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heights[9] := [* *];
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// ** 707A **: Elliptic Curve defined by y^2 + y = x^3 + x^2 - 12*x + 12 over Rational Field
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// Q(sqrt(-20))
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heights[10] := [* <"707A", -20, [*
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<2863442244926934637*3^-1 + O(p^39), false>,
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<618680693762694655*3^-1 + O(p^39), 4967281568111578394*3^-2 + O(p^38)>,
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<253124222861028157*3^-1 + O(p^39), 2167093244472804311*3^-2 + O(p^38)>,
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<-2182919647058111837*3^-1 + O(p^39), -2963606426795524531*3^-2 + O(p^38)>
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*]> *];
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// ** 709A **: Elliptic Curve defined by y^2 + y = x^3 - x^2 - 2*x over Rational Field
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// Q(sqrt(-11))
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heights[11] := [* <"709A", -11, [* <false, false>, <false, false>, <false, false>, <false, false> *]> *];
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// ** 718B **: Elliptic Curve defined by y^2 + x*y + y = x^3 - 5*x over Rational Field
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// Q(sqrt(-71))
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heights[12] := [* <"718B", -71, [* <false, false>, <-3499752323227473272*3^-3 + O(p^37), 1187547108495858778*3^-3 + O(p^37)>,
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<1462089231022375192*3^-3 + O(p^37), 3000668223899152171*3^-3 + O(p^37)>,
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<1030150665179043736*3^-3 + O(p^37), 670642666274802250*3^-3 + O(p^37)>
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*]> *];
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// ** 794A **: Elliptic Curve defined by y^2 + x*y + y = x^3 - 3*x + 2 over Rational Field
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// Q(sqrt(-23))
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heights[13] := [* <"794A", -23, [*
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<4462182510702605230*3^-2 + O(p^38), -3507347402157431810*3^-2 + O(p^38)>,
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<-5329915211526847534*3^-2 + O(p^38), 3598606149419360830*3^-3 + O(p^37)>,
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<2495277593850692735*3^-2 + O(p^38), -5260340722418140451*3^-3 + O(p^37)>,
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<1465109382275761220*3^-2 + O(p^38), -394586787821713007*3^-3 + O(p^37)>
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*]> *];
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// ** 817A **: Elliptic Curve defined by y^2 + y = x^3 + x^2 + x + 6 over Rational Field
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// Q(sqrt(-8))
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heights[14] := [* <"817A", -8, [* <false, false>, <-1354176743485423888*3^-3 + O(p^37), -2053462216707674134*3^-3 + O(p^37)>,
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<5938091252285788547*3^-3 + O(p^37), 1372550707058219822*3^-3 + O(p^37)>,
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<1891907623753773605*3^-3 + O(p^37), 3060040305083914487*3^-3 + O(p^37)>
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*]> *];
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// ** 916C **: Elliptic Curve defined by y^2 = x^3 - 4*x + 1 over Rational Field
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// Q(sqrt(-11))
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heights[15] := [* <"916C", -11, [* <false, false>, <false, 2575759341996527507*3^-2 + O(p^38)>,
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<false, false>, <false, false> *]> *];
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// ** 944E **: Elliptic Curve defined by y^2 = x^3 - 19*x + 34 over Rational Field
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// Q(sqrt(-11))
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heights[16] := [* <"944E", -11, [* <false, false>, <false, false>, <-5647105271707110206*3^-5 + O(p^35), false>,
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<false, false> *]> *];
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// ** 997B **: Elliptic Curve defined by y^2 + y = x^3 - x^2 - 5*x - 3 over Rational Field
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// Q(sqrt(-23))
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heights[17] := [**];
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// ** 997C **: Elliptic Curve defined by y^2 + y = x^3 - x^2 - 24*x + 54 over Rational Field
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// Q(sqrt(-23))
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heights[18] := [* <"997C", -23, [*
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<false, -1892039863366021720*3^-1 + O(p^39)>,
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<false, false>, <false, 5661372025716929659*3^-3 + O(p^37)>,
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<false, false> *]> *];
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procedure Entry(n)
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if #heights[n] eq 0 then
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return;
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end if;
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D := heights[n][1][2];
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E := heights[n][1][1];
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printf "E: %o, K=Q(sqrt(%o))\n", E, D;
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for a in [1,2] do
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for i in [1..4] do
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val := heights[n][1][3][i][a];
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printf "H_2(3^%o*P_%o)/3^%o := ", i,a,2*i;
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if Type(val) eq BoolElt then
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print "undef";
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else
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print val + O(p^5);
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end if;
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end for;
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print "";
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end for;
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end procedure;
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