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On the parity of the index of ramified Heegner divisors

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Abstract

Let E be an elliptic curve defined over the rationals and let N be its conductor. Assume N is prime. In this paper, we prove that the index on E of the Heegner divisor of discriminant \(D=-~4N\) is even provided \(N\equiv 7\pmod {8}\) and discuss some conjectures on further parity properties for the indexes on E of Heegner divisors of discriminant D dividing 4N. One of these conjectures suggests a possible link between the parity of the eigenvalue \(a_A(2)\) and the parity of the Šafarevič-Tate group of certain elliptic curves A of square conductor.

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Notes

  1. To enhance convergence one has to chose the representative \(Q_i\) with \(\mathfrak {I}(\tau _i)\) as large as possible.

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Acknowledgements

I would like to thank the Mathematics Section at ICTP for their hospitality while preparing the final version of the manuscript. The elliptic curve data used here come from Cremona’s Tables [5]. Computations were performed with the help of a GNU/Linux computer running Pari [20].

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  1. Centro Universitario de Investigación en Ciencias Básicas, Universidad de Colima, Bernal Díaz del Castillo 340, Villa San Sebastián, 28045, Colima, Mexico

    Carlos Castano-Bernard

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  1. Carlos Castano-Bernard
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Correspondence to Carlos Castano-Bernard.

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Castano-Bernard, C. On the parity of the index of ramified Heegner divisors. Bol. Soc. Mat. Mex. 25, 1–11 (2019). https://doi.org/10.1007/s40590-017-0192-4

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