Navegar

Colección

Datos Estadísticas

Artículo

Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

For a prime number p, we study the zeros modulo p of divisor polynomials of rational elliptic curves E of conductor p. Ono (CBMS regional conference series in mathematics, 2003, vol 102, p. 118) made the observation that these zeros are often j-invariants of supersingular elliptic curves over Fp¯. We show that these supersingular zeros are in bijection with zeros modulo p of an associated quaternionic modular form vE. This allows us to prove that if the root number of E is - 1 then all supersingular j-invariants of elliptic curves defined over Fp are zeros of the corresponding divisor polynomial. If the root number is 1, we study the discrepancy between rank 0 and higher rank elliptic curves, as in the latter case the amount of supersingular zeros in Fp seems to be larger. In order to partially explain this phenomenon, we conjecture that when E has positive rank the values of the coefficients of vE corresponding to supersingular elliptic curves defined over Fp are even. We prove this conjecture in the case when the discriminant of E is positive, and obtain several other results that are of independent interest. © 2017, The Author(s).

Registro:

Documento: Artículo
Título:Supersingular zeros of divisor polynomials of elliptic curves of prime conductor
Autor:Kazalicki, M.; Kohen, D.
Filiación:Department of Mathematics, University of Zagreb, Bijenička cesta 30, Zagreb, 10000, Croatia
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina
IMAS, CONICET, Buenos Aires, Argentina
Palabras clave:Brandt module; Divisor polynomial; Supersingular elliptic curves
Año:2017
Volumen:4
Número:1
DOI: http://dx.doi.org/10.1186/s40687-017-0099-8
Título revista:Research in Mathematical Sciences
Título revista abreviado:Res. Math. Sci.
ISSN:25220144
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_25220144_v4_n1_p_Kazalicki

Referencias:

  • Agashe, A., Ribet, K.A., Stein, W.A., The modular degree, congruence primes, and multiplicity one (2012) Number Theory, Analysis and Geometry, pp. 19-49. , Springer, New York
  • Edixhoven, B., Serre’s conjecture (1997) Modular Forms and Fermat’s Last Theorem, pp. 209-242. , Springer, New York
  • Emerton, M., Supersingular elliptic curves, theta series and weight two modular forms (2002) J. Am. Math. Soc., 15 (3), pp. 671-714
  • Gross, B., Heights and the special values of L-series. Number theory (1987) CMS Conference Proceedings, 7, pp. 115-187
  • Katz, N.M., P -adic properties of modular schemes and modular forms. In: Modular Functions of One Variable, III (Proceedings of the International Summer School (1973) Lecture Notes in Mathematics, 350, pp. 60-190. , University of Antwerp 1972
  • Le Hung, B., Li, C., Level raising mod 2 and arbitrary 2 -Selmer ranks (2016) Compos. Math., 152 (8), pp. 1576-1608
  • Mestre, J.F., La méthode des graphes. Exemples et applications (1986) In: Proceedings of the International Conference on Class Numbers and Fundamental Units of Algebraic Number Fields, pp. 217-242. , Nagoya University, Katata
  • Ono, K., The web of modularity: Arithmetic of the coefficients of modular forms and q-series (2003) CBMS Regional Conference Series in Mathematics, 102, p. 118
  • Robert, G., Congruences entre séries d’Eisenstein, dans le cas supersingulier (1980) Invent. Math., 61 (2), pp. 103-158
  • Serre, J.-P., Congruences et formes modulaires [d’après H. P. F. Swinnerton-Dyer] (1973) Séminaire Bourbaki, 24E année, 317, pp. 319-338. , 1971/1972, Exp. No. 416. Lecture Notes in Mathematics
  • Serre, J.-P., Formes modulaires et fonctions zêta p -adiques. In: Modular Functions of One Variable. In: III Proceedings of the International Summer School (1973) Lecture Notes in Mathematics, 350, pp. 191-268. , University of Antwerp 1972
  • Serre, J.-P., Sur les représentations modulaires de degré 2 de Gal(Q-/ Q) (1987) Duke Math. J., 54 (1), pp. 179-230
  • Serre, J.-P., Two letters on quaternions and modular forms (mod p) (1996) Isr. J. Math., 95, pp. 281-299
  • Watkins, M., Explicit lower bounds on the modular degree of an elliptic curve (2004) Mathematics Faculty Scholarship, 104. , Paper
  • Zagier, D., Modular parametrizations of elliptic curves (1985) Can. Math. Bull., 28, pp. 372-384

Citas:

---------- APA ----------
Kazalicki, M. & Kohen, D. (2017) . Supersingular zeros of divisor polynomials of elliptic curves of prime conductor. Research in Mathematical Sciences, 4(1).
http://dx.doi.org/10.1186/s40687-017-0099-8
---------- CHICAGO ----------
Kazalicki, M., Kohen, D. "Supersingular zeros of divisor polynomials of elliptic curves of prime conductor" . Research in Mathematical Sciences 4, no. 1 (2017).
http://dx.doi.org/10.1186/s40687-017-0099-8
---------- MLA ----------
Kazalicki, M., Kohen, D. "Supersingular zeros of divisor polynomials of elliptic curves of prime conductor" . Research in Mathematical Sciences, vol. 4, no. 1, 2017.
http://dx.doi.org/10.1186/s40687-017-0099-8
---------- VANCOUVER ----------
Kazalicki, M., Kohen, D. Supersingular zeros of divisor polynomials of elliptic curves of prime conductor. Res. Math. Sci. 2017;4(1).
http://dx.doi.org/10.1186/s40687-017-0099-8