Artículo
Abstract:
Given a Hecke eigenform f of weight 2 and square-free level N, by the work of Kohnen, there is a unique weight 3/2 modular form of level 4N mapping to f under the Shimura correspondence. Furthermore, by the work of Waldspurger the Fourier coefficients of such a form are related to the quadratic twists of the form f. Gross gave a construction of the half integral weight form when N is prime, and such construction was later generalized to square-free levels. However, in the non-square free case, the situation is more complicated since the natural construction is vacuous. The problem being that there are too many special points so that there is cancellation while trying to encode the information as a linear combination of theta series. In this paper, we concentrate in the case of level p2, for p > 2 a prime number, and show how the set of special points can be split into subsets (indexed by bilateral ideals for an order of reduced discriminant p2) which gives two weight 3/2 modular forms mapping to f under the Shimura correspondence. Moreover, the splitting has a geometric interpretation which allows to prove that the forms are indeed a linear combination of theta series associated to ternary quadratic forms. Once such interpretation is given, we extend the method of Gross-Zagier to the case where the level and the discriminant are not prime to each other to prove a Gross-type formula in this situation. © 2014 World Scientific Publishing Company.
Registro:
| Documento: | Artículo |
| Título: | Shimura correspondence for level p2 and the central values of L-series, II |
| Autor: | Pacetti, A.; Tornaría, G. |
| Filiación: | Departamento de Matemática, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, Buenos Aires, C.P.: 1428, Argentina Centro de Matemática, Facultad de Ciencias, Universidad de la República, Iguá 4225 esq. Mataojo, Montevideo, Uruguay |
| Palabras clave: | L-series special values; Shimura correspondence |
| Año: | 2014 |
| Volumen: | 10 |
| Número: | 7 |
| Página de inicio: | 1595 |
| Página de fin: | 1635 |
| DOI: | http://dx.doi.org/10.1142/S179304211450047X |
| Título revista: | International Journal of Number Theory |
| Título revista abreviado: | Int. J. Number Theory |
| ISSN: | 17930421 |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_17930421_v10_n7_p1595_Pacetti |
Referencias:
- Böcherer, S., Schulze-Pillot, R., The Dirichlet series of Koecher and Maass and modular forms of weight (1992) Math. Z., 209 (2), pp. 273-287
- Brzezínski, J., On orders in quaternion algebr (1983) Comm. Algebra, 11 (5), pp. 501-522
- Eichler, M., The basis problem for modular forms and the traces of the Hecke operato (1973) Modular Functions of One Variable, I, Lecture Notes in Mathematics, 320, pp. 75-151. , Springer, Berlin
- Gross, B.H., Heights and the special values of L-seri (1987) Number Theory, CMS Conference Proceedings, 7, pp. 115-187. , American Mathematical Society, Providence, RI
- Gross, B.H., Zagier, D.B., Heegner points and derivatives of L-seri (1986) Invent. Math., 84 (2), pp. 225-320
- Hijikata, H., Explicit formula of the traces of Hecke operators for ?0( (1974) J. Math. Soc. Japan, 26, pp. 56-82
- Pacetti, A., Rodriguez Villegas, F., Computing weight 2 modular forms of level (2005) Math. Comp., 74 (251), pp. 1545-1557. , (electronic); With an appendix by B. Gross
- Pacetti, A., Tornariá, G., Examples of the Shimura correspondence fo r level p2 and real quadratic twis (2007) Ranks of Elliptic Curves and Random Matrix Theory, London Mathematical Society Lecture Note Series, 341, pp. 289-314. , (Cambridge University Press, Cambridge
- Shimura correspondence for level p2 and the central values of L-seri (2007) J. Number Theory, 124 (2), pp. 396-414
- Pizer, A., Theta series and modular forms of level p (1980) Compos. Math., 40 (2), pp. 177-241
- Shimura, G., On modular forms of half integral weig (1973) Ann. of Math., 2 (97), pp. 440-481
- Tornarí, G., (2004) Data about the Central Values of the L-series of (Imaginary and Real) Quadratic Twists of Elliptic Curves, , http://www.ma.utexas.edu/users/tornaria/cnt, preprint
- Vignéras, M.-F., Arithmétique des algebres de quaternio (1980) Lecture Notes in Mathematics, 800. , Springer, Berlin
- Waldspurger, J.-L., Sur les coefficients de Fourier des formes modulaires de poids demi-enti (1981) J. Math. Pures Appl., 60 (4-9), pp. 375-484
- Correspondances de Shimura et quaternio (1991) Forum Math., 3 (3), pp. 219-307
Citas:
---------- APA ----------
Pacetti, A. & Tornaría, G. (2014) . Shimura correspondence for level p2 and the central values of L-series, II. International Journal of Number Theory, 10(7), 1595-1635.http://dx.doi.org/10.1142/S179304211450047X
---------- CHICAGO ----------
Pacetti, A., Tornaría, G. "Shimura correspondence for level p2 and the central values of L-series, II" . International Journal of Number Theory 10, no. 7 (2014) : 1595-1635.http://dx.doi.org/10.1142/S179304211450047X
---------- MLA ----------
Pacetti, A., Tornaría, G. "Shimura correspondence for level p2 and the central values of L-series, II" . International Journal of Number Theory, vol. 10, no. 7, 2014, pp. 1595-1635.http://dx.doi.org/10.1142/S179304211450047X
---------- VANCOUVER ----------
Pacetti, A., Tornaría, G. Shimura correspondence for level p2 and the central values of L-series, II. Int. J. Number Theory. 2014;10(7):1595-1635.http://dx.doi.org/10.1142/S179304211450047X
