On the Titchmarsh divisor problem for abelian varieties
Author:
Cristian Virdol
Journal:
Proc. Amer. Math. Soc. 145 (2017), 3681-3687
MSC (2010):
Primary 11G10, 11G15
DOI:
https://doi.org/10.1090/proc/13519
Published electronically:
February 15, 2017
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Abstract | References | Similar Articles | Additional Information
Abstract: In this article we study the Titchmarsh divisor problem in the context of abelian varieties. Under the Generalized Riemann Hypothesis we obtain an asymptotic formula.
- [AG] Amir Akbary and Dragos Ghioca, A geometric variant of Titchmarsh divisor problem, Int. J. Number Theory 8 (2012), no. 1, 53-69. MR 2887882, https://doi.org/10.1142/S1793042112500030
- [BK] Armand Brumer and Kenneth Kramer, Non-existence of certain semistable abelian varieties, Manuscripta Math. 106 (2001), no. 3, 291-304. MR 1869222, https://doi.org/10.1007/PL00005885
- [H] H. Halberstam, Footnote to the Titchmarsh-Linnik divisor problem, Proc. Amer. Math. Soc. 18 (1967), 187-188. MR 0204379
- [L] Ju. V. Linnik, The dispersion method in binary additive problems, Translated by S. Schuur, American Mathematical Society, Providence, R.I., 1963. MR 0168543
- [M] David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi, 2008. With appendices by C. P. Ramanujam and Yuri Manin; Corrected reprint of the second (1974) edition. MR 2514037
- [R] Gaetano Rodriquez, Sul problema dei divisori di Titchmarsh, Boll. Un. Mat. Ital. (3) 20 (1965), 358-366 (Italian, with English summary). MR 0197409
- [S] J-P. Serre, Résumé des cours de l'année scolaire, 1977-1978 (College de France, Paris, 1979), see J-P. Serre's Collected Papers vol. III, 1972-1984.
- [SI] Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094
- [T] E.C. Titchmarsh, A divisor problem, Rend. di. Palermo 54 (1931), 414-429.
- [V] Cristian Virdol, Titchmarsh divisor problem for abelian varieties of types I, II, III, and IV, Trans. Amer. Math. Soc. 368 (2016), no. 11, 8011-8028. MR 3546791, https://doi.org/10.1090/tran/6748
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Additional Information
Cristian Virdol
Affiliation:
Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul 03722, Korea
Email:
cristian.virdol@gmail.com
DOI:
https://doi.org/10.1090/proc/13519
Keywords:
Abelian varieties,
Titchmarsh divisor problem,
asymptotic formulas
Received by editor(s):
September 29, 2015
Received by editor(s) in revised form:
September 21, 2016
Published electronically:
February 15, 2017
Additional Notes:
This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2015R1D1A1A01056643)
Communicated by:
Mathew A. Papanikolas
Article copyright:
© Copyright 2017
American Mathematical Society