Kernels of $L$-functions and shifted convolutions
HTML articles powered by AMS MathViewer
- by Nikolaos Diamantis PDF
- Proc. Amer. Math. Soc. 148 (2020), 5059-5070 Request permission
Abstract:
We propose a characterisation of the field into which quotients of non-critical values of $L$-functions associated to a cusp form belong. The construction involves shifted convolution series of divisor sums and a certain double Eisenstein series that induces a kernel of $L$-functions.References
- A. A. Beĭlinson, Higher regulators of modular curves, Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 1–34. MR 862627, DOI 10.1090/conm/055.1/862627
- F. Brown Modular Values and the relative completion of the fundamental group of M1,1, arXiv:1407.5167
- François Brunault, Valeur en 2 de fonctions $L$ de formes modulaires de poids 2: théorème de Beilinson explicite, Bull. Soc. Math. France 135 (2007), no. 2, 215–246 (French, with English and French summaries). MR 2430191, DOI 10.24033/bsmf.2532
- Henri Cohen and Fredrik Strömberg, Modular forms, Graduate Studies in Mathematics, vol. 179, American Mathematical Society, Providence, RI, 2017. A classical approach. MR 3675870, DOI 10.1090/gsm/179
- P. Deligne, Valeurs de fonctions $L$ et périodes d’intégrales, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 313–346 (French). With an appendix by N. Koblitz and A. Ogus. MR 546622
- Nikolaos Diamantis and Cormac O’Sullivan, Kernels of $L$-functions of cusp forms, Math. Ann. 346 (2010), no. 4, 897–929. MR 2587096, DOI 10.1007/s00208-009-0419-4
- Nikolaos Diamantis and Cormac O’Sullivan, Kernels for products of $L$-functions, Algebra Number Theory 7 (2013), no. 8, 1883–1917. MR 3134038, DOI 10.2140/ant.2013.7.1883
- M. Kıral, Shifted Divisor Sum Dirichlet Series, preprint
- N. I. Koblic, Non-integrality of the periods of cusp forms outside the critical strip, Funkcional. Anal. i Priložen. 9 (1975), no. 3, 52–55 (Russian). MR 0404144
- Ju. I. Manin, Periods of cusp forms, and $p$-adic Hecke series, Mat. Sb. (N.S.) 92(134) (1973), 378–401, 503 (Russian). MR 0345909
- Mathew Rogers and Wadim Zudilin, From $L$-series of elliptic curves to Mahler measures, Compos. Math. 148 (2012), no. 2, 385–414. MR 2904192, DOI 10.1112/S0010437X11007342
- Goro Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976), no. 6, 783–804. MR 434962, DOI 10.1002/cpa.3160290618
- Maxim Kontsevich and Don Zagier, Periods, Mathematics unlimited—2001 and beyond, Springer, Berlin, 2001, pp. 771–808. MR 1852188
- Vicenţiu Paşol and Alexandru A. Popa, Modular forms and period polynomials, Proc. Lond. Math. Soc. (3) 107 (2013), no. 4, 713–743. MR 3108829, DOI 10.1112/plms/pdt003
- D. Zagier, Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields, Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Lecture Notes in Math., Vol. 627, Springer, Berlin, 1977, pp. 105–169. MR 0485703
Additional Information
- Nikolaos Diamantis
- Affiliation: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
- ORCID: 0000-0002-3670-278X
- Email: nikolaos.diamantis@nottingham.ac.uk
- Received by editor(s): September 2, 2019
- Received by editor(s) in revised form: February 9, 2020
- Published electronically: September 17, 2020
- Communicated by: Benjamin Brubaker
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 5059-5070
- MSC (2010): Primary 11F67, 11F68
- DOI: https://doi.org/10.1090/proc/15182
- MathSciNet review: 4163822