Kernels of 𝐿-functions and shifted convolutions

Diamantis, Nikolaos

doi:10.1090/proc/15182

Kernels of $L$-functions and shifted convolutions
HTML articles powered by AMS MathViewer

by Nikolaos Diamantis

Proc. Amer. Math. Soc. 148 (2020), 5059-5070

DOI: https://doi.org/10.1090/proc/15182

Published electronically: September 17, 2020

PDF | 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