In this paper, we study the Eichler cohomology associated with half-integral weight cusp forms using the Dedekind eta function and the theta function . We prove that -multiplication (resp. -multiplication) gives an isomorphism between the space of cusp forms of a half-integral weight and the cohomology group associated with the space (resp. ). We also show that there is an isomorphism between the direct sum of two spaces of cusp forms of half-integral weights and the cohomology group.
Dans cet article, we étudions la cohomologie d’Eichler associée aux formes cupsidales de poids demi-entiers en utilisant la fonction zêta de Dedekind . Nous montrons que la -multiplication (resp. -multiplication) induit un isomorphisme entre l’espace des formes cupsidales de poids demi-entiers et le groupe de cohomologie associé à l’espace (resp. ). Nous montrons aussi qu’il existe un isomorphisme entre la somme directe de deux espaces de telles formes et le groupe de cohomologie.
Keywords: Period functions, Eichler cohomology
Dohoon Choi 1; Subong Lim 2; Wissam Raji 3
@article{JTNB_2015__27_1_33_0,
author = {Dohoon Choi and Subong Lim and Wissam Raji},
title = {Period functions of half-integral weight modular forms},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {33--45},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {27},
number = {1},
year = {2015},
doi = {10.5802/jtnb.891},
mrnumber = {3346962},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.891/}
}
TY - JOUR AU - Dohoon Choi AU - Subong Lim AU - Wissam Raji TI - Period functions of half-integral weight modular forms JO - Journal de théorie des nombres de Bordeaux PY - 2015 SP - 33 EP - 45 VL - 27 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.891/ DO - 10.5802/jtnb.891 LA - en ID - JTNB_2015__27_1_33_0 ER -
%0 Journal Article %A Dohoon Choi %A Subong Lim %A Wissam Raji %T Period functions of half-integral weight modular forms %J Journal de théorie des nombres de Bordeaux %D 2015 %P 33-45 %V 27 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.891/ %R 10.5802/jtnb.891 %G en %F JTNB_2015__27_1_33_0
Dohoon Choi; Subong Lim; Wissam Raji. Period functions of half-integral weight modular forms. Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 1, pp. 33-45. doi : 10.5802/jtnb.891. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.891/
[1] K. Bringmann, P. Guerzhoy, Z. Kent and K. Ono, Eichler-Shimura theory for mock modular forms, Math. Ann., 355, (2013), 1085–1121. | MR
[2] K. Bringmann and B. Kane, Sums of class numbers and mixed mock modular forms, arXiv:1305.0112v1.
[3] R. Bruggeman, Harmonic lifts of modular forms, to appear in Ramanujan Journal. | MR
[4] M. Eichler, Eine Verallgemeinerung der Abelschen Integrale, Math. Z. 67, (1957), 267–298. | MR | Zbl
[5] R. Gunning, The Eichler cohomology groups and automorphic forms, Trans. Amer. Math. Soc., 100, (1961), 44–62. | MR | Zbl
[6] S.Y. Husseini and M. I. Knopp, Eichler cohomology and automorphic forms, Illinois J. Math., 15, (1971), 565–577. | MR | Zbl
[7] H. Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, 17, (1997). | MR | Zbl
[8] M. I. Knopp, Some new results on the Eichler cohomology of automorphic forms, Bull. Amer. Math. Soc., 80, (1974), 607–632. | MR | Zbl
[9] M. I. Knopp and H. Mawi, Eichler cohomology theorem for automorphic forms of small weights, Proc. Amer. Math. Soc., 138, 2 (2010), 395–404. | MR | Zbl
[10] W. Kohnen and D. Zagier, Modular forms with rational periods, Modular Forms (Durham, 1983), (1984), 197–249. | MR | Zbl
[11] I. Kra, On cohomology of Kleinian groups, Ann. of Math. (2) 89, (1969), 533–556. | MR | Zbl
[12] J. Manin, Periods of parabolic forms and -adic Hecke series, Math. Sbornik (1973), AMS translation 371–393. | Zbl
[13] G. Shimura, Sur les intégrales attachées aux formes automorphes, J. Math. Soc. Japan, 11, (1959), 291–311. | MR | Zbl
[14] D. Zagier, Traces of singular moduli, Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), Int. Press Lect. Ser., 3, Int. Press, Somerville, MA, 2002, 211–244. | MR | Zbl
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