Period functions of half-integral weight modular forms
Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 1, pp. 33-45.

In this paper, we study the Eichler cohomology associated with half-integral weight cusp forms using the Dedekind eta function η(z) and the theta function θ(z). We prove that η-multiplication (resp. θ-multiplication) gives an isomorphism between the space of cusp forms of a half-integral weight and the cohomology group associated with the space η𝒫 (resp. θ𝒫). 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 η(z). Nous montrons que la η-multiplication (resp. θ-multiplication) induit un isomorphisme entre l’espace des formes cupsidales de poids demi-entiers et le groupe de cohomologie associé à l’espace η𝒫 (resp. θ𝒫). Nous montrons aussi qu’il existe un isomorphisme entre la somme directe de deux espaces de telles formes et le groupe de cohomologie.

DOI: 10.5802/jtnb.891
Classification: 11F37, 32N10
Keywords: Period functions, Eichler cohomology
Dohoon Choi 1; Subong Lim 2; Wissam Raji 3

1 School of liberal arts and sciences Korea Aerospace University 200-1, Hwajeon-dong, Goyang Gyeonggi 412-791, Republic of Korea
2 Department of Mathematics Education Sungkyunkwan University Joongno-gu Seoul 110-745, Republic of Korea
3 Department of mathematics American University of Beirut Beirut, Lebanon
     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 = {}
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  -
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
%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.

[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 p-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

Cited by Sources: