## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### $p$-adic properties of coefficients of basis for the space of weakly holomorphic modular forms of weight 2

Soyoung Choi

#### Abstract

We observe properties of coefficients of certain basis elements for the space of weakly holomorphic modular forms of weight 2 for $SL_{2}(\mathbf{Z})$. Moreover we show that these coefficients are often highly divisible by the primes 2, 3, 5, 7, 11.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 88, Number 1 (2012), 11-15.

Dates
First available in Project Euclid: 30 December 2011

https://projecteuclid.org/euclid.pja/1325264390

Digital Object Identifier
doi:10.3792/pjaa.88.11

Mathematical Reviews number (MathSciNet)
MR2872209

Zentralblatt MATH identifier
1282.11031

#### Citation

Choi, Soyoung. $p$-adic properties of coefficients of basis for the space of weakly holomorphic modular forms of weight 2. Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 1, 11--15. doi:10.3792/pjaa.88.11. https://projecteuclid.org/euclid.pja/1325264390

#### References

• T. M. Apostol, Modular functions and Dirichlet series in number theory, 2nd ed., Graduate Texts in Mathematics, 41, Springer, New York, 1990.
• D. Doud and P. Jenkins, $p$-adic properties of coefficients of weakly holomorphic modular forms, Int. Math. Res. Not. IMRN 2010, no. 16, 3184–3206.
• W. Duke and P. Jenkins, On the zeros and coefficients of certain weakly holomorphic modular forms, Pure Appl. Math. Q. 4 (2008), no. 4, Special Issue: In honor of Jean-Pierre Serre. Part 1, 1327–1340.
• K. Ono, The web of modularity: arithmetic of the coefficients of modular forms and $q$-series, CBMS Regional Conference Series in Mathematics, 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2004.
• J. Lehner, Ramanujan identities involving the partition function for the moduli $11^{\alpha}$, Amer. J. Math. 65 (1943), 492–520.
• J. Lehner, Divisibility properties of the Fourier coefficients of the modular invariant $j(\tau)$, Amer. J. Math. 71 (1949), 136–148.
• J. Lehner, Further congruence properties of the Fourier coefficients of the modular invariant $j(\tau)$, Amer. J. Math. 71 (1949), 373–386.
• S. Ramanujan, Congruence properties of partitions [Proc. London Math. Soc. (2) 18 (1920), Records for 13 March 1919], in Collected papers of Srinivasa Ramanujan, 230, AMS Chelsea Publ., Providence, RI, 2000.