CONGRUENCE PROPERTIES OF BORCHERDS PRODUCT EXPONENTS

In his striking 1995 paper, Borcherds [Automorphic forms on Os+2,2(ℝ) and infinite products, Invent. Math.120 (1995) 161–213] found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant -d evaluated at the modular j-function. Among a number of powerful generalizations of Borcherds' work, Zagier made an analogous statement for twisted versions of this polynomial. He proves that the exponents of these product expansions, A(n,d), are the coefficients of certain special half-integral weight modular forms. We study the congruence properties of A(n,d) modulo a prime ℓ by relating it to a modular representation of the logarithmic derivative of the Hilbert class polynomial.

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THE SIGN OF FOURIER COEFFICIENTS OF HALF-INTEGRAL WEIGHT CUSP FORMS

International Journal of Number Theory ◽

10.1142/s179304211250042x ◽

2012 ◽

Vol 08 (03) ◽

pp. 749-762 ◽

Cited By ~ 11

Author(s):

THOMAS A. HULSE ◽

E. MEHMET KIRAL ◽

CHAN IEONG KUAN ◽

LI-MEI LIM

Keyword(s):

Cusp Form ◽

Modular Forms ◽

Fourier Coefficients ◽

Cusp Forms ◽

Square Root ◽

Critical Strip ◽

Half Integral Weight ◽

Integral Weight ◽

Finite Set ◽

Invent Math

From a result of Waldspurger [W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math.64 (1981) 175–198], it is known that the normalized Fourier coefficients a(m) of a half-integral weight holomorphic cusp eigenform 𝔣 are, up to a finite set of factors, one of [Formula: see text] when m is square-free and f is the integral weight cusp form related to 𝔣 by the Shimura correspondence [G. Shimura, On modular forms of half-integral weight, Ann. of Math.97 (1973) 440–481]. In this paper we address a question posed by Kohnen: which square root is a(m)? In particular, if we look at the set of a(m) with m square-free, do these Fourier coefficients change sign infinitely often? By partially analytically continuing a related Dirichlet series, we are able to show that this is so.

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ARITHMETIC TRACES OF NON-HOLOMORPHIC MODULAR INVARIANTS

International Journal of Number Theory ◽

10.1142/s1793042110002818 ◽

2010 ◽

Vol 06 (01) ◽

pp. 69-87 ◽

Cited By ~ 11

Author(s):

ALISON MILLER ◽

AARON PIXTON

Keyword(s):

Modular Form ◽

Modular Forms ◽

Fourier Coefficients ◽

Algebraic Numbers ◽

Heegner Points ◽

Positive Weight ◽

Poincare Series ◽

Half Integral Weight ◽

Integral Weight ◽

Modular Invariants

We extend results of Bringmann and Ono that relate certain generalized traces of Maass–Poincaré series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences [Formula: see text] and [Formula: see text]. We show that if f is a modular form of non-positive weight 2 - 2 λ and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms [Formula: see text].

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