Abstract
Let g be a Hecke cusp form of half-integral weight, level 4 and belonging to Kohnen’s plus subspace. Let c(n) denote the nth Fourier coefficient of g, normalized so that c(n) is real for all n≥ 1. A theorem of Waldspurger determines the magnitude of c(n) at fundamental discriminants n by establishing that the square of c(n) is proportional to the central value of a certain L-function. The signs of the sequence c(n) however remain mysterious. Conditionally on the Generalized Riemann Hypothesis, we show that c(n) < 0 and respectively c(n) > 0 holds for a positive proportion of fundamental discriminants n. Moreover we show that the sequence { c(n) } where n ranges over fundamental discriminants changes sign a positive proportion of the time. Unconditionally, it is not known that a positive proportion of these coefficients are non-zero and we prove results about the sign of c(n) which are of the same quality as the best known non-vanishing results. Finally we discuss extensions of our result to general half-integral weight forms g of level 4N with N odd, square-free.
Original language | English |
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Pages (from-to) | 1553-1604 |
Number of pages | 52 |
Journal | Mathematische Annalen |
Volume | 379 |
Issue number | 3-4 |
Early online date | 9 Jan 2021 |
DOIs | |
Publication status | Published - Apr 2021 |