We prove that Hermitian cusp forms of weight k for the Hermitian modular group of degree 2 are determined by their Fourier coefficients indexed by matrices whose determinants are essentially square-free. Moreover, we give a quantitative version of the above result. This is a consequence of the corresponding results for integral weight elliptic cusp forms, which are also treated in this paper.

Using the amplification technique, we prove that 'mass' of the pullback of the Saito-Kurokawa lift of a Hecke eigen form g ∈ S 2k is bounded by k 1− 1 210 +ε . This improves the previously known bound k for this quantity.where v 1 = vol.(SL 2 (Z)\H) and v 2 = vol.(Sp 2 (Z)\H 2 ). Here F g | z=0 , F g | z=0 denotes the Petersson norm of F g | z=0 on SL 2 (Z)\H × SL 2 (Z)\H (see section 2 for more details).

We formulate a precise conjecture about the size of the L ∞ -mass of the space of Jacobi forms on H n × C g×n of matrix index S of size g. This L ∞ -mass is measured by the size of the Bergman kernel of the space. We prove the conjectured lower bound for all such n, g, S and prove the upper bound in the k aspect when n = 1, g ≥ 1. When n = 1 and g = 1, we make a more refined study of the sizes of the index-(old and) new spaces, the latter via the Waldspurger's formula. Towards this and with independent interest, we prove a power saving asymptotic formula for the averages of the twisted central L-values L(1/2, f ⊗ χ D ) with f varying over newforms of level a prime p and even weight k as k, p → ∞ and D being (explicitly) polynomially bounded by k, p. Here χ D is a real quadratic Dirichlet character. We also prove that the size of the space of Saito-Kurokawa lifts (of even weight k) is k 5/2 by three different methods (with or without the use of central L-values), and show that the size of their pullbacks to the diagonally embedded H × H is k 2 . In an appendix, the same question is answered for the pullbacks of the whole space S 2 k , the size here being k 3 .

We prove that a non-zero F in the Atkin-Lehner type Siegel newspace of degree 2 and an odd level N is determined by fundamental Fourier coefficients up to a divisor of N . In particular when the level is odd and square-free we show that F is determined by its fundamental Fourier coefficients.

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