31 research outputs found
Mass equidistribution of Hilbert modular eigenforms
Let F be a totally real number field, and let f traverse a sequence of
non-dihedral holomorphic eigencuspforms on GL(2)/F of weight (k_1,...,k_n),
trivial central character and full level. We show that the mass of f
equidistributes on the Hilbert modular variety as max(k_1,...,k_n) tends to
infinity.
Our result answers affirmatively a natural analogue of a conjecture of
Rudnick and Sarnak (1994). Our proof generalizes the argument of
Holowinsky-Soundararajan (2008) who established the case F = Q. The essential
difficulty in doing so is to adapt Holowinsky's bounds for the Weyl periods of
the equidistribution problem in terms of manageable shifted convolution sums of
Fourier coefficients to the case of a number field with nontrivial unit group.Comment: 40 pages; typos corrected, nearly accepted for
Equidistribution of zeros of holomorphic sections in the non compact setting
We consider N-tensor powers of a positive Hermitian line bundle L over a
non-compact complex manifold X. In the compact case, B. Shiffman and S.
Zelditch proved that the zeros of random sections become asymptotically
uniformly distributed with respect to the natural measure coming from the
curvature of L, as N tends to infinity. Under certain boundedness assumptions
on the curvature of the canonical line bundle of X and on the Chern form of L
we prove a non-compact version of this result. We give various applications,
including the limiting distribution of zeros of cusp forms with respect to the
principal congruence subgroups of SL2(Z) and to the hyperbolic measure, the
higher dimensional case of arithmetic quotients and the case of orthogonal
polynomials with weights at infinity. We also give estimates for the speed of
convergence of the currents of integration on the zero-divisors.Comment: 25 pages; v.2 is a final update to agree with the published pape
On the sup-norm of SL(3) Hecke-Maass cusp forms
This work contains a proof of a non-trivial explicit quantitative bound in the eigenvalue aspect for the sup-norm of a SL(3,Z) Hecke-Maass cusp form restricted to a compact set