Journal of the Ramanujan Mathematical Society
Volume 40, Issue 1, March 2025 pp. 29–41.
A central limit theorem for Hilbert modular forms
Authors:
Jishu Das and Neha Prabhu
Author institution:Indian Institute of Science Education and Research Thiruvananthapuram, Kerala, India.
Summary:
For a prime ideal 𝔭 in a totally real number field L with the adele
ring 𝔸, we study the distribution of angles θ{π}(𝔭)
coming from Satake parameters corresponding to unramified π{𝔭}
where π{𝔭} comes from a global π ranging over a certain
finite set Π{k}(𝔫) of cuspidal automorphic representations of
GL{2}(𝔸) with trivial central character. For such a representation
π, it is known that the angles θ{π}(𝔭) follow the
Sato-Tate distribution. Fixing an interval I ⊆ [0,{π}], we prove a
central limit theorem for the number of angles θ{π}(𝔭) that
lie in I, as N(𝔭) → ∞. The result assumes 𝔫 to be
a squarefree integral ideal, and that the components in the weight vector k
grow suitably fast as a function of x.
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