Universality of the cokernels of random 𝑝-adic Hermitian matrices

• DOI: 10.1090/tran/9031
• Publication Date: 2023-07-27T17:39:35Z
• Authors: Jungin Lee
• Keywords: Mathematics - Number Theory; Probability (math.PR); FOS: Mathematics; Number Theory (math.NT); 0101 mathematics; 01 natural sciences; Mathematics - Probability

In this paper, we study the distribution of cokernel a general random Hermitian matrix over ring integers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {O}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> quadratic extension alttext="upper K"> <mml:mi>K</mml:mi> encoding="application/x-tex">K</mml:annotation> alttext="double-struck Q Subscript p"> <mml:msub> mathvariant="double-struck">Q</mml:mi> <mml:mi>p</mml:mi> </mml:msub> encoding="application/x-tex">\mathbb {Q}_p</mml:annotation> </inline-formula>. For each positive integer alttext="n"> <mml:mi>n</mml:mi> encoding="application/x-tex">n</mml:annotation> </inline-formula>, let X n"> <mml:mi>X</mml:mi> encoding="application/x-tex">X_n</mml:annotation> be alttext="n times <mml:mrow> <mml:mo>×<!-- × --></mml:mo> encoding="application/x-tex">n \times n</mml:annotation> whose triangular entries are independent and their reductions not too concentrated on certain values. We show that always converges to same which does depend choices as right-arrow normal infinity"> <mml:mo stretchy="false">→<!-- → mathvariant="normal">∞<!-- ∞ --></mml:mi> \rightarrow \infty</mml:annotation> provide an explicit formula for limiting distribution. This answers Open Problem 3.16 from ICM 2022 lecture note Wood [<italic>Probability theory groups arising in number theory</italic>, 2022] case