MOMENTS OF MARGULIS FUNCTIONS AND INDEFINITE TERNARY QUADRATIC FORMS
김성민
129동 301호
0
1709
2025.08.11 13:48
| 구분 | 정수론-동역학 |
|---|---|
| 일정 | 2025-08-12(화) 10:00~12:30 |
| 세미나실 | 129동 301호 |
| 강연자 | 김우연 (고등과학원) |
| 담당교수 | 임선희 |
| 기타 |
* 시간: 10:30-12:00
In this paper, we prove a quantitative version of the Oppenheim conjecture for indefinite ternary quadratic forms: for any indefinite irrational ternary quadratic form Q that is not extremely well approxiable by rational forms, and for a ă b the number of integral vectors of norm at most T satisfying a ă Qpvq ă b is asymptotically equivalent to `CQpb ́ aq ` IQpa, bq ̆Tas T tends to infinity, where the constant CQ ą 0 depends only on Q, and the term IQpa, bqTaccounts for the contribution from rational isotropic lines and degenerate planes.
The main technical ingredient is a uniform bound for the λ-moment of the Margulis α-function along expanding translates of a unipotent orbit in SL3pRq{ SL3pZq, for some λ ą 1. To establish this, we introduce a new height function rα on the space of lattices, which captures the failure of the classical Margulis inequality. This moment bound implies equidistribution of such translates with respect to a class of unbounded test functions, including the Siegel transform.
