Counting Lattice Points on Determinant Surfaces

Guria, Rachita (2023) Counting Lattice Points on Determinant Surfaces. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Rachita Guria (18RS067)) 18RS067.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

In this thesis we obtain asymptotic formulae for three counting problems. First, for the number of integer solutions (a, b, c, d) to the determinant equations xy −zw = r ̸= 0 lying in an expanding box [−X,X]⁴ with X −→ ∞. Second, for the same counting problem, where we now count the solutions with smooth weights of the form w(a/X),w(b/X), · · · , where w is a compactly supported smooth function; and third, for the number of 2 × 2 integer matrices with fixed trace and determinant where we count with smooth weights as in the second problem.

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