Larger sieve with height function and uniform bounds for integral points on curves over number fields

2026/07/27 by Saunak Bhattacharjee
#math.NT

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Abstract

Let A ⊆ O_K be a set of algebraic integers of height up to H such that |A \mod\mathfrakp|≤ α|O_K/\mathfrakp| for every prime ideal \mathfrakp with N\mathfrakp>c for some α∈ (0,1). It follows from a larger sieve due to Ellenberg, Elsholtz, Hall and Kowalski that |A| ≪_K,c,αH^2α. In this paper, we improve on this larger sieve bound by showing that |A|≪_K,c,αH^α(log H)^r. We also obtain a two-dimensional larger sieve of Helfgott and Venkatesh type over O_K × O_K and apply it to produce a Bombieri-Pila type bound over O_K.

Citations

• Polynomial values modulo primes on average and sharpness of the larger sieve
• Inverse questions for the large sieve
• Bounded rational points on curves
• Hilbert's irreducibility theorem and the larger sieve

Related

• Integral Points on Elliptic Curves and the Bombieri-Pila Bounds
• Polynomial values modulo primes on average and sharpness of the larger sieve
• The principle of the large sieve
• A quadratic large sieve inequality over number fields
• Geometric generalizations of the square sieve, with an application to cyclic covers