Normal distribution of bad reduction

Robert J. Lemke Oliver; Daniel Loughran; Ari Shnidman

doi:10.1007/s11139-025-01108-4

Normal distribution of bad reduction

Robert J. Lemke Oliver, Daniel Loughran, Ari Shnidman

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

We prove normal distribution laws for primes of bad semistable reduction in families of curves. As a consequence, we deduce that when ordered by height, 100% of curves in these families have, in a precise sense, many such primes.

Original language	English
Article number	52
Journal	The Ramanujan Journal
Volume	67
Issue number	3
Early online date	22 May 2025
DOIs	https://doi.org/10.1007/s11139-025-01108-4
Publication status	Published - Jul 2025

Bibliographical note

Publisher Copyright:
© The Author(s) 2025.

Data Availability Statement

Not applicable.

Funding

Robert Lemke Oliver is supported by the National Science Foundation grant DMS-2200760 and by a Simons Fellowship in Mathematics. Daniel Loughran is supported by UKRI Future Leaders Fellowship MR/V021362/1. Ari Shnidman was funded by the European Union (ERC, CurveArithmetic, 101078157) and the Israel Science Foundation (grant No. 2301/20).

Funders	Funder number
European Commission
National Science Foundation	DMS-2200760
UK Research & Innovation	MR/V021362/1
European Research Council	101078157
Israel Science Foundation	2301/20

Keywords

Arithmetic of curves
Bad reduction
Probabilistic number theory
Rational points

ASJC Scopus subject areas

Algebra and Number Theory

Access to Document

10.1007/s11139-025-01108-4Licence: CC BY

https://arxiv.org/abs/2305.15874Licence: CC BY

Cite this

@article{badda7c0d97e422fb5876fe6d342a212,

title = "Normal distribution of bad reduction",

abstract = "We prove normal distribution laws for primes of bad semistable reduction in families of curves. As a consequence, we deduce that when ordered by height, 100\% of curves in these families have, in a precise sense, many such primes.",

keywords = "Arithmetic of curves, Bad reduction, Probabilistic number theory, Rational points",

author = "\{Lemke Oliver\}, \{Robert J.\} and Daniel Loughran and Ari Shnidman",

note = "Publisher Copyright: {\textcopyright} The Author(s) 2025.",

year = "2025",

month = jul,

doi = "10.1007/s11139-025-01108-4",

language = "English",

volume = "67",

journal = "The Ramanujan Journal",

issn = "1382-4090",

publisher = "Springer Netherlands",

number = "3",

}

TY - JOUR

T1 - Normal distribution of bad reduction

AU - Lemke Oliver, Robert J.

AU - Loughran, Daniel

AU - Shnidman, Ari

PY - 2025/7

Y1 - 2025/7

N2 - We prove normal distribution laws for primes of bad semistable reduction in families of curves. As a consequence, we deduce that when ordered by height, 100% of curves in these families have, in a precise sense, many such primes.

AB - We prove normal distribution laws for primes of bad semistable reduction in families of curves. As a consequence, we deduce that when ordered by height, 100% of curves in these families have, in a precise sense, many such primes.

KW - Arithmetic of curves

KW - Bad reduction

KW - Probabilistic number theory

KW - Rational points

UR - https://www.scopus.com/pages/publications/105005806041

U2 - 10.1007/s11139-025-01108-4

DO - 10.1007/s11139-025-01108-4

M3 - Article

SN - 1382-4090

VL - 67

JO - The Ramanujan Journal

JF - The Ramanujan Journal

IS - 3

M1 - 52

ER -

Normal distribution of bad reduction