---
title: "Un peu d'effectivite pour les varietes modulaires de Hilbert-Blumenthal"
authors:
  - "Levent Alpoge"
year: 2021
venue: "arXiv preprint"
arxiv: "2111.12466"
doi: null
source: "https://arxiv.org/abs/2111.12466"
topics:
  - "arithmetic-geometry"
  - "number-theory"
seed_rank: 1203
seed_batch: "alpoge-math-2026-08-24"
reviewed: "2026-08-24"
pool: "maths-foundations"
relevance_score: 9
lineage: arithmetic-geometry
cites:
  - title: "Modularity and effective Mordell I"
    url: "https://arxiv.org/abs/2109.07917"
    year: 2021
    arxiv: "2109.07917"
    doi: null
see:
  []
---

# Un peu d'effectivite pour les varietes modulaires de Hilbert-Blumenthal

## One-sentence takeaway

Height-free effective isogeny bounds for abelian varieties of GL_2-type imply an effective upper bound on the number of S-integral K-points on a Hilbert modular variety.

## Why it's lovely

Effectivity without heights: the isogeny degree between K-isogenous GL_2-type A,B with good reduction outside S is bounded by g,K,S only.

## Problem

Effective Shafarevich / integral points on Hilbert-Blumenthal modular varieties.

## Design

Isogeny estimate for GL_2-type, then deduce the integral-point bound.

## Evidence

arXiv:2111.12466, 24 Nov 2021. Solo preprint.

## Limitations

GL_2-type only. Companion to 1206.

## Implications for Broadside

The Hilbert-modular effectivity card. See 1206 for Faltings on curves mapping to those stacks.

## Bottom line

Mint 2111.12466.

## Links

- arXiv: [2111.12466](https://arxiv.org/abs/2111.12466)