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The Hodge Conjecture for CM abelian varieties

Primarily human-written textHuman understanding: all partsmath.AG — Algebraic Geometrymath.NT — Number Theory

Contributed by Pol van Hoften ↗, Jack Sempliner ↗

Version 1 / Oct 15, 2026 / CC BY 4.0

Abstract

Following the recent work [Ope26], we give a proof of the Hodge conjecture for CM abelian varieties.

Provenance statement

This article is entirely human written. That said, we made extensive use of ChatGPT Pro during the preparation of this argument. We asked the model to give a version of the arguments in [Ope26, Section 2] that relies on [Haz03b], and this formed the starting point of our Section 2. Moreover, Lemma 2.3.2 and its proof were suggested to us by that model. We asked the model for a more direct proof of [Ope26, Proposition 3.1], which lead our proof of Theorem 2.3.4. When we asked the model for a reference to compute the first Betti cohomology of Picard modular surfaces, it suggested [Liu21], and we wrote Section 3 without further assistance except for the proof of Lemma 3.1.5 and Corollary 3.1.13, which were also suggested to us by the model. Section 4 was written after extensive back-and-forth with the model, in which the authors insisted that the corresponding calculation in [Ope26] should be done instead using the doubling method and recourse to the Siegel–Weil formula. In particular, Lemma 4.2.7 was eventually output after this back and forth with the model, the current proof was then worked out after a further extensive back and forth. The basic idea of varying the test function as is carried out in Lemma 4.2.9 was also suggested by the model. In addition we used the model in more quotidian ways: to proofread the document and check the arguments once the paper was assembled, and to do a literature search for the introduction.

Tools used

OpenAI
ChatGPTVersion 6.1

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Version history

  1. v1Submitted by Pol van HoftenInitial depositCurrentOct 15, 2026