[Paper Review] A survey around the Hodge, Tate and Mumford-Tate conjectures for abelian varieties

Victoria Cantoral‐Farfán|arXiv (Cornell University)|Feb 26, 2016

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This survey provides a comprehensive overview of the Hodge, Tate, and Mumford-Tate conjectures for abelian varieties, establishing their interconnections and summarizing known results. It demonstrates that the conjunction of the Mumford-Tate and Hodge conjectures implies the Tate conjecture for abelian varieties over number fields, offering a unifying framework for these central conjectures in arithmetic and algebraic geometry.

ABSTRACT

This survey describe Hodge, Tate and Mumford-Tate conjectures for abelian varieties. After some preliminaries on endomorphism ring, polarization and algebraic cycles, we state the three conjectures and provide a list of know results. Finally, we explain some links between these conjectures.

Motivation & Objective

• To synthesize and clarify the Hodge, Tate, and Mumford-Tate conjectures specifically for abelian varieties.
• To explain the deep analogies between these conjectures, particularly the role of Galois and Hodge groups in cohomological invariants.
• To summarize known results and conditions under which the conjectures hold, especially for abelian varieties of low dimension or special type.
• To establish the logical equivalence between the Tate conjecture and the conjunction of the Mumford-Tate and Hodge conjectures for abelian varieties over number fields.
• To provide a unified perspective on how these three conjectures interrelate through cohomological and Galois-theoretic structures.

Proposed method

• Systematically introduces foundational concepts: endomorphism rings, polarizations, Rosati involutions, and Albert's classification of abelian varieties by type.
• States the Hodge conjecture as the claim that Hodge classes in singular cohomology are rational linear combinations of algebraic cycles.
• States the Tate conjecture as the claim that Tate classes in étale cohomology are ℓ-adic linear combinations of algebraic cycles.
• Introduces the Mumford-Tate conjecture as the assertion that the ℓ-adic Galois monodromy group equals the Hodge group base-changed to ℚℓ.
• Uses the analogy between Hodge and Tate classes as invariants under Hg(X) and Hℓ respectively, and compares the groups Hg(X) ⊗ ℚℓ and Hℓ.
• Applies results from Shimura varieties, motives, and the theory of absolute Hodge classes to derive implications between the conjectures.

Experimental results

Research questions

• RQ1 Under what conditions do the Hodge and Mumford-Tate conjectures imply the Tate conjecture for abelian varieties over number fields?
• RQ2 How do the Hodge, Tate, and Mumford-Tate conjectures relate through the structure of Galois and Hodge groups in cohomology?
• RQ3 In which classes of abelian varieties (e.g., CM-type, simple, low-dimensional) are these conjectures known to hold?
• RQ4 What is the role of the Rosati involution and Albert’s classification in understanding the endomorphism structure relevant to the conjectures?
• RQ5 How does the theory of absolute Hodge classes support the implication (T) ⇒ (H), and how does this relate to the full equivalence (MT)+(H) ⇔ (T)?

Key findings

• The Mumford-Tate and Hodge conjectures together imply the Tate conjecture for abelian varieties over number fields, as shown via the equivalence (MT)+(H) ⇔ (T).
• For simple abelian varieties of prime dimension over a number field, the Mumford-Tate conjecture holds, and combined with Tankeev’s result on the Hodge conjecture, this implies the Tate conjecture.
• For CM-type abelian varieties over ℂ with Mumford-Tate group a torus of dimension g+1, the Hodge conjecture holds, and thus the Tate conjecture holds over number fields.
• For abelian varieties of type I or II with odd relative dimension, the Mumford-Tate conjecture holds, and since the Hodge conjecture holds for such varieties (by Murty and Hazama), the Tate conjecture follows.
• The implication (T) ⇒ (H) was proven by Piatetskii-Shapiro using absolute Hodge classes, and this, combined with (H)+(T) ⇒ (MT), establishes the full equivalence (MT)+(H) ⇔ (T).
• The conjectures are deeply linked through the cohomological invariants: Hodge classes are Hg(X)-invariant classes in H^2p(X_ℂ, ℚ), and Tate classes are Hℓ-invariant classes in H^2p(X_k, ℚℓ), with Hℓ ⊆ Hg(X) ⊗ ℚℓ.