Rasmussen–Tamagawa Conjecture

• The Rasmussen–Tamagawa Conjecture is a finiteness prediction asserting that for fixed field and dimension, only finitely many primes allow abelian varieties with constrained ℓ-power torsion fields.
• It employs ℓ-adic and residual Galois representations, particularly for CM abelian varieties, using techniques like compatible systems and congruence analysis.
• The proof in the abelian (CM) case applies Faltings’ finiteness theorem and Raynaud’s criterion to show that such varieties vanish for sufficiently large primes.

The Rasmussen–Tamagawa Conjecture is a central finiteness conjecture in arithmetic geometry regarding the existence of abelian varieties over number fields with highly constrained torsion and Galois behavior. It predicts that, for fixed field and dimension, such abelian varieties can only occur for finitely many primes, and for sufficiently large primes, no such varieties exist. The conjecture, and especially its CM (complex multiplication) case, has motivated deep investigations into the synergy between Galois representations, compatible systems, geometry of abelian varieties, and reduction theory.

1. Statement and Formulation

Given a number field $K$ and integer $g \geq 1$, let $A(K,g,\ell)$ denote the set of isomorphism classes $[A]$ of $g$-dimensional abelian varieties $A/K$ for which the $\ell$-power torsion fields satisfy

$$K(A[\ell^n]) \subset \text{an } \ell\text{-extension of } K(\mu_\ell)$$

for some $n \geq 1$. The Rasmussen–Tamagawa Conjecture asserts the finiteness of

$A(K,g) = \bigcup_{\ell} A(K,g,\ell)$

and, more precisely, that $A(K,g,\ell)$ is empty for all sufficiently large primes $\ell$.

In the context of abelian varieties with complex multiplication (CM), these are abelian varieties $A$ for which the $\ell$-adic Galois representation

$$\rho_{A,\ell} : G_K \longrightarrow \mathrm{GL}(T_\ell(A)) \simeq \mathrm{GL}_{2g}(\mathbb{Z}_\ell)$$

has abelian image and decomposes as a direct sum of $\mathbb{Z}_\ell$-valued characters: $$\rho_{A,\ell} \simeq \bigoplus_{i=1}^{2g} \chi_\ell^{a_i}.$$ The conjecture then specializes to asking whether there could be such $A$ where the constraint

$$K(A[\ell]) \subset \text{an } \ell\text{-extension of } K(\mu_\ell)$$

holds for infinitely many primes $\ell$.

2. Main Results for Abelian and CM Types

The main theorem of (Ozeki, 2011) proves that if $A$ is a $g$-dimensional abelian variety over $K$ whose $\ell$-adic Galois representation becomes abelian after possibly a finite unramified extension $L/K$ at $\ell$ (i.e., $\rho_{A,\ell}$ has abelian image over $L$) and

$$L(A[\ell]) \text{ is an } \ell\text{-extension of } L(\mu_\ell)$$

then for $\ell$ sufficiently large, such $A$ do not exist. In particular, this encompasses all CM abelian varieties, since their $\ell$-adic Galois representations are always abelian after base change to the field of definition of the endomorphisms.

The proof analyzes the "abelian" case

$A'(K,g,\ell)^{ab} = \{ A/K : \text{%%%%26%%%% abelian, %%%%27%%%% as above} \},$

demonstrating that $A'(K,g,\ell)^{ab} = \varnothing$ for $\ell \gg 0$. This establishes the conjectured finiteness—i.e., that "bad" abelian varieties with constrained $\ell$-power torsion and abelian Galois action can occur only for a finite set of primes.

3. Structural Methods and Compatible Systems

Central to the argument is a fine analysis of compatible systems of $\ell$‑adic representations:

• Strict compatibility: For abelian varieties with abelian Galois action, the compatible system $\{ \rho_{A,\ell} \}$ arises from collections of Hecke characters, each with precise local and global behavior. Specifically, one has decompositions

$$\rho_{A,\ell} \simeq \bigoplus_{i=1}^{2g} \chi_\ell^{m_i}$$

with the $m_i$ determined by the CM type.

• Residual representations: The reduction $$\overline{\rho}_{A,\ell}: G_K \to \mathrm{GL}_{2g}(\mathbb{F}_\ell)$$ splits as a direct sum of powers of the mod $\ell$ cyclotomic character:

$$\overline{\rho}_{A,\ell} \simeq \bigoplus_{i=1}^{2g} \chi_\ell^{a_i},$$

with $a_i \in \mathbb{Z}/(\ell-1)\mathbb{Z}$. This explicit description allows direct control over the eigenvalues of Frobenius elements at various places.

• Rigidity and congruence techniques: By comparing characteristic polynomials and congruence relations for eigenvalues modulo $\ell$, combined with bounds on tame inertia weights (e.g., Caruso's results), the argument shows that the structure of the representation is so rigid that, for large $\ell$, incompatible properties must arise, forcing the set to be empty.

4. Finiteness via Faltings' Theorem and Raynaud's Criterion

The proof tightly integrates geometric and representation-theoretic tools:

• Faltings’ finiteness: The set of isomorphism classes of abelian varieties with good reduction outside a fixed set $S$ is finite. This result, originally used in the proof of the Shafarevich Conjecture, is applied here to control the possibilities for $A$.
• Raynaud’s criterion for semistable reduction: For abelian varieties with full level-$N$ structure at primes not dividing $N$, Raynaud's criterion ensures semistable reduction after a finite extension, thus controlling the reduction types and the behavior of torsion points.
• Translation of reduction properties: The local structure (especially the behavior at places above $\ell$) is translated into precise conditions on the global Galois representation, ultimately producing contradictions for large $\ell$.

5. Consequences, Limitations, and Broader Implications

• The theorem demonstrates that, for abelian varieties whose $\ell$-adic representations are abelian (including all CM varieties), the only possible primes $\ell$ for which prime power torsion fields are so constrained are finitely many.
• This directly validates the Rasmussen–Tamagawa Conjecture in the abelian (particularly the CM) context and gives concrete evidence for its assertion that such "exotic" torsion phenomena cannot persist indefinitely as $\ell$ grows.
• The use of compatible systems and residual congruence analysis not only solves the CM case but provides a methodological blueprint for attacking broader cases, especially where the Galois image control is harder.
• The limitation is that the methods in (Ozeki, 2011) are fundamentally specific to cases where the Galois representations are abelian; in the general non-abelian situation, further progress likely requires new advances in the understanding of $\ell$-adic monodromy and the Sato–Tate or Mumford–Tate groups of general abelian varieties.

6. Key Formulas and Invariant Structures

Some critical formulas and objects in the analysis include:

Object / Notation	Description
$\rho_{A,\ell}$	$\ell$-adic Galois representation on the Tate module
$\chi_\ell : G_K \to \mathbb{Z}_\ell^\times$	$\ell$-adic cyclotomic character
$\overline{\rho}_{A,\ell}$	Residual mod $\ell$ Galois representation
$K(A[\ell])$	Field generated by $\ell$-torsion points
$K(\mu_\ell)$	Cyclotomic extension (adjoining $\ell$-th roots of unity)
$$\overline{\rho}_{A,\ell} \simeq \bigoplus_{i=1}^{2g} \chi_\ell^{a_i}$$	Residual decomposition as direct sum of characters

These structures are fundamental in translating arithmetic and geometric data into the compatible system paradigm.

7. Outlook and Future Research Directions

• The use of compatible systems and structure theorems for Galois representations remains a powerful tool for uniformity and finiteness results in arithmetic geometry.
• Extending the methods to non-abelian settings—where the Galois image is only partially constrained—is a significant, open challenge.
• Further research could explore finer bounds or effectivity in the finiteness statement, particularly in the presence of extra structures such as additional endomorphisms, or for other types of motives.
• Cross-relations with conjectures about p-adic Hodge theory, the Sato–Tate group, and monodromy for families of abelian varieties may yield additional structural insights necessary for addressing the general case.

In summary, the non-existence theorem for CM abelian varieties with constrained prime power torsion for large primes $\ell$ solves the CM case of the Rasmussen–Tamagawa Conjecture and provides a template for further advances in the paper of finiteness phenomena for abelian varieties over number fields.