Walter Feit's Conjecture

• Walter Feit's Conjecture is a collection of problems in finite group representation theory that unifies modular forms and number theory by linking global invariants to local data.
• It refines classical results like Bang-Zsigmondy's theorem and extends to function field arithmetic via Carlitz modules and conditions on large Zsigmondy primes.
• The conjecture influences block theory and character invariants, prompting new bounds in decomposition numbers and inspiring reduction theorems that connect local symmetries with global properties.

Walter Feit's Conjecture refers to several closely related problems in finite group representation theory, number theory, and the arithmetic of modular forms, unified by the principle that certain global invariants of irreducible characters are determined—or at least constrained—by explicitly classifiable local data. The conjecture, articulated in different forms by Feit from the 1980s onward, appears in contexts ranging from character conductor bounds to arithmetic properties of prime divisors in exponential and module-theoretic settings. Its formulations and refinements have catalyzed significant research developments in block theory, character theory, modular forms, and the arithmetic of function fields.

1. Historical Background and Classical Setting

Feit’s work builds upon classical results such as Bang-Zsigmondy’s theorem, which concerns primitive prime divisors of integers of the form $u^m - 1$ and their existence outside certain exceptional cases. In 1988, Feit refined this theorem by introducing the notion of "large" Zsigmondy primes: a Zsigmondy prime $p$ for $(u, m)$ is called large if $p > m+1$ or $p^2$ divides $u^m - 1$. He then proved that for any integers $u, m > 1$, such large Zsigmondy primes exist, with finitely many, explicitly enumerated exceptions. This result exhibits a deep analogy between number theoretic properties and group-theoretic phenomena, especially concerning the existence and nature of new prime divisors related to cyclotomic constructions (Nguyen, 2015).

2. Formulations in Function Field Arithmetic

The function field analogue of Feit’s theorem is formulated in the framework of Carlitz modules over $\mathbb{F}_q[T]$. Here, the role of $a^n - 1$ is played by Carlitz polynomial values $C_m(u)$, with corresponding notions of Zsigmondy and large Zsigmondy primes. A monic prime $p$ (i.e., monic irreducible in $\mathbb{F}_q[T]$) is termed a Zsigmondy prime for $(u, m)$ if $p \mid C_m(u)$ while $p \nmid C_n(u)$ for all $n \neq 0$ with $\deg(n) < \deg(m)$. The "large" condition is that $\deg(p) > \deg(m)$ or $p^2 \mid C_m(u)$. The main theorem in this setting asserts:

$$\text{For } q > 2 \text{ and monic } m,u \in \mathbb{F}_q[T] \text{ with } \max\{\deg(m),\deg(u)\} > 0, \text{ there exists a large Zsigmondy prime for } (u, m),$$

except for a short, explicitly classified list of exceptional cases (labeled EC-I through EC-IX) (Nguyen, 2015).

This result not only provides a rigorous function field parallel to Feit’s refinement of Bang-Zsigmondy but also enumerates all scenarios in which the phenomenon fails, deepening the analogy between abelian extensions of number fields and Drinfeld modules.

3. Representation Theory and Character Invariants

Feit’s conjecture intersects modular representation theory most notably in problems involving decomposition numbers and Cartan invariants of $p$-blocks in finite group algebras. Initially, Brauer asked if Cartan invariants $c_{xy}$ are invariably less than $p^d$ (where $d$ is the defect of the block), but counterexamples were found. Feit then proposed a weaker bound, asking if all decomposition numbers $d_{xy}$ satisfy $(d_{xy})^2 < p^d$ (Feit’s Problem VIII). This, too, is now known to fail for $p=2$, as counterexamples in groups such as $\mathrm{PSP}_4(4).4$ and $\mathrm{Sz}(32).5$ demonstrate $(d_{xy})^2$ significantly exceeding $2^{10}$ (Navarro et al., 2016).

Table: Summary of Feit's Problem VIII and Its Failure

Quantity	Feit's Bound	Counterexample Value	Context
$(d_{xy})^2$ (PSP₄(4).4)	$< 2^{10}=1024$	$44^2=1936$	Principal 2-block
$(d_{xy})^2$ (Sz(32).5)	$< 2^{10}=1024$	$47^2=2209$	Principal 2-block

These findings necessitate more nuanced bounds and indicate structural differences in block and decomposition theory for small primes, especially $p=2$. Conversely, such counterexamples have not been observed for odd primes, suggesting potential asymmetry in the arithmetic of $p$-blocks (Navarro et al., 2016).

4. Local-Global Principles and Reduction Theorems

A contemporary generalization is articulated in the reduction theorem for Feit’s conjecture: if all simple groups associated with a finite group $G$ satisfy an "inductive Feit condition," then Feit's conjecture holds for $G$ (Boltje et al., 25 Jul 2025). The inductive Feit condition formalizes the principle that for every irreducible character $\chi$ of a quasi-simple group $X$, there exists a proper subgroup $U$ and a character $\mu$ such that key invariants (like the field of character values and Galois stabilizers) are matched locally between $(X,\chi)$ and $(U,\mu)$. This mechanism parallels the philosophy underpinning Broué's abelian defect group conjecture and the construction of perfect isometries in block theory, notably verified for double covers of symmetric and alternating groups in work by Livesey.

A perfect isometry is a group isomorphism between generalized character lattices that preserves scalar products up to sign and maintains the equality of character fields. Such isometries concretely demonstrate that conductors and related invariants of global characters are controlled by their local analogues in specific subgroups, strongly affirming the reductionist perspective advocated by Feit (Boltje et al., 25 Jul 2025).

5. Applications to Hermitian Lattice Theory and Automorphic Forms

Feit’s classification of unimodular Hermitian lattices, especially of rank 12 over Eisenstein integers, provides a geometric context for examining automorphic forms via algebraic modular forms on unitary groups. Recent research constructs bases of Hecke eigenforms on the genus of 20 such lattices and computes Hecke eigenvalues, which are then "explained" by conjectural global Arthur parameters. Explicit formulae such as

$$\lambda_i(T_p) = (Np)^{11/2}\cdot \mathrm{Tr}(t_p(\pi_i)) + \frac{p^{12}-1}{p+1}$$

connect these eigenvalues to the spectral data of automorphic representations. Furthermore, the paper of congruences among Hecke eigenvalues features analogues of classical modular phenomena (Ramanujan congruence, level-raising). Notably, congruences between automorphic forms created via Eisenstein series and genuine cusp forms—especially on unitary groups $\mathbb{U}(2,2)$—extend conjectures of Harder and suggest further arithmetic regularities aligned with Feit’s original intuition (Dummigan et al., 2018).

6. Exceptional Cases and Their Classification

In both number-theoretic and function field manifestations, Feit’s conjecture admits finitely many explicit exceptions. For instance, in the function field Carlitz module setting, exceptions occur only for small fields or special forms of $u$ and $m$:

• $q=3$, $u=1$, $m=(p-1)\cdot p$ for $p$ a monic degree one prime in $\mathbb{F}_3[T]$
• $q=22$, $u=1$, $m=p-1$ for $p$ a monic degree one prime in $\mathbb{F}_{22}[T]$

and several other enumerated cases labeled EC-III through EC-IX (Nguyen, 2015). Complete classification of these exceptions assures that the failure of the existence of "new" or "large" primes (or primitive divisors) is a rare, well-understood phenomenon, analogous to exceptional cases in cyclotomic and group-theoretic constructions.

7. Broader Implications and Future Directions

The suite of results around Feit’s conjecture demonstrates a profound unification of arithmetic and group-theoretical properties, where local data—in subgroup representations or function field sectors—controls global invariants. The current frontier involves extending reduction theorems to broader families of finite groups, refining bounds for decomposition numbers and Cartan invariants, and deepening the analogy between automorphic representations of function fields and number fields.

Advances in computational representation theory, explicit construction of isometries, analysis of modular congruences, and the enumeration of exceptions collectively inform a sharpened understanding of how local symmetries and arithmetic forms orchestrate the behavior of global invariants in algebraic systems. Continued progress is anticipated in the generalization of reduction theorems, modular representation theory for odd primes, and explicit arithmetic classification via automorphic and geometric methods.