Mod p Monodromy of Cyclic Covers of the Projective Line

Abstract: In this paper, we prove a big monodromy theorem for the monodromy of cyclic coverings of projective line for cohomology with Fp-coefficients. This is a direct generalization of the results of Achter and Pries, where such a theorem is proved for cyclic coverings of degree 2 and 3. Instead of generalizing their methods, we adapt the proof of the analogous theorem for integral cohomology. In our subsequent work, we will apply this theorem to construct in infinitely many cases Galois extensions of Q with Galois group PSL(n, q) and PSU(n, q), where q can be an arbitrarilty large prime power.

• The paper demonstrates that mod p cohomology of cyclic covers exhibits full monodromy, revealing a dichotomy between unitary and linear cases.
• It leverages the reduced Gassner representation to prove irreducibility and primitivity of the monodromy, ensuring comprehensive group actions.
• The findings have direct applications in inverse Galois theory by enabling explicit realizations of finite simple groups as Galois groups.

Mod $p$ Monodromy for Cyclic Covers of $\mathbf{P}^1$

Overview and Main Results

The paper "Mod $p$ Monodromy of Cyclic Covers of the Projective Line" (2604.27391) addresses the monodromy representations arising from the cohomology with $\mathbf{F}_p$-coefficients of cyclic branched covers of the complex projective line $\mathbf{P}^1$. The author generalizes big monodromy results known for cyclic covers of degree 2 and 3, extending them to arbitrary odd prime degree $l$. Notably, the work repurposes the proof techniques used in the integral cohomology case (as opposed to directly extending the mod-$p$ approaches used for small degrees) and establishes comprehensive monodromy theorems for mod-$p$ cohomological representations of the pure braid group acting via monodromy on families of cyclic covers.

A significant application envisaged for these results is in the explicit realization of certain finite simple groups, specifically $\operatorname{PSL}(n,q)$ and $\operatorname{PSU}(n,q)$, as Galois groups over $\mathbf{P}^1$0 through the inverse Galois correspondence, using families of Jacobians of these covers and their torsion subgroups.

Geometric and Cohomological Setting

Cyclic covers of $\mathbf{P}^1$1 over $\mathbf{P}^1$2 are considered, defined as smooth models of affine curves with equations of the form

$\mathbf{P}^1$3

with $\mathbf{P}^1$4 an odd prime, $\mathbf{P}^1$5 distinct and $\mathbf{P}^1$6 integers in $\mathbf{P}^1$7. The configuration space parametrizes the unordered tuples of branch points. For each choice of branch data, the universal family of such covers inherits a natural monodromy representation of the Artin pure braid group $\mathbf{P}^1$8 on $\mathbf{P}^1$9, where $p$0.

The work focuses on describing the image of this monodromy representation, accomplishing this by analyzing the specialized reduced Gassner representation associated with the action of $p$1.

Structure and Image of the Monodromy Representation

The key theorem states that for $p$2, the image of $p$3 acting on the relevant component of $p$4 is either the group

• $p$5 (the special unitary group) or
• $p$6 (the special linear group), depending on explicit arithmetic data relating to the residue field $p$7 arising from $p$8 modulo $p$9.

This dichotomy—unitary vs. linear case—is determined by whether the relevant prime splits or remains inert in the extension $\mathbf{F}_p$0. The analysis carefully discusses the reduction of the hermitian form and the explicit formation of the coefficient field $\mathbf{F}_p$1, which may be either $\mathbf{F}_p$2 or $\mathbf{F}_p$3.

Gassner Representation Approach and Methodology

Rather than generalizing the cohomological methods from the degree-2 and 3 case, the proof strategy adapts the integral approach using the Gassner representation, a well-understood linear representation of the pure braid group with deep connections to cohomology via its specialization.

A detailed analysis is performed regarding the irreducibility, invariants, and unipotent radical images within the specialized reduced Gassner representation. By investigating transvections in the monodromy group's image and exploiting the relationship between Gassner representations under degeneracy of hermitian forms, the author is able to utilize results of Zalesskii and others on groups generated by transvections to demonstrate the largeness (in fact, fullness) of the monodromy image under suitable hypotheses.

Numerical and Structural Claims

• Irreducibility and Primitivity: The paper provides rigorous proofs that, under the specified conditions, the relevant Gassner representation is absolutely irreducible and primitive, ruling out proper invariant subspaces or factorizations over subfields.
• Full Monodromy: For $\mathbf{F}_p$4, the monodromy image is the entire special unitary or special linear group (depending on the parity and splitting conditions of primes), with the determinants of the monodromy images exhaustively covering the corresponding group of $\mathbf{F}_p$5-th roots of unity.

Implications for Inverse Galois Theory

The results have direct implications for the inverse Galois problem. The constructed monodromy representations can be used to realize $\mathbf{F}_p$6 and $\mathbf{F}_p$7 as Galois groups over $\mathbf{F}_p$8 in infinitely many new cases, as the large monodromy ensures the surjectivity needed for the Galois realizations. The limitation is set by congruence and ramification conditions, but the techniques reach further than those depending solely on integral or small-degree methods.

Theoretical Significance and Future Directions

The paper not only provides a structural generalization but also a methodological advancement in approaching mod-$\mathbf{F}_p$9 monodromy problems, bridging techniques between integral and modular settings in a context of arithmetic algebraic geometry and group theory. The explicitness of the group-theoretic images strengthens the arsenal of tools available for constructing Galois representations with prescribed properties via geometry.

Further research directions include:

• The explicit construction and arithmetic study of the resulting Galois extensions;
• Refinements to monodromy computations for other types of covers or more general moduli problems;
• Applications to the study of arithmetic monodromy vs. geometric monodromy and related questions on the Galois action on torsion subgroups of Jacobians.

Conclusion

This work completes a thorough generalization of large monodromy theorems to the mod-$\mathbf{P}^1$0 cohomology for cyclic covers of the projective line of arbitrary prime degree, providing explicit structural descriptions of monodromy group images that in turn have concrete applications to the inverse Galois problem. By blending techniques from geometric group theory, representation theory, and arithmetic geometry, the paper advances both the theoretical framework and practical feasibility of constructing Galois extensions with large and controllable Galois groups (2604.27391).