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This article introduces a new family of weight-shifting differential operators, called theta linkage maps, on the Siegel threefold to study the weight part of Serre's conjecture for GSp4. These maps, constructed from maps of Verma modules in characteristic p, exhibit a linkage pattern between Serre weights and provide insights into the structure of Breuil-Mezard cycles. The author demonstrates a generic entailment of Serre weights, showing that a Hecke eigenform with a generic Serre weight in the lowest alcove also possesses a Serre weight in one of the upper alcoves. Additionally, the article presents progress towards proving a case of Serre's conjecture where the Serre weight is dependent on the local Galois representation, allowing for non-semisimple representations.

Abstract

**Bibliographic Information:**Ortiz, M. (2024). Theta linkage maps and a generic entailment for GSp4. arXiv preprint arXiv:2410.09602v1.**Research Objective:**This paper investigates the weight part of Serre's conjecture for GSp4 by introducing a new family of weight-shifting differential operators called "theta linkage maps." These maps are used to establish relationships between Serre weights and provide insights into the structure of Breuil-Mezard cycles.**Methodology:**The author employs techniques from algebraic geometry and representation theory. They construct theta linkage maps by relating them to maps of Verma modules in characteristic p. These maps are then used to analyze the decomposition of Weyl modules into Serre weights and study the weight part of Serre's conjecture.**Key Findings:**- The paper introduces a new family of weight-shifting differential operators called "theta linkage maps" on the Siegel threefold.
- These maps exhibit a linkage pattern between Serre weights, reflecting the linkage relations in representation theory.
- The author proves a generic entailment of Serre weights, demonstrating that a Hecke eigenform with a generic Serre weight in the lowest alcove also has a Serre weight in one of the upper alcoves.
- The paper makes progress towards proving a case of Serre's conjecture where the Serre weight depends on the local Galois representation, allowing for non-semisimple representations.

**Main Conclusions:**The theta linkage maps provide a powerful tool for studying the weight part of Serre's conjecture for GSp4. The generic entailment result and the progress made on the case with a non-semisimple Galois representation demonstrate the potential of this approach.**Significance:**This research contributes significantly to the understanding of Serre's conjecture, a fundamental problem in the Langlands program. The introduction of theta linkage maps and their connection to Breuil-Mezard cycles provide new avenues for future research in this area.**Limitations and Future Research:**The paper primarily focuses on GSp4, and further research is needed to extend these results to other groups. Additionally, investigating the finer properties of theta linkage maps, such as their kernels and behavior in cohomology, remains an open question.

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by Martin Ortiz at **arxiv.org** 10-15-2024

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The concept of theta linkage maps, as introduced in the context of GSp4, can be generalized to other reductive groups using the following blueprint:
Flag Variety and Line Bundles: Start with a Shimura variety associated with a reductive group $G$ and consider its flag variety. Define a suitable family of automorphic line bundles on this flag variety, analogous to the $L(k,l)$ for GSp4. These line bundles should capture the weight information relevant to the representation theory of $G$.
Basic Theta Operators: Construct "basic" theta operators between these line bundles. These operators, often arising from geometric constructions like Kodaira-Spencer maps or Hasse invariants, will typically induce weight shifts related to the roots of $G$. The specific form of these operators will depend on the geometry of the Shimura variety and its flag variety.
Verma Modules and Linkage: Establish a connection between maps of Verma modules for $G$ in characteristic $p$ and differential operators on the flag variety. This connection, generalizing the one described in the paper for GSp4, will likely involve the Harish-Chandra isomorphism and the distribution algebra of $G$. The linkage relations between weights in characteristic $p$ will then dictate the existence of "theta linkage maps" as the images of these maps between Verma modules.
Insights into Serre's Conjecture:
Generalizing theta linkage maps to other reductive groups could offer several new insights into the weight part of Serre's conjecture:
Generic Entailments: Similar to the GSp4 case, theta linkage maps could reveal "generic entailment" patterns among Serre weights for $G$. These patterns would provide information about which Serre weights are forced to appear in the weight set of a mod $p$ Galois representation given that another Serre weight appears.
Structure of Breuil-Mezard Cycles: The connection between theta linkage maps and Breuil-Mezard cycles might extend beyond GSp4. Understanding this connection for more general groups could provide a powerful tool for studying the geometry of moduli spaces of $p$-adic Galois representations and their relationship to automorphic forms.
New Cases of Serre's Conjecture: Explicit constructions of theta linkage maps could lead to proofs of new cases of the weight part of Serre's conjecture for $G$, particularly for non-ordinary Galois representations where traditional methods are less effective.
However, generalizing this approach to other groups comes with challenges:
Geometric Complexity: The geometry of Shimura varieties and their flag varieties can be significantly more complicated for groups beyond GSp4, making the construction of basic theta operators and the analysis of their properties more challenging.
Representation Theory: The representation theory of reductive groups in characteristic $p$ is generally more intricate than in characteristic zero, and the linkage relations between weights can be more subtle.
Despite these challenges, the potential rewards of generalizing theta linkage maps make it a promising avenue for future research in the Langlands program.

Yes, alternative constructions or interpretations of theta linkage maps could provide valuable insights:
Geometric Perspectives:
Crystalline Cohomology: Theta linkage maps might be related to morphisms between the crystalline cohomology groups of different strata in the Ekedahl-Oort stratification of the Shimura variety. This perspective could connect them to the geometry of these strata and potentially to the theory of $F$-crystals.
Moduli of $p$-Divisible Groups: Interpreting theta linkage maps in terms of the geometry of moduli spaces of $p$-divisible groups could provide a more concrete understanding of their action on modular forms. This approach might also connect them to the theory of Dieudonné modules and $p$-adic Hodge theory.
Geometric Langlands: Theta linkage maps could potentially be understood within the framework of the geometric Langlands program. This perspective might relate them to sheaves on moduli spaces of bundles and provide a more conceptual explanation for their existence and properties.
Representation-Theoretic Perspectives:
Hecke Algebras: Theta linkage maps should induce operators on spaces of automorphic forms, which are modules over global and local Hecke algebras. Studying these operators from a purely algebraic perspective, focusing on their interaction with Hecke operators, could reveal new information about their structure and properties.
$p$-Adic Hodge Theory: Theta linkage maps might have interpretations within $p$-adic Hodge theory, potentially relating them to period rings and comparison isomorphisms. This perspective could provide a deeper understanding of their connection to Galois representations.
Derived Categories: Working in the derived category of sheaves on the flag variety could provide a more flexible framework for studying theta linkage maps. This approach might reveal connections to other important objects in the derived category, such as perverse sheaves and intersection cohomology.
Exploring these alternative perspectives could lead to a more comprehensive understanding of theta linkage maps, their properties, and their applications within the Langlands program and beyond.

The connection between theta linkage maps and Breuil-Mezard cycles has profound implications for our understanding of:
Geometry of Moduli Spaces of Galois Representations:
Irreducible Components: The existence of theta linkage maps between weights $\lambda$ and $\mu$ suggests a close relationship between the irreducible components of Breuil-Mezard cycles $Z_\lambda$ and $Z_\mu$. This connection could help determine the structure and intersections of these components, providing insights into the geometry of the moduli space itself.
Cycles and Entailment Relations: The paper demonstrates that theta linkage maps can induce inclusions between certain Breuil-Mezard cycles, reflecting the "entailment" relations between Serre weights. This observation suggests that the geometry of these cycles encodes deep information about the representation theory of $G$ in characteristic $p$.
Stratification and Special Loci: Theta linkage maps, being defined through geometric constructions, might shed light on the relationship between the Breuil-Mezard stratification of the moduli space and other important stratifications, such as the Ekedahl-Oort stratification. This could lead to a more unified understanding of the geometry of these moduli spaces.
Relationship Between Galois Representations and Automorphic Forms:
Weight Part of Serre's Conjecture: The connection to Breuil-Mezard cycles provides a geometric interpretation of the weight part of Serre's conjecture. Theta linkage maps, by relating different cycles, could lead to new proofs of this conjecture and refine our understanding of the weights in which a given Galois representation can appear.
Congruences and Families: Theta linkage maps might help explain congruences between automorphic forms of different weights. By relating their associated Galois representations through Breuil-Mezard cycles, these maps could provide insights into the structure of $p$-adic families of automorphic forms.
Beyond GL(2): While the Breuil-Mezard conjecture has been proven for GL(2), it remains a conjecture for more general groups. The connection between theta linkage maps and Breuil-Mezard cycles, if it extends to other groups, could provide new tools for attacking this conjecture and deepening our understanding of the relationship between Galois representations and automorphic forms in a broader context.
In summary, the interplay between theta linkage maps and Breuil-Mezard cycles offers a powerful lens through which to study the intricate relationship between the arithmetic of Galois representations and the geometry of Shimura varieties. This connection has the potential to significantly advance our understanding of the Langlands program and related areas of number theory.

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