Primitive Hilbert Modular Forms

• Primitive Hilbert modular forms are cuspidal eigenforms on GL₂ over totally real fields that are new at their level and not obtained via base change.
• They enable the attachment of compatible l-adic Galois representations and yield analytic L-functions with significant arithmetic data.
• Their study uses analytic, algebraic, and geometric methods, impacting the Langlands program, modularity lifting, and computational advances.

Primitive Hilbert modular forms are distinguished members within the landscape of Hilbert modular forms that serve as "newforms" at their given level and weight; they are not obtained from forms of lower level or via base change from a proper subfield. These objects are central for understanding the arithmetic of totally real fields, Galois representations, $L$-functions, congruences, and Iwasawa theory. Their paper combines analytic, algebraic, and geometric techniques and has direct implications for modern questions in number theory, including the Langlands program.

1. Definition and Arithmetic Context

A primitive Hilbert modular form is a cuspidal eigenform on $GL_2$ over a totally real field $F$ that is new at its level, possesses specified weight vector (often parallel weight), and is not induced by base change from a subfield. Formally, it is an eigenvector for the Hecke algebra acting on the new part of the space $S_{\mathbf{k}}^{new}(\mathfrak{n})$ for some ideal $\mathfrak{n} \subset \mathcal{O}_F$ and weight vector $\mathbf{k}$. The new/primitive distinction is characterized by the decomposition into old and new subspaces via degeneracy maps and Hecke stability (see (Donnelly et al., 2016)).

The primitive forms are the focus of several key constructions:

• Galois Representations: To each primitive Hilbert eigenform, one can attach a compatible system of $l$-adic Galois representations $\rho_{\lambda}$, with traces and determinants of Frobenius at unramified primes matching the Hecke eigenvalues of the form (Nair et al., 4 Jan 2024).
• $L$-Functions and Special Values: Primitive forms admit analytic $L$-functions with Euler products determined by Hecke eigenvalues; their special values encode deep arithmetic data.

2. Construction and Identification

Hecke Theory and Newform Decomposition

Primitive forms are isolated by decomposing spaces of Hilbert modular forms into old and new components. For parallel weight $2$, one uses the Jacquet–Langlands correspondence to transfer the computation to spaces associated with quaternionic modular forms (Donnelly et al., 2016). Brandt matrices and Fuchsian group methods provide explicit algorithms for computing Hecke eigenvalues and determining simultaneous eigenspaces. The identification of primitive forms is crucial for applications to abelian varieties and for the integrity of extensive computation, as verified by cross-method consistency.

Exceptional Primes and Tamely Dihedral Primes

Primitive newforms over totally real fields can be constructed to avoid exceptional primes (projective images of Galois representations being "small"): via level raising and enforcing a tamely dihedral local condition at carefully chosen split primes. A form is primitive if it is not a base change, has non-induced weight, and its associated Galois representations have large image—either $\operatorname{PSL}_2$ or $\operatorname{PGL}_2$ over some finite field (Dieulefait et al., 2014).

Explicit Algorithms in Characteristic $p$ and Weight One

Primitive forms in mod $p$ settings or for parallel weight one demand geometric and algebraic methods. For instance, mod $p$ forms are reduced via division by partial Hasse invariants to a minimal cone in the weight space (Diamond et al., 2016). Primitive forms of weight one can arise as companion forms when local Galois representations are unramified and $p$--distinguished, directly addressing Serre's modularity conjecture for totally real fields (1711.01680).

3. Analytic Properties and $L$-Functions

Primitive Hilbert modular forms are uniquely characterized by the values of Rankin–Selberg convolution $L$-functions and their central values. The deep connection is made manifest by theorems stating that a primitive form $g$ is determined by $L(f \otimes g, \frac{1}{2})$ as $f$ varies over families with either varying level (Hamieh et al., 2016) or weight (Hamieh et al., 2016). This uniqueness principle is analogous to the strong multiplicity one theorem, with proofs relying on approximate functional equations, trace formulas, and careful error analysis.

In practice, the Euler factors at primes $\mathfrak{p}$ are given in parallel to the structure for classical modular forms:

$$L(s, f) = \prod_{\mathfrak{p}|\mathfrak{n}} (1 - a_{\mathfrak{p}} N(\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\nmid\mathfrak{n}} (1 - a_{\mathfrak{p}} N(\mathfrak{p})^{-s} + N(\mathfrak{p})^{-2s})^{-1}$$

with $a_{\mathfrak{p}}$ the Hecke eigenvalue attached to a primitive form $f$ (Donnelly et al., 2016).

4. Congruences, Iwasawa Theory, and $p$-adic Measures

Primitive Hilbert modular forms are central to non-commutative Iwasawa theory through their $p$-adic $L$-functions and congruences. The interpolation of Rankin convolution $L$-values by $p$-adic measures attached to a primitive form enables the construction of abelian $p$-adic $L$-functions and the paper of congruences:

$$\prod_{j=1}^n N_{j,n}\left( \frac{a_j}{N_{0,j}(a_0)} \cdot \frac{\phi\circ N_{0,j-1}(a_0)}{\phi(a_{j-1})} \right)^{p^j} \equiv 1 \pmod{p^{n+1}}$$

as predicted by the conjectures in non-commutative Iwasawa theory (Ward, 2012). The explicit interpolation formula involves local epsilon and Euler factors, Petersson product periods, and ensures that primitive forms contribute fundamentally to the $p$-adic analytic landscape.

5. Geometric and Representation-Theoretic Aspects

Primitive Hilbert modular forms are critical for geometric and automorphic constructions.

• Cohomological Interpretation: The periods and period polynomials of primitive forms are connected to Eichler–Shimura-type cohomology and Kronecker series (Choie, 2021). The rationality and closed formulas for generating functions of periods reveal deep ties to the topology of Hilbert modular varieties.
• Kac–Moody Algebras and Borcherds Products: Primitive Hilbert modular forms, especially those with explicit product expansions (Borcherds products), arise as denominator functions for automorphic corrections to rank-two symmetric hyperbolic Kac–Moody algebras. The Fourier coefficients encode root multiplicities in certain generalized Kac–Moody superalgebras (Kim et al., 2012).
• Theta Series and Lattice Correspondences: The construction of half-integral weight forms via generalized theta series lifts data from primitive newforms, giving explicit relations between Fourier coefficients and central $L$-values (Sirolli et al., 2021). This is effective for arithmetic applications including computations and non-vanishing results.

6. Congruences and Modularity Lifting

Primitive forms admit rich congruence properties, often realized via quaternionic $S$-ideal classes and Atkin–Lehner operators. Modulo small primes (notably $2$), one can "flip" Atkin–Lehner sign patterns and construct, for each such pattern, a newform congruent modulo $2$ to a given eigenform when certain admissibility criteria are met (Martin, 2017). Such results play a key role in level-raising and in the modularity lifting strategy, as congruence phenomena can imply the modularity of Galois representations.

Additionally, explicit upper bounds on the number of classical weight one specializations in non-CM primitive Hida families (via class number and local unit considerations) give control over the "sparseness" of weight one primitive Hilbert modular forms, reinforcing their arithmetic uniqueness (Ozawa, 2016).

7. Impact and Ongoing Directions

Primitive Hilbert modular forms underpin major advances in number theory:

• Their Galois representations are compatible systems matching Hecke eigenvalues (traces and determinants), serving as principal objects for the Langlands program and modularity conjectures (Nair et al., 4 Jan 2024).
• Their $L$-functions, periods, and congruence properties govern key phenomena in arithmetic geometry and Iwasawa theory.
• Computational advances (databases, explicit algorithms) for primitive forms are leveraged in arithmetic statistics and for the investigation of abelian varieties and their endomorphism algebras (Donnelly et al., 2016).

Future directions include generalizations to forms of non-parallel weight, extensions to other reductive groups, refinement of modularity lifting in the ramified and mod $p$ setting, and the further elucidation of congruence phenomena and their geometric interpretations.

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Primitive Hilbert modular forms are thus the analytic and arithmetic "atoms" whose properties and interactions inform broad swathes of modern algebraic number theory, arithmetic geometry, and automorphic representation theory.