Level Raising in Automorphic Forms

• Level raising is a set of techniques that uses congruence conditions, degeneracy maps, and duality to generate higher-level automorphic forms and associated Galois representations.
• It relies on completed (co)homology and Poincaré duality to relate lower-level modular forms to new forms through explicit cohomological and geometric constructions.
• The approach extends classical congruence methods by incorporating p-adic, motivic, and eigenvariety techniques, revealing singular intersection points that encode arithmetic information.

Level raising refers to a collection of techniques, congruence conditions, and geometric or cohomological criteria allowing one to produce automorphic forms or Galois representations of higher level that are congruent, in a suitable sense, to given forms of lower level. Originally developed in the context of modular forms and their Galois representations, level raising has evolved to include p-adic, motivic, and geometric realizations across a wide spectrum of objects including eigenvarieties, automorphic representations on various reductive groups, and cohomology of Shimura varieties. One unifying feature is the presence of certain local or congruence conditions (often involving Hecke eigenvalues or traces of Frobenius elements) that force the existence of new automorphic forms, eigenvariety points, or Galois representations with prescribed ramification properties at auxiliary primes.

1. Core Cohomological Framework and Poincaré Duality

Central to modern approaches to level raising is the use of completed (co)homology spaces, as in Emerton's framework for p-adic modular forms. The completed cohomology $\widehat{H}^i(K^p, \mathcal{O})$ is defined as

$$\widehat{H}^i(K^p, \mathcal{O}) = \varprojlim_s H^i(K^p, \mathcal{O}/\pi^s\mathcal{O}),$$

with $K^p$ a compact open away from $p$ and $\mathcal{O}$ a coefficient ring; dually, completed homology spaces appear as

$$\widehat{H}_i(K^p, \mathcal{O}) = \varprojlim_{K_p} H_i(Y(K_pK^p), \mathcal{O}),$$

where $K_p$ runs over compact open subgroups at $p$.

A key structural input is the application of Poincaré duality spectral sequences to relate compactly supported completed cohomology and Borel–Moore homology: $$\operatorname{Hom}_{\mathrm{cts}}(\widehat{H}^1_c(K^p, \mathcal{O}), \mathcal{O}) \cong \widehat{H}^{BM}_1(K^p, \mathcal{O}).$$ This duality underpins the analysis of level raising maps and their adjoints, specifically providing the functional-analytic bridge required to relate modular forms of different levels via their (co)homological invariants.

The completed homology naturally carries an action of the Iwasawa algebra of a pro-$p$ subgroup, which, together with the duality, is essential for analyzing new subspaces and the images of trace maps.

2. Level Raising Mechanism and Degeneracy Maps

Level raising is implemented by comparing completed cohomology at two tame levels, for example, at level $\Gamma_1(N)$ and at the level $\Gamma_1(N)\cap\Gamma_0(l)$, introducing a new prime $l$ at which the level structure changes. The mechanism is realized through degeneracy maps: $$i: \widehat{H}^1(K(N))^2 \rightarrow \widehat{H}^1(K(N, l)),$$ which, at the classical modular forms level, are induced by the two canonical maps from higher to lower level modular curves.

The adjoint map $i^\dagger$ is analyzed using the aforementioned dualities. The locus of "level raising" is then isolated: $$\widehat{H}^1_{\mathrm{new}}(K(N, l)) := \ker(i^\dagger),$$ serving as the $\ell$-new subspace in the higher level. On the eigencurve, this translates into a geometric criterion: a point satisfying

$T_l(x)^2 - (l+1)^2 S_l(x) = 0$

parametrizes forms such that one factor of the $l$-th Hecke polynomial at a classical (lower-level) point $x$ gives rise to an $\ell$-new form at the higher-level point $x_1$. The intersection points of old and new components are necessarily non-smooth (singular) points of the eigencurve at level $K(N, l)$.

3. Geometric and Functorial Interpretations: Eigencurves and Jacquet–Langlands Maps

A crucial refinement in the context of p-adic families is the interpretation of level raising within the structure of eigencurves. Level raising loci correspond to singular intersection points of "old" and "new" irreducible components. By identifying

$$D(N, l)^{\mathrm{new}}_c = \overline{\{\text{classical } l\text{-new points}\}},$$

the locus of points in the eigencurve new at $l$ is identified exactly as the Zariski closure of classical $l$-new forms. This has direct implications for the understanding of the geometric structure of eigenvarieties, highlighting how singularities encode key arithmetic information.

This perspective harmonizes with Chenevier's $p$-adic Jacquet–Langlands correspondence, which induces a map from the eigenvariety for a definite quaternion algebra to that for $\mathrm{GL}_2$. The image of this map is characterized as lying within the $l$-new component of the higher-level eigencurve, unifying algebraic, cohomological, and geometric points of view.

4. Duality, Hecke Algebras, and Trace Formulae

The analysis relies on the interplay of Hecke algebras at various levels, their modules, and degeneracy maps. The pairing mechanisms provided by duality (for example, via the Iwasawa algebra) allow for a precise control over new subspaces and congruence modules.

The degeneracy maps are compatible with the duality, and level raising can be interpreted through the trace of Hecke operators (e.g., via Jacquet modules or local models). This structure is essential for transporting newform conditions across levels and for identifying when congruent (or Galois-congruent) forms at higher levels exist, often characterized by explicit congruence conditions on Hecke eigenvalues.

5. Comparison With Classical and Modulo Prime Power Results

The analytic, cohomological, and geometric level raising constructions generalize the classical mod $\ell$ congruence-based theorems (e.g., Ribet's work), as well as their generalizations to higher prime powers and higher rank. In classical settings, level raising is often characterized by congruences

$a_l(f) \equiv \pm(l+1) \pmod{\lambda^n}$

for eigenvalues of modular forms or traces of Frobenius on associated Galois representations, which directly feed into the structure of Jacobians and their cohomology.

The completed cohomology approach—together with duality arguments—provides an effective way to treat such congruences in families, in both the classical and $p$-adic overconvergent contexts, illuminating the full structure of congruences and inter-level relationships.

6. Broader Implications and Future Directions

The synthetic approach combining completed cohomology, duality, and level raising has significant implications for the paper of $p$-adic eigenvarieties, the deformation theory of Galois representations, and the geometry of Shimura varieties. It provides new tools for:

• Analyzing singularities and intersections in eigenvarieties.
• Understanding the precise relationship between automorphic forms on different groups via $p$-adic Jacquet–Langlands correspondences.
• Advancing the paper of modularity lifting theorems, especially concerning the $p$-adic Langlands program.
• Providing a robust framework for constructing and characterizing new congruences and for generalizing classical results to settings involving completed (co)homology spaces, non-commutative Iwasawa algebras, and $p$-adic families.

The methods—especially those exploiting duality and completed cohomology with Iwasawa actions—are expected to have further application in the understanding of higher-rank cases, new constructions in the Lenstra program, and the exploration of non-abelian $p$-adic Hodge theory.

Summary Table: Key Features of Level Raising in Completed Cohomology (GL(2) Context)

Aspect	Key Construction / Result	Formula / Characterization
Completed (Co)homology	Inverse limits over finite level cohomology	$\widehat{H}^i(K^p,\mathcal{O})$, $\widehat{H}_i$
Poincaré Duality	Pairing between homology and cohomology	$\operatorname{Hom}_{\mathrm{cts}}(\ldots)$
Level Raising Map	Degeneracy/trace maps and their adjoints	$i$, $i^\dagger$, $\ker(i^\dagger)$
Level Raising Criteria	Hecke algebra congruences at auxiliary prime $l$	$T_l^2 - (l+1)^2 S_l = 0$
Eigenvariety Geometry	Intersection of old and new components	Singular (non-smooth) point on $D(N, l)$
Jacquet–Langlands Image	Zariski closure of classical newforms	$D(N, l)^{\mathrm{new}}_c = \overline{\{\ldots\}}$
Module-Theoretic Structure	Action of Iwasawa algebra and Hecke operators	$[[K_p]]$, Hecke module dualities

In essence, level raising within the framework of completed (co)homology and Poincaré duality offers a powerful, cohomologically robust, and geometrically meaningful enhancement of congruence-based techniques. It substantiates the connection between singularities on eigenvarieties, the structure of Hecke modules, and the arithmetic of automorphic forms in $p$-adic settings, advancing both theoretical understanding and practical applications in the arithmetic theory of modular forms and their generalizations.