Hecke Eigenclass Research Articles

Cohomology with twisted one-dimensional coefficients for congruence subgroups of [formula omitted] and Galois representations

We extend the computations in [2–4] to find the cohomology in degree five of a congruence subgroup of SL4(Z) with coefficients in a field twisted by a nebentype character, along with the action of the Hecke algebra on the cohomology. This is the top cuspidal degree. For each Hecke eigenclass we find, we produce the unique Galois representation that appears to be attached to it.The computations require serious modifications to our previous algorithms. Nontrivial coefficients add a layer of complication to our data structures. New possibilities must be taken into account in the Galois Finder, the code that finds the Galois representations. We have improved the Galois Finder to report when the attached Galois representation is uniquely determined by our data.

Sums of Galois representations and arithmetic homology

Let Γ 0 ( n , N ) \Gamma _0(n,N) denote the usual congruence subgroup of type Γ 0 \Gamma _0 and level N N in SL ( n , Z ) \text {SL}(n,{\mathbb Z}) . Suppose for i = 1 , 2 i=1,2 that we have an irreducible odd n n -dimensional Galois representation ρ i \rho _i attached to a homology Hecke eigenclass in H ∗ ( Γ 0 ( n , N i ) , M i ) H_*(\Gamma _0(n,N_i),M_i) , where the level N i N_i and the weight and nebentype making up M i M_i are as predicted by the Serre-style conjecture of Ash, Doud, Pollack, and Sinnott. We assume that n n is odd, that N 1 N 2 N_1N_2 is squarefree, and that ρ 1 ⊕ ρ 2 \rho _1\oplus \rho _2 is odd. We prove two theorems that assert that ρ 1 ⊕ ρ 2 \rho _1\oplus \rho _2 is attached to a homology Hecke eigenclass in H ∗ ( Γ 0 ( 2 n , N ) , M ) H_*(\Gamma _0(2n,N),M) , where N N and M M are as predicted by the Serre-style conjecture. The first theorem requires the hypothesis that the highest weights of M 1 M_1 and M 2 M_2 are small in a certain sense. The second theorem requires the truth of a conjecture as to what degrees of homology can support Hecke eigenclasses with irreducible Galois representations attached, but no hypothesis on the highest weights of M 1 M_1 and M 2 M_2 . This conjecture is known to be true for n = 3 n=3 , so we obtain unconditional results for GL ( 6 ) \text {GL}(6) . A similar result for GL ( 4 ) \text {GL}(4) appeared in an earlier paper.

Even Galois representations and the cohomology of $$\mathrm{GL}(2,\mathbb Z)$$ GL ( 2 , Z )

Let $$\rho $$ be an even two-dimensional representation of the Galois group $${{\mathrm{Gal}}}(\overline{\mathbb Q}/\mathbb Q)$$ which is induced from a character $$\chi $$ of odd order of the absolute Galois group of a real quadratic field K. After imposing some additional conditions on $$\chi $$ , we attach $$\rho $$ to a Hecke eigenclass in the cohomology of $$\mathrm{GL}(2,\mathbb Z)$$ with coefficients in a certain infinite-dimensional vector space V over an arbitrary field of characteristic not equal to 2. The space V is defined purely algebraically starting from the field K.

Reducible Galois representations and arithmetic homology for GL(4)

We prove that a sum of two odd irreducible two-dimensional Galois representations with squarefree relatively prime Serre conductors is attached to a Hecke eigenclass in the homology of a subgroup of GL(4,ℤ), with the level, nebentype, and coefficient module of the homology predicted by a generalization of Serre’s conjecture to higher dimensions. To do this we prove along the way that any Hecke eigenclass in the homology of a congruence subgroup of a maximal parabolic subgroup of GL(n,ℚ) has a reducible Galois representation attached, where the dimensions of the components correspond to the type of the parabolic subgroup. Our main new tool is a resolution of ℤ by GL(n,ℚ)-modules consisting of sums of Steinberg modules for all subspaces of ℚ n .

Toward an explicit eigencurve for GL(3)

Starting with a numerically noncritical (at [Formula: see text]) Hecke eigenclass [Formula: see text] in the homology of a congruence subgroup [Formula: see text] of [Formula: see text] (where [Formula: see text] divides the level of [Formula: see text]) with classical coefficients, we first show how to compute to any desired degree of accuracy a lift of [Formula: see text] to a Hecke eigenclass [Formula: see text] with coefficients in a module of [Formula: see text]-adic distributions. Then we show how to find to any desired degree of accuracy the germ of the projection [Formula: see text] to weight space of the eigencurve around the point [Formula: see text] corresponding to the system of Hecke eigenvalues of [Formula: see text]. We do this under the conjecturally mild hypothesis that [Formula: see text] is smooth at [Formula: see text].

Partial Hasse invariants on splitting models of Hilbert modular varieties

Let $F$ be a totally real field of degree $g$, and let $p$ be a prime number. We construct $g$ partial Hasse invariants on the characteristic $p$ fiber of the Pappas-Rapoport splitting model of the Hilbert modular variety for $F$ with level prime to $p$, extending the usual partial Hasse invariants defined over the Rapoport locus. In particular, when $p$ ramifies in $F$, we solve the problem of lack of partial Hasse invariants. Using the stratification induced by these generalized partial Hasse invariants on the splitting model, we prove in complete generality the existence of Galois pseudo-representations attached to Hecke eigenclasses of paritious weight occurring in the coherent cohomology of Hilbert modular varieties $\mathrm{mod}$ $p^m$, extending a previous result of M. Emerton and the authors which required $p$ to be unramified in $F$.

Relaxation of strict parity for reducible Galois representations attached to the homology of GL(3, ℤ)

In this paper, we prove the following theorem: Let [Formula: see text] be an algebraic closure of a finite field of characteristic [Formula: see text]. Let [Formula: see text] be a continuous homomorphism from the absolute Galois group of [Formula: see text] to [Formula: see text]) which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. We assume that the image of [Formula: see text] is contained in the intersection of the stabilizers of the line spanned by [Formula: see text] and the plane spanned by [Formula: see text], where [Formula: see text] denotes the standard basis. Such [Formula: see text] will not satisfy a certain strict parity condition. Under the conditions that the Serre conductor of [Formula: see text] is squarefree, that the predicted weight [Formula: see text] lies in the lowest alcove, and that [Formula: see text], we prove that [Formula: see text] is attached to a Hecke eigenclass in [Formula: see text], where [Formula: see text] is a subgroup of finite index in [Formula: see text] and [Formula: see text] is an [Formula: see text]-module. The particular [Formula: see text] and [Formula: see text] are as predicted by the main conjecture of the 2002 paper of the authors and David Pollack, minus the requirement for strict parity.

Highly reducible Galois representations attached to the homology of $\mathrm {GL}(n,\mathbb {Z})$

Let $n\ge 1$ and $\mathbb {F}$ be an algebraic closure of a finite field of characteristic $p>n+1$. Let $\rho :G_{\mathbb {Q}}\to \mathrm {GL}(n,\mathbb {F})$ be a Galois representation that is isomorphic to a direct sum of a collection of characters and an odd $m$-dimensional representation $\tau$. We assume that $m=2$ or $m$ is odd, and that $\tau$ is attached to a homology class in degree $m(m-1)/2$ of a congruence subgroup of $\mathrm {GL}(m,\mathbb {Z})$ in accordance with the main conjecture of an earlier work of the authors and Pollack. We also assume a certain compatibility of $\tau$ with the parity of the characters and that the Serre conductor of $\rho$ is square-free. We prove that $\rho$ is attached to a Hecke eigenclass in $H_t(\Gamma ,M)$, where $\Gamma$ is a subgroup of finite index in $\rm {SL}$$(n,\mathbb {Z})$, $t=n(n-1)/2$ and $M$ is an $\mathbb {F}\Gamma$-module. The particular $\Gamma$ and $M$ are as predicted by the main conjecture of an earlier work. The method uses modular cosymbols, as in a recent work of the first author.

Galois representations and torsion in the coherent cohomology of Hilbert modular varieties

Abstract Let F be a totally real number field and let p be a prime unramified in F. We prove the existence of Galois pseudo-representations attached to mod ⁡ p m ${\operatorname{mod}p^{m}}$ Hecke eigenclasses of paritious weight occurring in the coherent cohomology of Hilbert modular varieties for F of level prime to p.

Mod 2 homology for [formula omitted] and Galois representations

We extend the computations in [AGM11] to find the mod 2 homology in degree 1 of a congruence subgroup Γ of SL(4,Z) with coefficients in the sharbly complex, along with the action of the Hecke algebra. This homology group is related to the cohomology of Γ with F2 coefficients in the top cuspidal degree. These computations require a modification of the algorithm to compute the action of the Hecke operators, whose previous versions required division by 2. We verify experimentally that every mod 2 Hecke eigenclass found appears to have an attached Galois representation, giving evidence for a conjecture in [AGM11]. Our method of computation was justified in [AGM12].

Direct Sums of Mod p Characters of Gal and the Homology of GL(n, ℤ)

We prove the following theorem: Let 𝔽 be an algebraic closure of a finite field of characteristic p. Let ρ be a continuous homomorphism from the absolute Galois group of ℚ to GL(n, 𝔽) which is isomorphic to a direct sum of one-dimensional representations χ i where p > n + 1 and the product of the conductors of the χ i is squarefree and prime to p. If a certain parity condition holds, then ρ is attached to a Hecke eigenclass in the homology of an arithmetic subgroup Γ of SL(n, ℤ) with coefficients in a module V, where Γ and V are as predicted by Conjecture 2.2 of [5].

Reducible Galois Representations and the Homology of GL(3, ℤ)

Let Fp be an algebraic closure of a finite field of characteristic p. Let ρ be a continuous homomorphism from the absolute Galois group of Q to GL(3, Fp) which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. Under the condition that the Serre conductor of ρ is squarefree, we prove that ρ is attached to a Hecke eigenclass in the homology of an arithmetic subgroup Γ of GL(3,Z). In addition, we prove that the coefficient module needed is, in fact, predicted by the main conjecture of [3].

Hecke Operators and the Stable Homology of GL(n)

Let R be a field of any characteristic and A a principal ideal domain. We make a conjecture that asserts that any Hecke operator T acts punctually on any Hecke eigenclass in the stable homology with trivial coefficients R of a principal congruence subgroup Γ in GL(n, A), i.e., as multiplication by the number of single cosets contained in T. In the case where A = ℤ, this conjecture implies that the Galois representation consisting of the sum of the 0th, 1st, ⋅, n − 1st powers of the cyclotomic character is attached to any stable Hecke eigenclass. When A = ℤ and Γ =GL(n, ℤ), this conjecture was already made by Calegari and Venkatesh. We obtain partial results giving evidence for the conjecture. These results imply that some part of the Hecke algebra acts punctually. If the characteristic of R is 0, they show that the entire Hecke algebra acts punctually, which was already known in a completely different way using a result of A. Borel.

Resolutions of the Steinberg module for [formula omitted

We give several resolutions of the Steinberg representation St n for the general linear group over a principal ideal domain, in particular over Z . We compare them, and use these results to prove that the computations in Avner Ash et al. (2011) [AGM11] are definitive. In particular, in Avner Ash et al. (2011) [AGM11] we use two complexes to compute certain cohomology groups of congruence subgroups of SL ( 4 , Z ) . One complex is based on Voronoiʼs polyhedral decomposition of the symmetric space for SL ( n , R ) , whereas the other is a larger complex that has an action of the Hecke operators. We prove that both complexes allow us to compute the relevant cohomology groups, and that the use of the Voronoi complex does not introduce any spurious Hecke eigenclasses.

Torsion in the cohomology of congruence subgroups of [formula omitted] and Galois representations

We report on the computation of torsion in certain homology theories of congruence subgroups of SL ( 4 , Z ) . Among these are the usual group cohomology, the Tate–Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2, 3, 5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels ⩽31.

FINDING GALOIS REPRESENTATIONS CORRESPONDING TO CERTAIN HECKE EIGENCLASSES

In 1992, Ash and McConnell presented computational evidence of a connection between three-dimensional Galois representations and certain arithmetic cohomology classes. For some examples, they were unable to determine the attached representation. For several Hecke eigenclasses (including one for which Ash and McConnell did not find the Galois representation), we find a Galois representation which appears to be attached and show strong evidence for the uniqueness of this representation. The techniques that we use to find defining polynomials for the Galois representations include a targeted Hunter search, class field theory and elliptic curves.

The Steinberg symbol and special values of $L$-functions

The main results of this article concern the definition of a compactly supported cohomology class for the congruence group Γ 0 (p n ) with values in the second Milnor K-group (modulo 2-torsion) of the ring of p-integers of the cyclotomic extension Q(μ p n). We endow this cohomology group with a natural action of the standard Hecke operators and discuss the existence of special Hecke eigenclasses in its parabolic cohomology. Moreover, for n = 1, assuming the non-degeneracy of a certain pairing on p-units induced by the Steinberg symbol when (p, k) is an irregular pair, i.e. p| B k /k, we show that the values of the above pairing are congruent mod p to the L-values of a weight k, level 1 cusp form which satisfies Eisenstein-type congruences mod p, a result that was predicted by a conjecture of R. Sharifi.

Rigidity of p -adic cohomology classes of congruence subgroups of GL(n , ℤ)

We extend the work of Ash and Stevens [Ash-Stevens 97] on p-adic analytic families of p-ordinary arithmetic cohomology classes for GL(N,Q) by introducing and investigating the concept of p-adic rigidity of arithmetic Hecke eigenclasses. An arithmetic eigenclass is said to be "rigid" if (modulo twisting) it does not admit a nontrivial p-adic deformation containing a Zariski dense set of arithmetic specializations. This paper develops tools for explicit investigation into the structure of eigenvarieties for GL(N). We use these tools to prove that known examples of non-sefldual cohomological cuspforms for GL(3) are rigid. Moreover, we conjecture that for GL(3), rigidity is equivalent to non-selfduality.

Smith theory and Hecke operators

Of the connections between the cohomology of arithmetic groups and representations of Galois groups, some are known and more are conjectured. This paper proves one of these conjectures for certain monomial Galois representations. Let p be a prime. We show that the representation of the absolute Galois group of Q induced from an F p × -valued ray class character of the cyclotomic field Q (ζ p ) is attached to a Hecke eigenclass in the mod p cohomology of a torsion-free congruence subgroup Γ ( M ) of GL(p−1, Z ) . The method involves constructing an action of the Hecke algebra on the fundamental exact sequence of Smith theory arising from the action by conjugation of an element of order p in GL(p−1, Z ) on Γ ( M ).

An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the mod p$ cohomology of GL(n,ℤ)

An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the mod p$ cohomology of GL(n,ℤ)