Supersingular Representations Research Articles

On some mod p representations of quaternion algebra over ℚ p

Let $F$ be a totally real field in which $p$ is unramified and let $B$ be a quaternion algebra over $F$ which splits at at most one infinite place. Let $\overline {r}:\operatorname {{\mathrm {Gal}}}(\overline {F}/F)\rightarrow \mathrm {GL}_2(\overline {\mathbb {F}}_p)$ be a modular Galois representation which satisfies the Taylor–Wiles hypotheses. Assume that for some fixed place $v|p$ , $B$ ramifies at $v$ and $F_v$ is isomorphic to $\mathbb {Q}_p$ and $\overline {r}$ is generic at $v$ . We prove that the admissible smooth representations of the quaternion algebra over $\mathbb {Q}_p$ coming from mod $p$ cohomology of Shimura varieties associated to $B$ have Gelfand–Kirillov dimension $1$ . As an application we prove that the degree-two Scholze's functor (which is defined by Scholze [On the $p$ -adic cohomology of the Lubin–Tate tower, Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), 811–863]) vanishes on generic supersingular representations of $\mathrm {GL}_2(\mathbb {Q}_p)$ . We also prove some finer structure theorems about the image of Scholze's functor in the reducible case.

On irreducible supersingular representations of GL2(F)

Let $F$ be a non-archimedean local field of residual characteristic $p>3$ and residue degree $f>1$. We study a certain type of diagram, called \emph{cyclic diagrams}, and use them to show that the universal supersingular modules of $\mathrm{GL}_{2}(F)$ admit infinitely many non-isomorphic irreducible admissible quotients.

A family of irreducible supersingular representations of GL2(F) for some ramified p-adic fields

A family of irreducible supersingular representations of GL2(F) for some ramified p-adic fields

Iwahori–Hecke model for mod p representations of GL(2,F)

For a $p$-adic field $F$, the space of pro-$p$-Iwahori invariants of a universal supersingular mod $p$ representation $\tau$ of ${\rm GL}_2(F)$ is determined in the works of Breuil, Schein, and Hendel. The representation $\tau$ is introduced by Barthel and Livn\'e and this is defined in terms of the spherical Hecke operator. In earlier work of Anandavardhanan-Borisagar, an Iwahori-Hecke approach was introduced to study these universal supersingular representations in which they can be characterized via the Iwahori-Hecke operators. In this paper, we construct a certain quotient $\pi$ of $\tau$, making use of the Iwahori-Hecke operators. When $F$ is not totally ramified over $\mathbb Q_p$, the representation $\pi$ is a non-trivial quotient of $\tau$. We determine a basis for the space of invariants of $\pi$ under the pro-p Iwahori subgroup. A pleasant feature of this "new" representation $\pi$ is that its space of pro-$p$-Iwahori invariants admits a more uniform description vis-\`a-vis the description of the space of pro-$p$-Iwahori invariants of $\tau$.

A note on presentations of supersingular representations of $${\text {GL}}_2(F)$$

We prove that any smooth irreducible supersingular representation with central character of $${\text {GL}}_2(F)$$ is never of finite presentation when F is a finite field extension of $$\mathbb {Q}_p$$ such that $$F\ne \mathbb {Q}_p$$ , extending a result of Schraen in (J Reine Angew Math (Crelle’s J) 2015(704):187–208, 2015) for quadratic extensions.

A vanishing result for higher smooth duals

In this paper we prove a general vanishing result for Kohlhaase’s higher smooth duality functors Si. If G is any unramified connected reductive p-adic group, K is a hyperspecial subgroup, and V is a Serre weight, we show that Si(indKGV)=0 for i> dim(G∕B), where B is a Borel subgroup and the dimension is over ℚp. This is due to Kohlhaase for GL2(ℚp), in which case it has applications to the calculation of Si for supersingular representations. Our proof avoids explicit matrix computations by making use of Lazard theory, and we deduce our result from an analogous statement for graded algebras via a spectral sequence argument. The graded case essentially follows from Koszul duality between symmetric and exterior algebras.

On extensions of supersingular representations of [formula omitted

In this note, for p>3, we calculate the dimensions of ExtSL2(Qp)1(τ,σ), for any two irreducible supersingular representations τ and σ of SL2(Qp).

On the universal mod p supersingular quotients for GL2(F) over [formula omitted] for a general [formula omitted

Let F/Qp be a finite extension. We explore the universal supersingular mod p representations of GL2(F) by computing a basis for their spaces of invariants under the pro-p Iwahori subgroup. This generalizes works of Breuil and Schein (from Qp and the totally ramified cases to an arbitrary extension F/Qp). Using these results we then construct, for an unramified F/Qp, quotients of the universal supersingular modules which have as quotients all the supersingular representations of GL2(F) with a GL2(OF)-socle that is expected to appear in the mod p local Langlands correspondence. A construction in the case of an extension of Qp with inertia degree 2 and suitable ramification index is also presented.

Parabolic induction and extensions

Let $G$ be a $p$-adic reductive group. We determine the extensions between admissible smooth mod $p$ representations of $G$ parabolically induced from supersingular representations of Levi subgroups of $G$, in terms of extensions between representations of Levi subgroups of $G$ and parabolic induction. This proves for the most part a conjecture formulated by the author in a previous article and gives some strong evidence for the remaining part. In order to do so, we use the derived functors of the left and right adjoints of the parabolic induction functor, both related to Emerton's $\delta$-functor of derived ordinary parts. We compute the latter on parabolically induced representations of $G$ by pushing to their limits the methods initiated and expanded by the author in previous articles.

Cocenters and representations of pro-𝑝 Hecke algebras

In this paper, we study the relation between the cocenter H ~ ¯ \overline {{\tilde {\mathcal H}}} and the representations of an affine pro- p p Hecke algebra H ~ = H ~ ( 0 , − ) {\tilde {\mathcal H}}={\tilde {\mathcal H}}(0, -) . As a consequence, we obtain a new criterion on supersingular representations: a (virtual) representation of H ~ {\tilde {\mathcal H}} is supersingular if and only if its character vanishes on the non-supersingular part of the cocenter H ~ ¯ \overline {\tilde {\mathcal H}} .

Modulo p parabolic induction of pro-p-Iwahori Hecke algebra

Abstract We study the structure of parabolic inductions of a pro-p-Iwahori Hecke algebra. In particular, we give a classification of irreducible modulo p representations of pro-p-Iwahori Hecke algebras in terms of supersingular representations. Since supersingular representations are classified by Ollivier and Vignéras, it completes the classification of irreducible modulo p representations.

A classification of irreducible admissible mod 𝑝 representations of 𝑝-adic reductive groups

Let F F be a locally compact non-archimedean field, p p its residue characteristic, and G \textbf {G} a connected reductive group over F F . Let C C be an algebraically closed field of characteristic p p . We give a complete classification of irreducible admissible C C -representations of G = G ( F ) G=\mathbf {G}(F) , in terms of supercuspidal C C -representations of the Levi subgroups of G G , and parabolic induction. Thus we push to their natural conclusion the ideas of the third author, who treated the case G = G L m \mathbf {G}=\mathrm {GL}_m , as further expanded by the first author, who treated split groups G \mathbf {G} . As in the split case, we first get a classification in terms of supersingular representations of Levi subgroups, and as a consequence show that supersingularity is the same as supercuspidality.

Hecke modules and supersingular representations of U(2,1)

Let F F be a nonarchimedean local field of odd residual characteristic p p . We classify finite-dimensional simple right modules for the pro- p p -Iwahori-Hecke algebra H C ( G , I ( 1 ) ) \mathcal {H}_C(G,I(1)) , where G G is the unramified unitary group U ( 2 , 1 ) ( E / F ) \textrm {U}(2,1)(E/F) in three variables. Using this description when C = F ¯ p C = \overline {\mathbb {F}}_p , we define supersingular Hecke modules and show that the functor of I ( 1 ) I(1) -invariants induces a bijection between irreducible nonsupersingular mod- p p representations of G G and nonsupersingular simple right H C ( G , I ( 1 ) ) \mathcal {H}_C(G,I(1)) -modules. We then use an argument of Paškūnas to construct supersingular representations of G G .

On non-abelian Lubin–Tate theory for

We analyse the$\text{mod}~p$étale cohomology of the Lubin–Tate tower both with compact support and without support. We prove that there are no supersingular representations in the$H_{c}^{1}$of the Lubin–Tate tower. On the other hand, we show that in$H^{1}$of the Lubin–Tate tower appears the$\text{mod}~p$local Langlands correspondence and the$\text{mod}~p$local Jacquet–Langlands correspondence, which we define in the text. We discuss the local-global compatibility part of the Buzzard–Diamond–Jarvis conjecture which appears naturally in this context.

Compatibility between Satake and Bernstein isomorphisms in characteristicp

We study the center of the pro-p Iwahori-Hecke ring H of a connected split p-adic reductive group G. For k an algebraically closed field with characteristic p, we prove that the center of the k-algebra H_k:= H\otimes_Z k contains an affine semigroup algebra which is naturally isomorphic to the Hecke algebra attached to any irreducible smooth k-representation of a given hyperspecial maximal compact subgroup of G. This isomorphism is obtained using the inverse Satake isomorphism constructed in arXiv:1207.5557. We apply this to classify the simple supersingular H_k-modules, study the supersingular block in the category of finite length H_k-modules, and relate the latter to supersingular representations of G.

On the universal supersingular mod p representations of [formula omitted

The irreducible supersingular mod p representations of G=GL2(F), where F is a finite extension of Qp, are the building blocks of the mod p representation theory of G. They all arise as irreducible quotients of certain universal supersingular representations. We investigate the structure of these universal modules in the case when F/Qp is totally ramified.

On a classification of irreducible admissible modulo representations of a -adic split reductive group

AbstractWe give a classification of irreducible admissible modulo $p$ representations of a split$p$-adic reductive group in terms of supersingular representations. This is a generalization of a theorem of Herzig.

Explicit description of irreducible [formula omitted]-representations over [formula omitted

Let p be an odd prime number. The classification of irreducible representations of GL 2 ( Q p ) over F ¯ p is known thanks to the works of Barthel and Livné (1995) [BL95] and Breuil (2003) [Bre03a]. In the present paper we illustrate an exhaustive description of such irreducible representations, through the study of certain functions on the Bruhat–Tits tree of GL 2 ( Q p ) . In particular, we are able to detect the socle filtration for the KZ-restriction of supersingular representations, principal series and special series.

The classification of irreducible admissible mod p representations of a p-adic GL n

Let F be a finite extension of ℚ p . Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over \(\overline{ \mathbb{F}}_{p}\) to be supersingular. We then give the classification of irreducible admissible smooth GL n (F)-representations over \(\overline{ \mathbb{F}}_{p}\) in terms of supersingular representations. As a consequence we deduce that supersingular is the same as supercuspidal. These results generalise the work of Barthel–Livné for n=2. For general split reductive groups we obtain similar results under stronger hypotheses.

Coefficient systems and supersingular representations of $\mathrm {GL}_2(F)$

Coefficient systems and supersingular representations of $\mathrm {GL}_2(F)$