Non-archimedean Local Field Research Articles

Deligne–Lusztig constructions for division algebras and the local Langlands correspondence, II

In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive p-adic groups, analogous to Deligne–Lusztig theory for finite reductive groups. In this paper we establish a new instance of Lusztig’s program. Precisely, let X be the Deligne–Lusztig (ind-pro-)scheme associated to a division algebra D over a non-Archimedean local field K of positive characteristic. We study the $$D^\times $$ -representations $$H_\bullet (X)$$ by establishing a Deligne–Lusztig theory for families of finite unipotent groups that arise as subquotients of $$D^\times $$ . There is a natural correspondence between quasi-characters of the (multiplicative group of the) unramified degree-n extension of K and representations of $$D^{\times }$$ given by $$\theta \mapsto H_\bullet (X)[\theta ]$$ . For a broad class of characters $$\theta ,$$ we show that the representation $$H_\bullet (X)[\theta ]$$ is irreducible and concentrated in a single degree. After explicitly constructing a Weil representation from $$\theta $$ using $$\chi $$ -data, we show that the resulting correspondence matches the bijection given by local Langlands and therefore gives a geometric realization of the Jacquet–Langlands transfer between representations of division algebras.

On the exactness of ordinary parts over a local field of characteristic p

Let $G$ be a connected reductive group over a non-archimedean local field $F$ of residue characteristic $p$, $P$ be a parabolic subgroup of $G$, and $R$ be a commutative ring. When $R$ is artinian, $p$ is nilpotent in $R$, and $\mathrm{char}(F)=p$, we prove that the ordinary part functor $\mathrm{Ord}_P$ is exact on the category of admissible smooth $R$-representations of $G$. We derive some results on Yoneda extensions between admissible smooth $R$-representations of $G$.

On the sharpness of the bound for the Local Converse Theorem of 𝑝-adic 𝐺𝐿_{𝑝𝑟𝑖𝑚𝑒}

We introduce a novel ultrametric on the set of equivalence classes of cuspidal irreducible representations of a general linear group GL N {\operatorname {GL}}_{N} over a non-archimedean local field, based on distinguishability by twisted gamma factors. In the case that N N is prime and the residual characteristic is greater than or equal to ⌊ N 2 ⌋ \left \lfloor \frac {N}{2}\right \rfloor , we prove that, for any natural number i ≤ ⌊ N 2 ⌋ i\le \left \lfloor \frac {N}{2}\right \rfloor , there are pairs of cuspidal irreducible representations whose logarithmic distance in this ultrametric is precisely − i -i . This implies that, under the same conditions on N N , the bound ⌊ N 2 ⌋ \left \lfloor \frac {N}{2}\right \rfloor in the Local Converse Theorem for GL N \operatorname {GL}_N is sharp.

On the unicity of types in special linear groups

Let F be a non-archimedean local field. We show that any representation of a maximal compact subgroup of mathbf {SL}_N(F) which is typical for an essentially tame supercuspidal representation must be induced from a Bushnell–Kutzko maximal simple type. From this, we explicitly count and describe the conjugacy classes of such typical representations, and give an explicit description of an inertial Langlands correspondence for essentially tame irreducible N-dimensional projective representations of the Weil group of F.

On the local converse theorem for p-adic GLn

In this paper, we completely prove a standard conjecture on the local converse theorem for generic representations of ${\rm GL}_n(F)$, where $F$ is a non-archimedean local field.

Igusa’s Local Zeta Functions and Exponential Sums for Arithmetically Non Degenerate Polynomials

We study the twisted local zeta function associated to a polynomial in two variables with coefficients in a non-Archimedean local field of arbitrary characteristic. Under the hypothesis that the polynomial is arithmetically non degenerate, we obtain an explicit list of candidates for the poles in terms of geometric data obtained from a family of arithmetic Newton polygons attached to the polynomial. The notion of arithmetical non degeneracy due to Saia and Zúñiga-Galindo is weaker than the usual notion of non degeneracy due to Kouchnirenko. As an application we obtain asymptotic expansions for certain exponential sums attached to these polynomials.

On the Gross-Keating invariant of a quadratic form over a non-archimedean local field

Let $B$ be a half-integral symmetric matrix of size $n$ defined over ${\Bbb Q}_p$. The Gross-Keating invariant of $B$ was defined by Gross and Keating, and has important applications to arithmetic geometry. But the nature of the Gross-Keating invariant was not understood very well for $n\geq 4$. In this paper, we establish basic properties of the Gross-Keating invariant of a half-integral symmetric matrix of general size over an arbitrary non-archimedean local field of characteristic zero.

Integration of functions of motivic exponential class, uniform in all non-archimedean local fields of characteristic zero

Through a cascade of generalizations, we develop a theory of motivic integration which works uniformly in all non-archimedean local fields of characteristic zero, overcoming some of the difficulties related to ramification and small residue field characteristics. We define a class of functions, called functions of motivic exponential class, which we show to be stable under integration and under Fourier transformation, extending results and definitions from previous papers. We prove uniform results related to rationality and to various kinds of loci. A key ingredient is a refined form of Denef-Pas quantifier elimination which allows us to understand definable sets in the value group and in the valued field.

Explicit determination of root numbers of abelian varieties

Let A A be an abelian variety over a nonarchimedean local field of definition K K and let W ( A ) W(A) be the root number of A A . Let F F be a Galois extension of K K over which A A has semistable reduction, allowing F = K F = K . We analyze W ( A ) W(A) in terms of contributions from the toric and abelian variety components of the closed fibers of the Néron models of A A over the ring of integers of K K and of F F . In particular, our results can be used to calculate W ( A ) W(A) in two main instances: first, for abelian varieties with additive reduction over K K and totally toroidal reduction over F F , provided that the residue characteristic of K K is odd; second, for the Jacobian A = J ( C ) A = J(C) of a stable curve C C over K K .

An explicit formula of the unramified Shintani functions for $(\mathbf{GSp}_{4},\mathbf{GL}_{2} \times_{\mathbf{GL}_{1}} \mathbf{GL}_{2})$ and its application

Let $F$ be a non-archimedean local field of arbitrary characteristic. In this paper, we announce an explicit formula of the unramified Shintani functions for $(\mathbf{GSp}_{4}(F),(\mathbf{GL}_{2} \times_{\mathbf{GL}_{1}} \mathbf{GL}_{2})(F))$. As an application, we compute a local zeta integral, which represents the spin $L$-factor of $\mathbf{GSp}_{4}$.

The regular representations of GLN over finite local principal ideal rings

Let $\mathfrak{o}$ be the ring of integers in a non-Archimedean local field with finite residue field, $\mathfrak{p}$ its maximal ideal, and $r\geq2$ an integer. An irreducible representation of the finite group $G_{r}=\mathrm{GL}_{N}(\mathfrak{o}/\mathfrak{p}^{r})$ is called regular if its restriction to the principal congruence kernel $K^{r-1}=1+\mathfrak{p}^{r-1}\mathrm{M}_{N}(\mathfrak{o}/\mathfrak{p}^{r})$ consists of representations whose stabilisers modulo $K^{1}$ are centralisers of regular elements in $\mathrm{M}_{N}(\mathfrak{o}/\mathfrak{p})$. The regular representations form the largest class of representations of $G_{r}$ which is currently amenable to explicit construction. Their study, motivated by constructions of supercuspidal representations, goes back to Shintani, but the general case remained open for a long time. In this paper we give an explicit construction of all the regular representations of $G_{r}$.

On L-packets and depth for SL2(K) and its inner form

We consider the group [Formula: see text], where [Formula: see text] is a local non-archimedean field of characteristic two. We prove that the depth of any irreducible representation of [Formula: see text] is larger than the depth of the corresponding Langlands parameter, with equality if and only if the [Formula: see text]-parameter is essentially tame. We also work out a classification of all [Formula: see text]-packets for [Formula: see text] and for its non-split inner form, and we provide explicit formulae for the depths of their [Formula: see text]-parameters.

Computation of the Lambda function for a finite Galois extension

By Langlands [13], and Deligne [4] we know that the local constants are extendible functions. Therefore, to give an explicit formula of the local constant of an induced representation of a local Galois group of a non-Archimedean local field F of characteristic zero, we have to compute the lambda function λK/F for a finite extension K/F. In this paper, when a finite extension K/F is Galois, we give a formula for λK/F.

Local zeta functions, pseudodifferential operators and Sobolev-type spaces over non-Archimedean local fields

In this articlewe introduce a new type of local zeta functions and study some connections with pseudodifferential operators in the framework of non-Archimedean fields. The new local zeta functions are defined by integrating complex powers of norms of polynomials multiplied by infinitely pseudo-differentiable functions. In characteristic zero, the new local zeta functions admit meromorphic continuations to the whole complex plane, but they are not rational functions. The real parts of the possible poles have a description similar to the poles of Archimedean zeta functions. But they can be irrational real numbers while in the classical case are rational numbers. We also study, in arbitrary characteristic, certain connections between local zeta functions and the existence of fundamental solutions for pseudodifferential equations.

TEST VECTORS FOR LOCAL CUSPIDAL RANKIN–SELBERG INTEGRALS

Let $\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2}$be a pair of cuspidal complex, or $\ell$-adic, representations of the general linear group of rank $n$over a nonarchimedean local field $F$of residual characteristic $p$, different to $\ell$. Whenever the local Rankin–Selberg $L$-factor $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$is nontrivial, we exhibit explicit test vectors in the Whittaker models of $\unicode[STIX]{x1D70B}_{1}$and $\unicode[STIX]{x1D70B}_{2}$such that the local Rankin–Selberg integral associated to these vectors and to the characteristic function of $\mathfrak{o}_{F}^{n}$is equal to $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$. As an application we prove that the $L$-factor of a pair of banal $\ell$-modular cuspidal representations is the reduction modulo $\ell$of the $L$-factor of any pair of $\ell$-adic lifts.

An explicit formula of the unramified Shintani functions for [formula omitted

Let F be a non-archimedean local field of arbitrary characteristic. We shall prove the uniqueness of the unramified Shintani functions for (GSp4(F),(GL2×GL1GL2)(F)) and give an explicit formula of those. Our result is a generalization of the result for the split special orthogonal group SO5 due to Kato–Murase–Sugano.

A VARIANT OF HARISH-CHANDRA FUNCTORS

Harish-Chandra induction and restriction functors play a key role in the representation theory of reductive groups over finite fields. In this paper, extending earlier work of Dat, we introduce and study generalisations of these functors which apply to a wide range of finite and profinite groups, typical examples being compact open subgroups of reductive groups over non-archimedean local fields. We prove that these generalisations are compatible with two of the tools commonly used to study the (smooth, complex) representations of such groups, namely Clifford theory and the orbit method. As a test case, we examine in detail the induction and restriction of representations from and to the Siegel Levi subgroup of the symplectic group $\text{Sp}_{4}$ over a finite local principal ideal ring of length two. We obtain in this case a Mackey-type formula for the composition of these induction and restriction functors which is a perfect analogue of the well-known formula for the composition of Harish-Chandra functors. In a different direction, we study representations of the Iwahori subgroup $I_{n}$ of $\text{GL}_{n}(F)$, where $F$ is a non-archimedean local field. We establish a bijection between the set of irreducible representations of $I_{n}$ and tuples of primitive irreducible representations of smaller Iwahori subgroups, where primitivity is defined by the vanishing of suitable restriction functors.

Twisting formula of epsilon factors

For characters of a non-Archimedean local field we have explicit formula for epsilon factors. But in general, we do not have any generalized twisting formula of epsilon factors. In this paper, we give a generalized twisting formula of epsilon factors via local Jacobi sums.

On the order of vanishing of newforms at cusps

Let $E$ be an elliptic curve over $\mathbb{Q}$ of conductor $N$. We obtain an explicit formula, as a product of local terms, for the ramification index at each cusp of a modular parametrization of $E$ by $X_0(N)$. Our formula shows that the ramification index always divides 24, a fact that had been previously conjectured by Brunault as a result of numerical computations. In fact, we prove a more general result which gives the order of vanishing at each cusp of a holomorphic newform of arbitary level, weight and character, provided its field of rationality satisfies a certain condition. The above result relies on a purely $p$-adic computation of possibly independent interest. Let $F$ be a non-archimedean local field and $\pi$ an irreducible, admissible, generic representation of $\mathrm{GL}_2(F)$. We introduce a new integral invariant, which we call the \emph{vanishing index} and denote $e_\pi(l)$, that measures the degree of "extra vanishing" at matrices of level $l$ of the Whittaker function associated to the newvector of $\pi$. Our main local result writes down the value of $e_\pi(l)$ in every case.

Parahoric Restriction for GSp(4)

We determine the parahoric restriction of non-cuspidal irreducible smooth representations of GSp(4,F) for a local non-archimedean number field F.