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Nadir Matringe

Publications and source records attributed to Nadir Matringe.

At least 55 records · Page 3Linked to original sources
arXiv2017

Gamma factors root numbers and distinction

We study a relation between distinction and special values of local invariants for representations of the general linear group over a quadratic extension of $p$-adic fields. We show that the local Rankin-Selberg root number of any pair of distinguished representation is trivial and as a corollary we obtain an analogue for the global root number of any pair of distinguished cuspidal representations. We further study the extent to which the gamma factor at $1/2$ is trivial for distinguished representations as well as the converse problem.

math.RT↗
arXiv2016

Distinction of the Steinberg representation for inner forms of $GL(n)$

Let $F$ be a non archimedean local field of characteristic not $2$. Let $D$ be a division algebra of dimension $d^2$ over its center $F$, and $E$ a quadratic extension of $F$. If $m$ is a positive integer, to a character $χ$ of $E^*$, one can attach the Steinberg representation $St(χ)$ of $G=GL(m,D\otimes_F E)$. Let $H$ be the group $GL(m,D)$, we prove that $St(χ)$ is $H$-distinguished if and only if $χ_{|F^*}$ is the quadratic character $η_{E/F}^{md-1}$, where $η_{E/F}$ is the character of $F^*$ with kernel the norms of $E^*$. We also get multiplicity one for the space of invariant linear forms. As a corollary, we see that the Jacquet-Langlands correspondence preserves distinction for Steinberg representations.

math.RT↗
arXiv2016

Test vectors for local periods

Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic zero. An irreducible admissible representation $π$ of $GL(n,E)$ is said to be distinguished with respect to $GL(n,F)$ if it admits a non-trivial linear form that is invariant under the action of $GL(n,F)$. It is known that there is exactly one such invariant linear form up to multiplication by scalars, and an explicit linear form is given by integrating Whittaker functions over the $F$-points of the mirabolic subgroup when $π$ is unitary and generic. In this paper, we prove that the essential vector of [JPSS81] is a test vector for this standard distinguishing linear form and that the value of this form at the essential vector is a local $L$-value. As an application we determine the value of a certain proportionality constant between two explicit distinguishing linear forms. We then extend all our results to the non-unitary generic case.

math.NT↗
arXiv2016

Shalika periods and parabolic induction for GL(n) over a non archimedean local field

Let $F$ be a non archimedean local field, and $n_1$ and $n_2$ two positive even integers. We prove that if $π_1$ and $π_2$ are two smooth representations of $GL(n_1,F)$ and $GL(n_2,F)$ respectively, both admitting a Shalika period, then the normalised parabolically induced representation $π_1\times π_2$ also admits a Shalika period. Combining this with the results of \cite{M-localBF}, we obtain as a corollary the classification of generic representations of $GL(n,F)$ admitting a Shalika period when $F$ has characteristic zero. This result is relevant to the study of the Jacquet-Shalika exterior square $L$ factor.

math.RT↗
arXiv2016

Derivatives and asymptotics of Whittaker functions

Let $F$ be a $p$-adic field, and $G_n$ one of the groups $GL(n,F)$, $GSO(2n-1,F)$, $GSp(2n,F)$, or $GSO(2(n-1),F)$. Using the mirabolic subgroup or analogues of it, and related "derivative" functors, we give an asymptotic expansion of functions in the Whittaker model of generic representations of $G_n$, with respect to a minimal set of characters of subgroups of the maximal torus. Denoting by $Z_n$ the center of $G_n$, and by $N_n$ the unipotent radical of its standard Borel subgroup, we characterize generic representations occurring in $L^2(Z_nN_n\backslash G_n)$ in terms of these characters. This is related to a conjecture of Lapid and Mao for general split groups, asserting that the generic representations occurring in $L^2(Z_nN_n\backslash G_n)$ are the generic discrete series; we prove it for the group $G_n$.

math.RT↗
arXiv2016

Derivatives and Asymptotics of Whittaker functions

Let F be a p-adic field, and Gn one of the groups GL(n, F), GSO(2n-1, F), GSp(2n, F), or GSO(2(n - 1), F). Using the mirabolic subgroup or analogues of it, and related "derivative" functors, we give an asymptotic expansion of functions in the Whittaker model of generic representations of Gn, with respect to a minimal set of characters of subgroups of the maximal torus. Denoting by Zn the center of Gn, and by Nn the unipotent radical of its standard Borel subgroup, we characterize generic representations occurring in L2(ZnNn\Gn) in terms of these characters. This is related to a conjecture of Lapid and Mao for general split groups, asserting that the generic representations occurring in L2(ZnNn\Gn) are the generic discrete series; we prove it for the group Gn.

math.RT↗
arXiv2016

On the local Bump-Friedberg L function II

Let $F$ be a $p$-adic field with residue field of cardinality $q$. To each irreducible representation of $GL(n,F)$, we attach a local Euler factor $L^{BF}(q^{-s},q^{-t},π)$ via the Rankin-Selberg method, and show that it is equal to the expected factor $L(s+t+1/2,ϕ_π)L(2s,Λ^2\circ ϕ_π)$ of the Langlands' parameter $ϕ_π$ of $π$. The corresponding local integrals were introduced in [BF], and studied in [M15]. This work is in fact the continuation of [M15]. The result is a consequence of the fact that if $δ$ is a discrete series representation of $GL(2m,F)$, and $χ$ is a character of Levi subgoup $L=GL(m,F)\times GL(m,F)$, trivial on $GL(m,F)$ embedded diagonally, then $δ$ is $(L,χ)$-distinguished if an only if it admits a Shalika model, a result which was only established for $χ=1$ before.

math.NT↗
arXiv2015

Rankin-Selberg local factors modulo $\ell$

After extending the theory of Rankin-Selberg local factors to pairs of $\ell$-modular representations of Whittaker type, of general linear groups over a non-archimedean local field, we study the reduction modulo $\ell$ of $\ell$-adic local factors and their relation to these $\ell$-modular local factors. While the $\ell$-modular local $γ$-factor we associate to such a pair turns out to always coincide with the reduction modulo $\ell$ of the $\ell$-adic $γ$-factor of any Whittaker lifts of this pair, the local $L$-factor exhibits a more interesting behaviour; always dividing the reduction modulo-$\ell$ of the $\ell$-adic $L$-factor of any Whittaker lifts, but with the possibility of a strict division occurring. In our main results, we completely describe $\ell$-modular $L$-factors in the generic case. We obtain two simple to state nice formulae: Let $π,π'$ be generic $\ell$-modular representations; then, writing $π_b,π'_b$ for their banal parts, we have \[L(X,π,π')=L(X,π_b,π_b').\] Using this formula, we obtain the inductivity relations for local factors of generic representations. Secondly, we show that \[L(X,π,π')=\mathbf{GCD}(r_{\ell}(L(X,τ,τ'))),\] where the divisor is over all integral generic $\ell$-adic representations $τ$ and $τ'$ which contain $π$ and $π'$, respectively, as subquotients after reduction modulo $\ell$.

math.RT↗
arXiv2015

A specialisation of the Bump-Friedberg $L$-function

We study the restriction of the Bump-Friedberg integrals to affine lines $\{(s+α,2s),s\in\C\}$. It has a simple theory, very close to that of the Asai $L$-function. It is an integral representation of the product $L(s+α,π)L(2s,Λ^2,π)$ which we denote by $L^{lin}(s,π,α)$ for this abstract, when $π$ is a cuspidal automorphic representation of $GL(k,A)$ for $A$ the adeles of a number field. When $k$ is even, we show that for a cuspidal automorphic representation $π$, the partial $L$-function $L^{lin,S}(s,π,α)$ has a pole at 1/2, if and only if $π$ admits a (twisted) global period, this gives a more direct proof of a theorem of Jacquet and Friedberg, asserting that $π$ has a twisted global period if and only if $L(α+1/2,π)\neq 0$ and $L(1,Λ^2,π)=\infty$. When $k$ is odd, the partial $L$-function is holmorphic in a neighbourhood of $Re(s)\geq 1/2$ when $Re(α)$ is $\geq 0$.

math.NT↗
arXiv2014

On the Termination of Linear and Affine Programs over the Integers

The termination problem for affine programs over the integers was left open in\cite{Braverman}. For more that a decade, it has been considered and cited as a challenging open problem. To the best of our knowledge, we present here the most complete response to this issue: we show that termination for affine programs over Z is decidable under an assumption holding for almost all affine programs, except for an extremely small class of zero Lesbegue measure. We use the notion of asymptotically non-terminating initial variable values} (ANT, for short) for linear loop programs over Z. Those values are directly associated to initial variable values for which the corresponding program does not terminate. We reduce the termination problem of linear affine programs over the integers to the emptiness check of a specific ANT set of initial variable values. For this class of linear or affine programs, we prove that the corresponding ANT set is a semi-linear space and we provide a powerful computational methods allowing the automatic generation of these $ANT$ sets. Moreover, we are able to address the conditional termination problem too. In other words, by taking ANT set complements, we obtain a precise under-approximation of the set of inputs for which the program does terminate.

cs.DM↗
arXiv2014

Unitary representations of GL(n,K) distinguished by a Galois involution, for K a p-adic field

Let $F$ be a $p$-adic field, and $K$ a quadratic extension of $F$. Using Tadic's classification of the unitary dual of $GL(n,K)$, we give the list of irreducible unitary representations of this group distinguished by $GL(n,F)$, in terms of distinguished discrete series. As it is known that a generalised Steinberg representation $St(k,ρ)$ is distinguished if and only if the cuspidal representation $ρ$ is $η^{k-1}$-distinguished, for $η$ the character of $F^*$ with kernel the norms of $K^*$, this actually gives a classification of distinguished unitary representations in terms of distinguished cuspidal representations.

math.RT↗
arXiv2014

Characterization of Termination for Linear Loop Programs

We present necessary and sufficient conditions for the termination of linear homogeneous programs. We also develop a complete method to check termination for this class of programs. Our complete characterization of termination for such programs is based on linear algebraic methods. We reduce the verification of the termination problem to checking the orthogonality of a well determined vector space and a certain vector, both related to loops in the program. Moreover, we provide theoretical results and symbolic computational methods guaranteeing the soundness, completeness and numerical stability of the approach. Finally, we show that it is enough to interpret variable values over a specific countable number field, or even over its ring of integers, when one wants to check termination over the reals.

cs.PL↗
arXiv2014

Generating Asymptotically Non-Terminating Initial Values for Linear Programs

We present the notion of asymptotically non-terminating initial variable values for linear loop programs. Those values are directly associated to initial variable values for which the corresponding program does not terminate. Our theoretical contributions provide us with powerful computational methods for automatically generating sets of asymptotically non-terminating initial variable values. Such sets are represented symbolically and exactly by a semi-linear space, e.g., characterized by conjunctions and disjunctions of linear equalities and inequalities. Moreover, by taking their complements, we obtain a precise under-approximation of the set of inputs for which the program does terminate. We can then reduce the termination problem of linear programs to the emptiness check of a specific set of asymptotically non-terminating initial variable values. Our static input data analysis is not restricted only to programs where the variables are interpreted over the reals. We extend our approach and provide new decidability results for the termination problem of affine integer and rational programs.

cs.DM↗
arXiv2013

On the local Bump-Friedberg L-function

Let $F$ be a $p$-adic field. If $π$ be an irreducible representation of $GL(n,F)$, Bump and Friedberg associated to $π$ an Euler fator $L(π,BF,s_1,s_2)$ in \cite{BF}, that should be equal to $L(ϕ(π),s_1)L(ϕ(π),Λ^2,s_2)$, where $ϕ(π)$ is the Langlands' parameter of $π$. The main result of this paper is to show that this equality is true when $(s_1,s_2)=(s+1/2,2s)$, for $s$ in $\C$. To prove this, we classify in terms of distinguished discrete series, generic representations of $GL(n,F)$ which are $χ_α$-distinguished by the Levi subgroup $GL([(n+1)/2],F) \times GL([n/2],F)$, for $χ_α(g_1,g_2)=α(det(g_1)/det(g_2))$, where $α$ is a character of $F^*$ of real part between -1/2 and 1/2. We then adapt the technique of \cite{CP} to reduce the proof of the equality to the case of discrete series. The equality for discrete series is a consequence of the relation between linear periods and Shalika periods for discrete series, and the main result of \cite{KR}.

math.NT↗
arXiv2009

Distinction and Asai $L$-functions for generic representations of general linear groups over p-adic fields

Let $K/F$ be a quadratic extension of $p$-adic fields, and $n$ a positive integer. A smooth irreducible representation of the group $GL(n,K)$ is said to be distinguished, if it admits on its space a nonzero $GL(n,F)$-invariant linear form. In the present work, we classify genric distinguished representations of the group $GL(n,K)$ in terms of inducing quasi-square-integrable representations. This has as a consequence the truth of the expected equality between the Rankin-Selberg type Asai $L$-function of a generic representation, and the Asai $L$-function of its Langlands parameter.

math.RT↗