Epsilon Factors Research Articles

On Continuity of Local Epsilon Factors of ℓ-adic Sheaves

Abstract Let $S$ be a noetherian scheme and $ Y\to S$ be a smooth morphism of relative dimension $1$. For a locally constant sheaf on the complement of a divisor in $Y$ flat over $S$, Deligne and Laumon proved that the universal local acyclicity is equivalent to the local constancy of Swan conductors. In this article, assuming the universal local acyclicity, we show an analogous result of continuity of local epsilon factors. We also give a generalization of this result to a family of isolated singularities.

Galois self-dual cuspidal types and Asai local factors

Let \mathrm{F}/\mathrm{F}_{\mathsf{o}} be a quadratic extension of non-archimedean locally compact fields of odd residual characteristic and \sigma be its non-trivial automorphism. We show that any \sigma -self-dual cuspidal representation of \operatorname{GL}_n(\mathrm{F}) contains a \sigma -self-dual Bushnell–Kutzko type. Using such a type, we construct an explicit test vector for Flicker's local Asai L-function of a \operatorname{GL}_n(\mathrm{F}_\mathsf{o}) -distinguished cuspidal representation and compute the associated Asai root number. Finally, by using global methods, we compare this root number to Langlands–Shahidi's local Asai root number, and more generally we compare the corresponding epsilon factors for any cuspidal representation.

Epsilon factors of symplectic type characters in the wild case

Abstract By work of John Tate we can associate an epsilon factor with every multiplicative character of a local field. In this paper, we determine the explicit signs of the epsilon factors for symplectic type characters of K × {K^{\times}} , where K / F {K/F} is a wildly ramified quadratic extension of a non-Archimedean local field F of characteristic zero.

Epsilon factors of representations of finite general linear groups

We define epsilon factors for irreducible representations of finite general linear groups using Macdonald's correspondence. These epsilon factors satisfy multiplicativity, and are expressible as products of Gauss sums. The tensor product epsilon factors are related to the Rankin-Selberg gamma factors, by which we prove that the Rankin-Selberg gamma factors can be written as products of Gauss sums. The exterior square epsilon factors relate the Jacquet-Shalika exterior square gamma factors and the Langlands-Shahidi exterior square gamma factors for level zero supercuspidal representations. We prove that these exterior square factors coincide in a special case.

A note on quadratic twisting of epsilon factors for modular forms with arbitrary nebentypus

In this article, we investigate the variance of local $\varepsilon$-factor for a modular form with arbitrary nebentypus with respect to twisting by a quadratic character. We detect the type of the supercuspidal representation from that. For modular forms with trivial nebentypus, similar results are proved by Pacetti. Our method however is completely different from that of Pacetti and we use representation theory crucially. For ramified principal series (with $\infdiv{p}{N}$ and $p$ odd) and unramified supercuspidal representations of level zero, we relate these numbers with the Morita's $p$-adic Gamma function.

Epsilon factors as algebraic characters on the smooth dual of $\mathrm {GL}_n$

Let $K$ be a non-archimedean local field and let $G = \mathrm{GL}_n(K)$. We have shown in previous work that the smooth dual $\mathbf{Irr}(G)$ admits a complex structure: in this article we show how the epsilon factors interface with this complex structure. The epsilon factors, up to a constant term, factor as invariant characters through the corresponding complex tori. For the arithmetically unramified smooth dual of $\mathrm{GL}_n$, we provide explicit formulas for the invariant characters.

On a localization formula of epsilon factors via microlocal geometry

Given a lisse l -adic sheaf G on a smooth proper variety X and a lisse sheaf ℱ on an open dense U in X , Kato and Saito conjectured a localization formula for the global l -adic epsilon factor e l ( X , ℱ ⊗ G ) in terms of the global epsilon factor of ℱ and a certain intersection number associated to det ( G ) and the Swan class of ℱ . In this article, we prove an analog of this conjecture for global de Rham epsilon factors in the classical setting of D X -modules on smooth projective varieties over a field of characteristic zero.

The effect of interpersonal communication skills and work motivation on performance of marketing employee

To improve and maintain the performance of the marketing department employee, the company should improve the capacity and capability of the employees. This research aimed to investigate the influence of interpersonal communication and work motivation on employee performance marketing. This research uses a quantitative approach method with path analysis method. This study uses survey technique with the primary data collection tool in the form of a questionnaire. The study was conducted at the Jabar Banten Shariah Bank Bandung branch, with respondents as many as 33 people. The test results show that partial and or simultaneously, interpersonal communication and work motivation have significant positive effect on employee performance. The results show that the value of the determinant obtained for 0808, this means that the variables of interpersonal communication and motivation can explain the employees’ performance by 80.0 %, while 19.2 % is influenced by other variables that are not included in the model. Epsilon factors are leadership, organizational culture, facilities and infrastructure in the works, and others. The result show that performance of the marketing department can be improved through the development of interpersonal communication capacity of employees and strengthening of motivation to work.

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.

Triple product formula and the subconvexity bound of Triple product L-function in level aspect

In this paper we derive a nice general formula for the local integrals of triple product formula whenever one of the representations has sufficiently higher level than the other two. As an application we generalize Venkatesh and Woodbury's work on the subconvexity bound of triple product $L$-function in level aspect, allowing joint ramifications, higher ramifications, general unitary central characters, and general special values of local epsilon factors.

Endoscopie et conjecture locale raffinée de Gan–Gross–Prasad pour les groupes unitaires

Under endoscopic assumptions about $L$-packets of unitary groups, we prove the local Gan–Gross–Prasad conjecture for tempered representations of unitary groups over $p$-adic fields. Roughly, this conjecture says that branching laws for $U(n-1)\subset U(n)$ can be computed using epsilon factors.

Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic $\mathcal D$-modules

The aim of this paper is to compute the Frobenius structures of some cohomological operators of arithmetic {\mathcal D} -modules. To do this, we calculate explicitly an isomorphism between canonical sheaves defined abstractly. Using this calculation, we establish the relative Poincaré duality in the style of SGA4. As another application, we compare the push-forward as arithmetic {\mathcal D} -modules and the rigid cohomologies taking Frobenius into account. These theorems will be used to prove " p -adic Weil II" and a product formula for p -adic epsilon factors.

Product formula forp-adic epsilon factors

AbstractLet$X$be a smooth proper curve over a finite field of characteristic$p$. We prove a product formula for$p$-adic epsilon factors of arithmetic$\mathscr{D}$-modules on$X$. In particular we deduce the analogous formula for overconvergent$F$-isocrystals, which was conjectured previously. The$p$-adic product formula is a counterpart in rigid cohomology of the Deligne–Laumon formula for epsilon factors in$\ell$-adic étale cohomology (for$\ell \neq p$). One of the main tools in the proof of this$p$-adic formula is a theorem of regular stationary phase for arithmetic$\mathscr{D}$-modules that we prove by microlocal techniques.

On epsilon factors attached to supercuspidal representations of unramified 𝑈(2,1)

Let G G be the unramified unitary group in three variables defined over a p p -adic field F F with p ≠ 2 p \neq 2 . Gelbart, Piatetski-Shapiro and Baruch attached zeta integrals of Rankin-Selberg type to irreducible generic representations of G G . In this paper, we formulate a conjecture on L L - and ε \varepsilon -factors defined through zeta integrals in terms of newforms for G G , which is an analogue of the result by Casselman and Deligne for G L ( 2 ) \mathrm {GL}(2) . We prove our conjecture for the generic supercuspidal representations of G G .

Local L and epsilon factors in Hecke eigenvalues

Formulas (Theorems 3.5 and 4.1) which express the local L -factor and the local epsilon factor of an irreducible admissible representation of GL d over a non-archimedean local field in terms of the eigenvalues of some explicitly given Hecke operators are derived.

Theta dichotomy and doubling epsilon factors for Sp͠ n ( F )

In this work, we aim to prove the theta dichotomy conjecture for dual pairs of type $(\widetilde{{\rm Sp}}(W),{\rm O}(V))$ with $\dim(V)$ odd. The crux of the proof relies on the analysis of the reducibility of a certain degenerate principal series representation of the group $\widetilde{{\rm Sp}}(W')$ having double the rank of $\widetilde{{\rm Sp}}(W)$. This analysis is carried out by way of a normalized standard intertwining operator which is also used to define an epsilon factor coming from the doubling integral construction of Piatetski-Shapiro and Rallis. In this paper, we also illustrate the connection between the doubling epsilon factor and the theta dichotomy conjecture.

Hasse-Arf filtrations in 𝑝-adic analytic geometry

We develop a theory of local Fourier transforms for abelian sheaves on the étale site of a p p -adic punctured disc, and we prove a principle of stationary phase linking these local Fourier transforms to the global Fourier transform that was introduced in one of our earlier works. We use this theory to study the local monodromy of abelian sheaves on the étale site of a p p -adic punctured disc, and in particular we exhibit a natural slope decomposition for (germs of) such sheaves, with properties that are wholly analogous to those of the slope decomposition for representation of the Galois group of a local field. We conclude with an application to the study of the so-called cohomological epsilon factor of a locally constant abelian sheaf on the étale site of an affine curve defined over a p p -adic field: namely, we exhibit a decomposition of such factor as a tensor product of “local epsilon factors” that depend only on the local monodromies of the abelian sheaf.

On counting twists of a character appearing in its associated Weil representation

Consider an irreducible, admissible representation $\pi$ of GL(2,$F$) whose restriction to GL(2,$F)^+$ breaks up as a sum of two irreducible representations $\pi_+ + \pi_-$. If $\pi=r_{\theta}$, the Weil representation of GL(2,$F$) attached to a character $\theta$ of $K^*$ which does not factor through the norm map from $K$ to $F$, then $\chi\in \hat{K^*}$ with $(\chi >. \theta ^{-1})|_{F^{*}}=\omega_{{K/F}}$ occurs in ${r_{\theta}}_+$ if and only if $\epsilon(\theta\chi^{-1},\psi_0)=\epsilon(\bar \theta\chi^{-1},\psi_0)=1$ and in ${r_{\theta}}_-$ if and only if both the epsilon factors are -1. But given a conductor $n$, can we say precisely how many such $\chi$ will appear in $\pi$? We calculate the number of such characters at each given conductor $n$ in this work.

Bessel models for GSp(4)

Methods of theta correspondence are used to analyze local and global Bessel models for GSp 4 proving a conjecture of Gross and Prasad which describes these models in terms of local epsilon factors in the local case, and the non-vanishing of central critical L-value in the global case.

Local Fourier transform and epsilon factors

AbstractLaumon introduced the local Fourier transform forℓ-adic Galois representations of local fields, of equal characteristicpdifferent fromℓ, as a powerful tool for studying the Fourier–Deligne transform ofℓ-adic sheaves over the affine line. In this article, we compute explicitly the local Fourier transform of monomial representations satisfying a certain ramification condition, and deduce Laumon’s formula relating the ε-factor to the determinant of the local Fourier transform under the same condition.