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In this paper, we give an explicit computable algorithm for the Zelevinsky–Aubert duals of irreducible representations of $p$-adic symplectic and odd special orthogonal groups. To do this, we establish explicit formulas for certain derivatives and socles. We also give a combinatorial criterion for the irreducibility of certain parabolically induced representations.
For each prime p, we show that there exist geometrically simple abelian varieties A over
${\mathbb Q}$
with . Specifically, for any prime
$N\equiv 1 \ \pmod p$
, let
$A_f$
be an optimal quotient of
$J_0(N)$
with a rational point P of order p, and let
$B = A_f/\langle P \rangle $
. Then the number of positive integers
$d \leq X$
with is
$ \gg X/\log X$
, where
$\widehat B_d$
is the dual of the dth quadratic twist of B. We prove this more generally for abelian varieties of
$\operatorname {\mathrm {GL}}_2$
-type with a p-isogeny satisfying a mild technical condition. In the special case of elliptic curves, we give stronger results, including many examples where for an explicit positive proportion of integers d.
Motivated by the p-adic approach in two of Mahler’s problems, we obtain some results on p-adic analytic interpolation of sequences of integers
$(u_n)_{n\geq 0}$
. We show that if
$(u_n)_{n\geq 0}$
is a sequence of integers with
$u_n = O(n)$
which can be p-adically interpolated by an analytic function
$f:\mathbb {Z}_p\rightarrow \mathbb {Q}_p$
, then
$f(x)$
is a polynomial function of degree at most one. The case
$u_n=O(n^d)$
with
$d>1$
is also considered with additional conditions. Moreover, if X and Y are subsets of
$\mathbb {Z}$
dense in
$\mathbb {Z}_p$
, we prove that there are uncountably many p-adic analytic injective functions
$f:\mathbb {Z}_p\to \mathbb {Q}_p$
, with rational coefficients, such that
$f(X)=Y$
.
Let X be a real prehomogeneous vector space under a reductive group G, such that X is an absolutely spherical G-variety with affine open orbit. We define local zeta integrals that involve the integration of Schwartz–Bruhat functions on X against generalized matrix coefficients of admissible representations of
$G(\mathbb {R})$
, twisted by complex powers of relative invariants. We establish the convergence of these integrals in some range, the meromorphic continuation, as well as a functional equation in terms of abstract
$\gamma $
-factors. This subsumes the archimedean zeta integrals of Godement–Jacquet, those of Sato–Shintani (in the spherical case), and the previous works of Bopp–Rubenthaler. The proof of functional equations is based on Knop’s results on Capelli operators.
We show that the theory of Galois actions of a torsion Abelian group A is companionable if and only if, for each prime p, the p-primary part of A is either finite or it coincides with the Prüfer p-group. We also provide a model-theoretic description of the model companions we obtain.
When $p$ is an odd prime, Delbourgo observed that any Kubota–Leopoldt $p$-adic $L$-function, when multiplied by an auxiliary Euler factor, can be written as an infinite sum. We shall establish such expressions without restriction on $p$, and without the Euler factor when the character is non-trivial, by computing the periods of appropriate measures. As an application, we will reprove the Ferrero–Greenberg formula for the derivative $L_p'(0,\chi )$. We will also discuss the convergence of sum expressions in terms of elementary $p$-adic analysis, as well as their relation to Stickelberger elements; such discussions in turn give alternative proofs of the validity of sum expressions.
Kottwitz’s conjecture describes the contribution of a supercuspidal representation to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze’s more general spaces of local shtukas. Using a new Lefschetz–Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that, for an irreducible smooth representation of an inner form of
$\operatorname {\mathrm {GL}}_n$
, the L-parameter constructed by Fargues–Scholze agrees with the usual semisimplified parameter arising from local Langlands.
We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse–Schmidt derivations [13] and about Galois actions [14]. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.
We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base monoidal differential graded (dg)-category. Moreover, we prove Künneth formulas for dg-category of singularities and for inertia-invariant vanishing cycles. As an application, we prove a categorical version of Bloch's conductor conjecture (originally stated by Spencer Bloch in 1985), under the additional hypothesis that the monodromy action of the inertia group is unipotent.
We show that a rational function f of degree
$>1$
on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value has no potentially good reductions if and only if the Berkovich Julia set of f is uniformly perfect. As an application, a uniform regularity of the boundary of each Berkovich Fatou component of f is also established.
We prove some qualitative results about the p-adic Jacquet–Langlands correspondence defined by Scholze, in the
$\operatorname {\mathrm {GL}}_2(\mathbb{Q}_p )$
residually reducible case, using a vanishing theorem proved by Judith Ludwig. In particular, we show that in the cases under consideration, the global p-adic Jacquet–Langlands correspondence can also deal with automorphic forms with principal series representations at p in a nontrivial way, unlike its classical counterpart.
For
$c \in \mathbb {Q}$
, consider the quadratic polynomial map
$\varphi _c(z)=z^2-c$
. Flynn, Poonen, and Schaefer conjectured in 1997 that no rational cycle of
$\varphi _c$
under iteration has length more than
$3$
. Here, we discuss this conjecture using arithmetic and combinatorial means, leading to three main results. First, we show that if
$\varphi _c$
admits a rational cycle of length
$n \ge 3$
, then the denominator of c must be divisible by
$16$
. We then provide an upper bound on the number of periodic rational points of
$\varphi _c$
in terms of the number s of distinct prime factors of the denominator of c. Finally, we show that the Flynn–Poonen–Schaefer conjecture holds for
$\varphi _c$
if
$s \le 2$
, i.e., if the denominator of c has at most two distinct prime factors.
We prove a local–global compatibility result in the mod $p$ Langlands program for $\mathrm {GL}_2(\mathbf {Q}_{p^f})$. Namely, given a global residual representation $\bar {r}$ appearing in the mod $p$ cohomology of a Shimura curve that is sufficiently generic at $p$ and satisfies a Taylor–Wiles hypothesis, we prove that the diagram occurring in the corresponding Hecke eigenspace of mod $p$ completed cohomology is determined by the restrictions of $\bar {r}$ to decomposition groups at $p$. If these restrictions are moreover semisimple, we show that the $(\varphi ,\Gamma )$-modules attached to this diagram by Breuil give, under Fontaine's equivalence, the tensor inductions of the duals of the restrictions of $\bar {r}$ to decomposition groups at $p$.
Let G be a connected reductive group over a p-adic number field F. We propose and study the notions of G-
$\varphi $
-modules and G-
$(\varphi ,\nabla )$
-modules over the Robba ring, which are exact faithful F-linear tensor functors from the category of G-representations on finite-dimensional F-vector spaces to the categories of
$\varphi $
-modules and
$(\varphi ,\nabla )$
-modules over the Robba ring, respectively, commuting with the respective fiber functors. We study Kedlaya’s slope filtration theorem in this context, and show that G-
$(\varphi ,\nabla )$
-modules over the Robba ring are “G-quasi-unipotent,” which is a generalization of the p-adic local monodromy theorem proved independently by Y. André, K. S. Kedlaya, and Z. Mebkhout.
Let
$N/K$
be a finite Galois extension of p-adic number fields, and let
$\rho ^{\mathrm {nr}} \colon G_K \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$
be an r-dimensional unramified representation of the absolute Galois group
$G_K$
, which is the restriction of an unramified representation
$\rho ^{\mathrm {nr}}_{{{\mathbb Q}}_{p}} \colon G_{{\mathbb Q}_{p}} \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$
. In this paper, we consider the
$\mathrm {Gal}(N/K)$
-equivariant local
$\varepsilon $
-conjecture for the p-adic representation
$T = \mathbb Z_p^r(1)(\rho ^{\mathrm {nr}})$
. For example, if A is an abelian variety of dimension r defined over
${{\mathbb Q}_{p}}$
with good ordinary reduction, then the Tate module
$T = T_p\hat A$
associated to the formal group
$\hat A$
of A is a p-adic representation of this form. We prove the conjecture for all tame extensions
$N/K$
and a certain family of weakly and wildly ramified extensions
$N/K$
. This generalizes previous work of Izychev and Venjakob in the tame case and of the authors in the weakly and wildly ramified case.
We prove a necessary and sufficient condition for isogenous elliptic curves based on the algebraic dependence of p-adic elliptic functions. As a consequence, we give a short proof of the p-adic analogue of Schneider’s theorem on the linear independence of p-adic elliptic logarithms of algebraic points on two nonisogenous elliptic curves defined over the field of algebraic numbers.
The plus and minus norm groups are constructed by Kobayashi as subgroups of the formal group of an elliptic curve with supersingular reduction, and they play an important role in Kobayashi’s definition of the signed Selmer groups. In this paper, we study the cohomology of these plus and minus norm groups. In particular, we show that these plus and minus norm groups are cohomologically trivial. As an application of our analysis, we establish certain (quasi-)projectivity properties of the non-primitive mixed signed Selmer groups of an elliptic curve with good reduction at all primes above p. We then build on these projectivity results to derive a Kida formula for the signed Selmer groups under a slight weakening of the usual µ = 0 assumption, and study the integrality property of the characteristic element attached to the signed Selmer groups.
Let G be a reductive p-adic group which splits over an unramified extension of the ground field. Hiraga, Ichino and Ikeda [24] conjectured that the formal degree of a square-integrable G-representation
$\pi $
can be expressed in terms of the adjoint
$\gamma $
-factor of the enhanced L-parameter of
$\pi $
. A similar conjecture was posed for the Plancherel densities of tempered irreducible G-representations.
We prove these conjectures for unipotent G-representations. We also derive explicit formulas for the involved adjoint
$\gamma $
-factors.
The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $\Pi$ of $\operatorname {GL}_{2n}$ over a totally real field $F$ admitting a Shalika model. We use a modular symbol approach, along the global lines of the work of Ash and Ginzburg, but our results are more definitive because we draw heavily upon the methods used in the recent and separate works of all three authors. By construction, our $p$-adic $L$-functions are distributions on the Galois group of the maximal abelian extension of $F$ unramified outside $p\infty$. Moreover, we work under a weaker Panchishkine-type condition on $\Pi _p$ rather than the full ordinariness condition. Finally, we prove the so-called Manin relations between the $p$-adic $L$-functions at all critical points. This has the striking consequence that, given a unitary $\Pi$ whose standard $L$-function admits at least two critical points, and given a prime $p$ such that $\Pi _p$ is ordinary, the central critical value $L(\frac {1}{2}, \Pi \otimes \chi )$ is non-zero for all except finitely many Dirichlet characters $\chi$ of $p$-power conductor.
Étant donné un groupe réductif $G$ sur une extension de degré fini de $\mathbb {Q}_p$ on classifie les $G$-fibrés sur la courbe introduite dans Fargues and Fontaine [Courbes et fibrés vectoriels en théorie de Hodge$p$-adique, Astérisque 406 (2018)]. Le résultat est interprété en termes de l'ensemble $B(G)$ de Kottwitz. On calcule également la cohomologie étale de la courbe à coefficients de torsion en lien avec la théorie du corps de classe local.