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We show that compatible systems of
$\ell$
-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on
$\ell$
-independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.
Let
$F$
be a non-Archimedean locally compact field of residual characteristic
$p$
with Weil group
${\mathcal{W}}_{F}$
. Let
$\unicode[STIX]{x1D70E}$
be an irreducible smooth complex representation of
${\mathcal{W}}_{F}$
, realized as the Langlands parameter of an irreducible cuspidal representation
$\unicode[STIX]{x1D70B}$
of a general linear group over
$F$
. In an earlier paper we showed that the ramification structure of
$\unicode[STIX]{x1D70E}$
is determined by the fine structure of the endo-class
$\unicode[STIX]{x1D6E9}$
of the simple character contained in
$\unicode[STIX]{x1D70B}$
, in the sense of Bushnell and Kutzko. The connection is made via the Herbrand function
$\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$
of
$\unicode[STIX]{x1D6E9}$
. In this paper we concentrate on the fundamental Carayol case in which
$\unicode[STIX]{x1D70E}$
is totally wildly ramified with Swan exponent not divisible by
$p$
. We show that, for such
$\unicode[STIX]{x1D70E}$
, the associated Herbrand function satisfies a certain functional equation, and that this property essentially characterizes this class of representations. We calculate
$\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$
explicitly, in terms of a classical Herbrand function arising naturally from the classification of simple characters. We describe exactly the class of functions arising as Herbrand functions
$\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6EF}}$
, as
$\unicode[STIX]{x1D6EF}$
varies over the set of totally wild endo-classes of Carayol type. In a separate argument, we derive a complete description of the restriction of
$\unicode[STIX]{x1D70E}$
to any ramification subgroup and hence a detailed interpretation of the Herbrand function. This gives concrete information concerning the Langlands correspondence.
A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz’s
$2$
-variable
$p$
-adic
$L$
-functions) and algebraic objects (two ‘everywhere unramified’ Iwasawa modules) involving codimension two cycles in a
$2$
-variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field
$K$
(where an odd prime
$p$
splits) of an elliptic curve
$E$
, defined over
$\mathbb{Q}$
, with good supersingular reduction at
$p$
. On the analytic side, we consider eight pairs of
$2$
-variable
$p$
-adic
$L$
-functions in this setup (four of the
$2$
-variable
$p$
-adic
$L$
-functions have been constructed by Loeffler and a fifth
$2$
-variable
$p$
-adic
$L$
-function is due to Hida). On the algebraic side, we consider modifications of fine Selmer groups over the
$\mathbb{Z}_{p}^{2}$
-extension of
$K$
. We also provide numerical evidence, using algorithms of Pollack, towards a pseudonullity conjecture of Coates–Sujatha.
We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa’s conjecture on exponential sums, with the log canonical threshold in the exponent of the estimates. We show that this covers optimally all situations of the conjectures for nonrational singularities by comparing the log canonical threshold with a local notion of the motivic oscillation index.
Given a property of representations satisfying a basic stability condition, Ramakrishna developed a variant of Mazur’s Galois deformation theory for representations with that property. We introduce an axiomatic definition of pseudorepresentations with such a property. Among other things, we show that pseudorepresentations with a property enjoy a good deformation theory, generalizing Ramakrishna’s theory to pseudorepresentations.
Let
$p$
be an odd prime number and
$E$
an elliptic curve defined over a number field
$F$
with good reduction at every prime of
$F$
above
$p$
. We compute the Euler characteristics of the signed Selmer groups of
$E$
over the cyclotomic
$\mathbb{Z}_{p}$
-extension. The novelty of our result is that we allow the elliptic curve to have mixed reduction types for primes above
$p$
and mixed signs in the definition of the signed Selmer groups.
For a character of the absolute Galois group of a complete discrete valuation field, we define a lifting of the refined Swan conductor, using higher dimensional class field theory.
Let
$p$
be a prime, let
$K$
be a complete discrete valuation field of characteristic
$0$
with a perfect residue field of characteristic
$p$
, and let
$G_{K}$
be the Galois group. Let
$\unicode[STIX]{x1D70B}$
be a fixed uniformizer of
$K$
, let
$K_{\infty }$
be the extension by adjoining to
$K$
a system of compatible
$p^{n}$
th roots of
$\unicode[STIX]{x1D70B}$
for all
$n$
, and let
$L$
be the Galois closure of
$K_{\infty }$
. Using these field extensions, Caruso constructs the
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$
-modules, which classify
$p$
-adic Galois representations of
$G_{K}$
. In this paper, we study locally analytic vectors in some period rings with respect to the
$p$
-adic Lie group
$\operatorname{Gal}(L/K)$
, in the spirit of the work by Berger and Colmez. Using these locally analytic vectors, and using the classical overconvergent
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$
-modules, we can establish the overconvergence property of the
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$
-modules.
We show that the Galois cohomology groups of
$p$
-adic representations of a direct power of
$\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$
can be computed via the generalization of Herr’s complex to multivariable
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$
-modules. Using Tate duality and a pairing for multivariable
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$
-modules we extend this to analogues of the Iwasawa cohomology. We show that all
$p$
-adic representations of a direct power of
$\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$
are overconvergent and, moreover, passing to overconvergent multivariable
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$
-modules is an equivalence of categories. Finally, we prove that the overconvergent Herr complex also computes the Galois cohomology groups.
We discuss the generalizations of the concept of Chebyshev’s bias from two perspectives. First, we give a general framework for the study of prime number races and Chebyshev’s bias attached to general L-functions satisfying natural analytic hypotheses. This extends the cases previously considered by several authors and involving, among others, Dirichlet L-functions and Hasse–Weil L-functions of elliptic curves over Q. This also applies to new Chebyshev’s bias phenomena that were beyond the reach of the previously known cases. In addition, we weaken the required hypotheses such as GRH or linear independence properties of zeros of L-functions. In particular, we establish the existence of the logarithmic density of the set
$ \{x \ge 2:\sum\nolimits_{p \le x} {\lambda _f}(p) \ge 0\}$
for coefficients (λf(p)) of general L-functions conditionally on a much weaker hypothesis than was previously known.
We generalize the Cohen–Lenstra heuristics over function fields to étale group schemes
$G$
(with the classical case of abelian groups corresponding to constant group schemes). By using the results of Ellenberg–Venkatesh–Westerland, we make progress towards the proof of these heuristics. Moreover, by keeping track of the image of the Weil-pairing as an element of
$\wedge ^{2}G(1)$
, we formulate more refined heuristics which nicely explain the deviation from the usual Cohen–Lenstra heuristics for abelian
$\ell$
-groups in cases where
$\ell \mid q-1$
; the nature of this failure was suggested already in the works of Malle, Garton, Ellenberg–Venkatesh–Westerland, and others. On the purely large random matrix side, we provide a natural model which has the correct moments, and we conjecture that these moments uniquely determine a limiting probability measure.
Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct
$p$
-adic
$L$
-functions for non-critical slope rational modular forms, the theory has been extended to construct
$p$
-adic
$L$
-functions for non-critical slope automorphic forms over totally real and imaginary quadratic fields by the first and second authors, respectively. In this paper, we give an analogous construction over a general number field. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspace. We then show that if one takes the modular symbol attached to a small slope cuspidal eigenform, then one can construct a ray class distribution from the corresponding overconvergent symbol, which moreover interpolates critical values of the
$L$
-function of the eigenform. We prove that this distribution is independent of the choices made in its construction. We define the
$p$
-adic
$L$
-function of the eigenform to be this distribution.
Let
$p$
be a prime and let
$G$
be a finite group. By a celebrated theorem of Swan, two finitely generated projective
$\mathbb{Z}_{p}[G]$
-modules
$P$
and
$P^{\prime }$
are isomorphic if and only if
$\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}P$
and
$\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}P^{\prime }$
are isomorphic as
$\mathbb{Q}_{p}[G]$
-modules. We prove an Iwasawa-theoretic analogue of this result and apply this to the Iwasawa theory of local and global fields. We thereby determine the structure of natural Iwasawa modules up to (pseudo-)isomorphism.
We establish some supercongruences for the truncated
$_{2}F_{1}$
and
$_{3}F_{2}$
hypergeometric series involving the
$p$
-adic gamma functions. Some of these results extend the four Rodriguez-Villegas supercongruences on the truncated
$_{3}F_{2}$
hypergeometric series. Related supercongruences modulo
$p^{3}$
are proposed as conjectures.
Let
$G$
be a
$p$
-adic group that splits over an unramified extension. We decompose
$\text{Rep}_{\unicode[STIX]{x1D6EC}}^{0}(G)$
, the abelian category of smooth level
$0$
representations of
$G$
with coefficients in
$\unicode[STIX]{x1D6EC}=\overline{\mathbb{Q}}_{\ell }$
or
$\overline{\mathbb{Z}}_{\ell }$
, into a product of subcategories indexed by inertial Langlands parameters. We construct these categories via systems of idempotents on the Bruhat–Tits building and Deligne–Lusztig theory. Then, we prove compatibilities with parabolic induction and restriction functors and the local Langlands correspondence.
A recent result by the authors gives an explicit construction for a universal deformation of a formal group
$\unicode[STIX]{x1D6F7}$
of finite height over a finite field
$k$
. This provides in particular a parametrization of the set of deformations of
$\unicode[STIX]{x1D6F7}$
over the ring
${\mathcal{O}}$
of Witt vectors over
$k$
. Another parametrization of the same set can be obtained through the Dieudonné theory. We find an explicit relation between these parameterizations. As a consequence, we obtain an explicit expression for the action of
$\text{Aut}_{k}(\unicode[STIX]{x1D6F7})$
on the set of
${\mathcal{O}}$
-deformations of
$\unicode[STIX]{x1D6F7}$
in the coordinate system defined by the universal deformation. This generalizes a formula of Gross and Hopkins and the authors’ result for one-dimensional formal groups.
We present a new construction of the
$p$
-adic local Langlands correspondence for
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
via the patching method of Taylor–Wiles and Kisin. This construction sheds light on the relationship between the various other approaches to both the local and the global aspects of the
$p$
-adic Langlands program; in particular, it gives a new proof of many cases of the second author’s local–global compatibility theorem and relaxes a hypothesis on the local mod
$p$
representation in that theorem.
Let
$K$
be a (non-archimedean) local field and let
$F$
be the function field of a curve over
$K$
. Let
$D$
be a central simple algebra over
$F$
of period
$n$
and
$\unicode[STIX]{x1D706}\in F^{\ast }$
. We show that if
$n$
is coprime to the characteristic of the residue field of
$K$
and
$D\cdot (\unicode[STIX]{x1D706})=0$
in
$H^{3}(F,\unicode[STIX]{x1D707}_{n}^{\otimes 2})$
, then
$\unicode[STIX]{x1D706}$
is a reduced norm from
$D$
. This leads to a Hasse principle for the group
$\operatorname{SL}_{1}(D)$
, namely, an element
$\unicode[STIX]{x1D706}\in F^{\ast }$
is a reduced norm from
$D$
if and only if it is a reduced norm locally at all discrete valuations of
$F$
.
Suppose that
$F/F^{+}$
is a CM extension of number fields in which the prime
$p$
splits completely and every other prime is unramified. Fix a place
$w|p$
of
$F$
. Suppose that
$\overline{r}:\operatorname{Gal}(\overline{F}/F)\rightarrow \text{GL}_{3}(\overline{\mathbb{F}}_{p})$
is a continuous irreducible Galois representation such that
$\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$
is upper-triangular, maximally non-split, and generic. If
$\overline{r}$
is automorphic, and some suitable technical conditions hold, we show that
$\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$
can be recovered from the
$\text{GL}_{3}(F_{w})$
-action on a space of mod
$p$
automorphic forms on a compact unitary group. On the way we prove results about weights in Serre’s conjecture for
$\overline{r}$
, show the existence of an ordinary lifting of
$\overline{r}$
, and prove the freeness of certain Taylor–Wiles patched modules in this context. We also show the existence of many Galois representations
$\overline{r}$
to which our main theorem applies.
We describe the ramification in cyclic extensions arising from the Kummer theory of the Weil restriction of the multiplicative group. This generalises the classical theory of Hecke describing the ramification of Kummer extensions.