Adic Field Research Articles

Modular Representations of ${\rm GL}{(n)}$ Distinguished by ${\rm GL}{(n-1)}$ over a $p$-Adic Field

Let F be a non-Archimedean locally compact field, q be the cardinality of its residue field, and R be an algebraically closed field of characteristic l not dividing q⁠. We classify all irreducible smooth R-representations of GLn(F) having a non-zero GLn−1(F)-invariant linear form, when q is not congruent to 1 mod l⁠. Partial results in the case when q is 1 mod l show that, unlike the complex case, the space of GLn−1(F)-invariant linear forms has dimension 2 for certain irreducible representations..

Finiteness of the Moderate Rational Points of Once-punctured Elliptic Curves

In the present paper, we prove the {\it finiteness} of the set of {\it moderate} rational points of a once-punctured elliptic curve over a number field. This {\it finiteness} may be regarded as an analogue for a once-punctured elliptic curve of the well-known {\it finiteness} of the set of torsion rational points of an abelian variety over a number field. In order to obtain the {\it finiteness}, we discuss the {\it center} of the image of the pro-$l$ outer Galois action associated to a hyperbolic curve. In particular, we give, under the assumption that $l$ is {\it odd}, a {\it necessary and sufficient condition} for a certain hyperbolic curve over a generalized sub-$l$-adic field to have {\it trivial center}.

Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the $\mathrm{AdS}_3 / \mathrm{CFT}_2$ correspondence

One of the many remarkable properties of conformal field theory in two dimensions is its connection to algebraic geometry. Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which a priori depend on the analytic structure of the spacetime) can be formulated in purely algebraic language. This opens the door to interesting generalizations, obtained by taking another choice of field: for instance, the p-adics. We generalize the AdS/CFT correspondence according to this principle; the result is a formulation of holography in which the bulk geometry is discrete—the Bruhat–Tits tree for PGL(2,Qp)—but the group of bulk isometries nonetheless agrees with that of boundary conformal transformations and is not broken by discretization. We suggest that this forms the natural geometric setting for tensor networks that have been proposed as models of bulk reconstruction via quantum error correcting codes; in certain cases, geodesics in the Bruhat–Tits tree reproduce those constructed using quantum error correction. Other aspects of holography also hold: Standard holographic results for massive free scalar fields in a fixed background carry over to the tree, whose vertical direction can be interpreted as a renormalization-group scale for modes in the boundary CFT. Higher-genus bulk geometries (the BTZ black hole and its generalizations) can be understood straightforwardly in our setting, and the Ryu–Takayanagi formula for the entanglement entropy appears naturally.

Solutions of the cubic Fermat equation in ring class fields of imaginary quadratic fields (as periodic points of a 3-adic algebraic function)

Explicit solutions of the cubic Fermat equation are constructed in ring class fields [Formula: see text], with conductor [Formula: see text] prime to [Formula: see text], of any imaginary quadratic field [Formula: see text] whose discriminant satisfies [Formula: see text] (mod [Formula: see text]), in terms of the Dedekind [Formula: see text]-function. As [Formula: see text] and [Formula: see text] vary, the set of coordinates of all solutions is shown to be the exact set of periodic points of a single algebraic function and its inverse defined on natural subsets of the maximal unramified, algebraic extension [Formula: see text] of the [Formula: see text]-adic field [Formula: see text]. This is used to give a dynamical proof of a class number relation of Deuring. These solutions are then used to give an unconditional proof of part of Aigner’s conjecture: the cubic Fermat equation has a nontrivial solution in [Formula: see text] if [Formula: see text] (mod [Formula: see text]) and the class number [Formula: see text] is not divisible by [Formula: see text]. If [Formula: see text], congruence conditions for the trace of specific elements of [Formula: see text] are exhibited which imply the existence of a point of infinite order in [Formula: see text].

Milnor $K$-groups attached to elliptic curves over a $p$-adic field

We study the Galois symbol map of the Milnor $K$-group attached to elliptic curves over a $p$-adic field. As by-products, we determine the structure of the Chow group for the product of elliptic curves over a $p$-adic field under some assumptions.

Yet another $p$-adic hyperbolic disc: Hilbert distance for $p$-adic fields

We describe in this paper a geometric construction in the projective p -adic plane that gives, together with a suitable notion of p -adic convexity, some open subsets of \mathbf P_2(\mathbf Q_p) naturally endowed with a "Hilbert" distance and a transitive action of PGL(2, \mathbf Q_p) by isometries. These open sets are natural analogues of the hyperbolic disc, more precisely of Klein's projective model. But, unlike the real case, there is not only one such hyperbolic disc. Indeed, we find three of them if p is odd (and seven if p=2 ). Let us stress out that neither the usual notion of convexity nor that of connectedness as known for the real case are meaningful in the p -adic case. Thus, there will be a rephrasing game for the definitions of real convexity until we reach a formulation suitable for other local fields. It will lead us to a definition of p -adic convexity by duality. Although we will not recover the beautiful behaviour of real convexity, we will still be able to define the most important tool for our goals, namely the Hilbert distance. We construct our analogues of the hyperbolic disc (once again, via the projective model of the hyperbolic plane) in a quite geometric, even naive, way. Our construction gives 2 -dimensional objects over \mathbf Q_p . It is very different, in spirit and in facts, of Drinfeld p -adic hyperbolic plane [1]. The possible relations between the two objects remain still unexplored. Another object often viewed as an analogue of the hyperbolic disc is the tree of PGL(2, \mathbf Q_p) [8]. We explore the relations between our discs and this tree, constructing a natural quasi-isometric projection from the discs to the tree. Eventually we explore the transformation groups of our discs. And, whereas the transformation group of the tree is huge, we prove that only PGL(2, \mathbf Q_p) acts on the discs preserving the convex structure.

Generalized pseudo-coefficients of discrete series of $p$-adic groups

Let $G$ be a connected reductive group over a $p$-adic field $F$ of characteristic 0 and let $M$ be an $F$-Levi subgroup of $G.$ Given a discrete series representation $\sigma$ of $M(F),$ we prove that there exists a locally constant and compactly supported function on $M(F),$ which generalizes a pseudo-coefficient of $\sigma.$ This function satisfies similar properties to the pseudo-coefficient, and its lifting to $G(F)$ is applied to the Plancherel formula.

THE BERNSTEIN CENTER OF THE CATEGORY OF SMOOTH -MODULES

We consider the category of smooth $W(k)[\text{GL}_{n}(F)]$-modules, where $F$ is a $p$-adic field and $k$ is an algebraically closed field of characteristic $\ell$ different from $p$. We describe a factorization of this category into blocks, and show that the center of each such block is a reduced, $\ell$-torsion free, finite type $W(k)$-algebra. Moreover, the $k$-points of the center of a such a block are in bijection with the possible ‘supercuspidal supports’ of the smooth $k[\text{GL}_{n}(F)]$-modules that lie in the block. Finally, we describe a large explicit subalgebra of the center of each block and give a description of the action of this algebra on the simple objects of the block, in terms of the description of the classical ‘characteristic zero’ Bernstein center of Bernstein and Deligne [Le ‘centre’ de Bernstein, in Representations des groups redutifs sur un corps local, Traveaux en cours (ed. P. Deligne) (Hermann, Paris), 1–32].

Ergodic Products and Powers on Compact Subsets of the $p$-adic Field

Ergodic Products and Powers on Compact Subsets of the $p$-adic Field

Positive definite *-spherical functions, property (T) and C*-completions of Gelfand pairs

AbstractThe study of existence of a universal C*-completion of the *-algebra canonically associated to a Hecke pair was initiated by Hall, who proved that the Hecke algebra associated to (SL2($\mathbb{Q}$p), SL2($\mathbb{Z}$p)) does not admit a universal C*-completion. Kaliszewski, Landstad and Quigg studied the problem by placing it in the framework of Fell–Rieffel equivalence, and highlighted the role of other C*-completions. In the case of the pair (SLn($\mathbb{Q}$p), SLn($\mathbb{Z}$p)) for n ⩾ 3 we show, invoking property (T) of SLn($\mathbb{Q}$p), that the C*-completion of the L1-Banach algebra and the corner of C*(SLn($\mathbb{Q}$p)) determined by the subgroup are distinct. In fact, we prove a more general result valid for a simple algebraic group of rank at least 2 over a $\mathfrak{p}$-adic field with a good choice of a maximal compact open subgroup.

On Descriptions of All Translation Invariant p−adic Gibbs Measures for the Potts Model on The Cayley Tree of Order Three

Unlike the real number field, a set of p−adic Gibbs measures of p−adic lattice models of statistical mechanics has a complex structure in a sense that it is strongly tied up with a Diophantine problem over p−adic fields. Recently, all translation-invariant p−adic Gibbs measures of the p−adic Potts model on the Cayley tree of order two were described by means of roots of a certain quadratic equation over some domain of the p−adic field. In this paper, we consider the same problem on the Cayley tree of order three. In this case, we show that all translation-invariant p−adic Gibbs measures of the p−adic Potts model can be described in terms of roots of some cubic equation over $\mathbb {Z}_{p}\setminus \mathbb {Z}_{p}^{*}$ . In own its turn, we also provide a solvability criterion of a general cubic equation over $\mathbb {Z}_{p}\setminus \mathbb {Z}_{p}^{*}$ for p > 3.

Formal Group Exponentials and Galois Modules in Lubin–Tate Extensions

Explicit descriptions of local integral Galois module generators in certain extensions of $p$-adic fields due to Pickett have recently been used to make progress with open questions on integral Galois module structure in wildly ramified extensions of number fields. In parallel, Pulita has generalised the theory of Dwork's power series to a set of power series with coefficients in Lubin-Tate extensions of $\Q_p$ to establish a structure theorem for rank one solvable p-adic differential equations. In this paper we first generalise Pulita's power series using the theories of formal group exponentials and ramified Witt vectors. Using these results and Lubin-Tate theory, we then generalise Pickett's constructions in order to give an analytic representation of integral normal basis generators for the square root of the inverse different in all abelian totally, weakly and wildly ramified extensions of a p-adic field. Other applications are also exposed.

SIMPLE SUPERCUSPIDAL REPRESENTATIONS OF $\operatorname{GL}(n)$

Following a construction of Gross and Reeder, we define simple supercuspidal representations of $\operatorname{GL}(n)$ over a $\mathfrak{p}$-adic field. We show that they have conductor $\mathfrak{p}^{n+1}$. We then give a general formula for the matrix coefficient attached to a new vector, and make it completely explicit when $n=2$.

Étale homotopy equivalence of rational points on algebraic varieties

It is possible to talk about the \'etale homotopy equivalence of rational points on algebraic varieties by using a relative version of the \'etale homotopy type. We show that over $p$-adic fields rational points are homotopy equivalent in this sense if and only if they are \'etale-Brauer equivalent. We also show that over the real field rational points on projective varieties are \'etale homotopy equivalent if and only if they are in the same connected component. We also study this equivalence relation over number fields and prove that in this case it is finer than the other two equivalence relations for certain generalised Ch\^atelet surfaces.

Counting arithmetic subgroups and subgroup growth of virtually free groups

Let K be a p -adic field, and let H=PSL_2(K) endowed with the Haar measure determined by giving a maximal compact subgroup measure 1 . Let AL_H(x) denote the number of conjugacy classes of arithmetic lattices in H with co-volume bounded by x . We show that under the assumption that K does not contain the element \zeta+\zeta^{-1} , where \zeta denotes the p -th root of unity over \mathbb{Q}_p , we have \lim_{x\rightarrow\infty}\frac{\log AL_H(x)}{x\log x}=q-1 where q denotes the order of the residue field of K . The main innovation of this paper is the proof of a sharp bound on subgroup growth of lattices in H as above.

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.

ON THE VANISHING OF COHOMOLOGIES OF <i>p</i>-ADIC GALOIS REPRESENTATIONS ASSOCIATED WITH ELLIPTIC CURVES

Let $K$ be a $p$-adic field and $E$ an elliptic curve over $K$ with potential good reduction. For some large Galois extensions $L$ of $K$ containing all $p$-power roots of unity, we show the vanishing of certain Galois cohomology groups of $L$ with values in the $p$-adic representation associated with $E$. This generalizes some results of Coates, Sujatha and Wintenberger.

Heights and totally p-adic numbers

We study the behavior of canonical height functions $\widehat{h}_f$, associated to rational maps $f$, on totally $p$-adic fields. In particular, we prove that there is a gap between zero and the next smallest value of $\widehat{h}_f$ on the maximal totall

Expression d'un facteur epsilon de paire par une formule intégrale

AbstractLet E/F be a quadratic extension of p–adic fields and let d, m be nonnegative integers of distinct parities. Fix admissible irreducible tempered representations π and σ of GLd(E) and GLm(E) respectively. We assume that π and σ are conjugate–dual. That is to say and where c is the nontrivial F–automorphism of E. This implies that we can extend π to an unitary representation π of a nonconnected group GLd(E) . Define the same way. We state and prove an integral formula for involving the characters of and . ˜˜This formula is related to the local Gan–Gross–Prasad conjecture for unitary groups.

Strongly positive representations of $$GSpin_{2n+1}$$ G S p i n 2 n + 1 and the Jacquet module method

We explicitly construct the structure of Jacquet modules of parabolically induced representations of $$GSpin_{2n+1}$$ over a $$p$$ -adic field $$F$$ of any characteristic. Using this construction of the Jacquet module, we obtain a classification of strongly positive representations of $$GSpin_{2n+1}$$ over $$F$$ and describe the general discrete series representations of $$GSpin_{2n+1}$$ over $$F$$ , assuming the half-integer conjecture. One application of this paper is the proof of the equality of $$L$$ -functions from the Langlands–Shahidi method and Artin $$L$$ -functions through the local Langlands correspondence (Kim in Langlands–Shahidi $$L$$ -functions for $$GSpin$$ groups and the generic Arthur packet conjecture, preprint).