Journal articles: 'Local and $p$-adic fields'

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Li, Yin, and Hua Qiu. "p-adic Laplacian in local fields." Nonlinear Analysis: Theory, Methods & Applications 139 (July 2016): 131–51. http://dx.doi.org/10.1016/j.na.2016.02.025.

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LI, YIN. "WEIERSTRASS-TYPE FUNCTIONS IN p-ADIC LOCAL FIELDS." Fractals 28, no. 03 (May 2020): 2050043. http://dx.doi.org/10.1142/s0218348x20500437.

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The Weierstrass nowhere differentiable function has been studied often as example of functions whose graphs are fractals in [Formula: see text]. This paper investigates the Weierstrass-type function in the [Formula: see text]-adic local field [Formula: see text] whose graph is a repelling set of a discrete dynamical system, and proves that there exists a linear connection between the orders of the [Formula: see text]-adic calculus and the dimensions of the corresponding graphs.

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Qiu, Hua, and Weiyi Su. "3-Adic Cantor function on local fields and its p-adic derivative." Chaos, Solitons & Fractals 33, no. 5 (August 2007): 1625–34. http://dx.doi.org/10.1016/j.chaos.2006.03.024.

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Harari, David, and Tamás Szamuely. "Local-global questions for tori over $p$-adic function fields." Journal of Algebraic Geometry 25, no. 3 (March 31, 2016): 571–605. http://dx.doi.org/10.1090/jag/661.

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Mochizuki, Shinichi. "A Version of the Grothendieck Conjecture for p-Adic Local Fields." International Journal of Mathematics 08, no. 04 (June 1997): 499–506. http://dx.doi.org/10.1142/s0129167x97000251.

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Hua, Qiu, and Su Weiyi. "Weierstrass-like functions on local fields and their p-adic derivatives." Chaos, Solitons & Fractals 28, no. 4 (May 2006): 958–65. http://dx.doi.org/10.1016/j.chaos.2005.09.017.

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Scholze, Peter. "The Local Langlands Correspondence for GL n over p-adic fields." Inventiones mathematicae 192, no. 3 (August 11, 2012): 663–715. http://dx.doi.org/10.1007/s00222-012-0420-5.

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Seveso, Marco Adamo. "p-adic L-functions and the Rationality of Darmon Cycles." Canadian Journal of Mathematics 64, no. 5 (October 1, 2012): 1122–81. http://dx.doi.org/10.4153/cjm-2011-076-8.

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Abstract Darmon cycles are a higher weight analogue of Stark–Heegner points. They yield local cohomology classes in the Deligne representation associated with a cuspidal form on Γ0(N) of even weight k0 ≥ 2. They are conjectured to be the restriction of global cohomology classes in the Bloch–Kato Selmer group defined over narrow ring class fields attached to a real quadratic field. We show that suitable linear combinations of them obtained by genus characters satisfy these conjectures. We also prove p-adic Gross–Zagier type formulas, relating the derivatives of p-adic L-functions of the weight variable attached to imaginary (resp. real) quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express the second derivative of the Mazur– Kitagawa p-adic L-function of the weight variable in terms of a global cycle defined over a quadratic extension of ℚ.

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Gras, Georges. "Les θ-régulateurs locaux d'un nombre algébrique : Conjectures p-adiques." Canadian Journal of Mathematics 68, no. 3 (June 1, 2016): 571–624. http://dx.doi.org/10.4153/cjm-2015-026-3.

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AbstractLet K/ℚ be Galois and let η K ×be such that Reg∞(η)=0 .We define the local θ–regulator for the ℚp–irreducible characters θ of G = Gal(Kℚ). Let Vθ be the θ-irreducible representation. A linear representation is associated with whose nullity is equivalent to δ≥1. Each yields Regθp modulo p in the factorization of (normalized p–adic regulator). From Prob f ≥ 1 is a residue degree) and the Borel–Cantelli heuristic, we conjecture that for p large enough, RegGp(η) is a p–adic unit (a single with f = δ=1); this obstruction may be led assuming the existence of a binomial probability law confirmed through numerical studies (groups C3, C5, D6) is conjecture would imply that for all p large enough, Fermat quotients, normalized p–adic regulators are p–adic units and that number fields are p-rational.We recall some deep cohomological results that may strengthen such conjectures.

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Bocardo-Gaspar, Miriam, Hugo García-Compeán, Edgar Y. López, and Wilson A. Zúñiga-Galindo. "Local Zeta Functions and Koba–Nielsen String Amplitudes." Symmetry 13, no. 6 (May 29, 2021): 967. http://dx.doi.org/10.3390/sym13060967.

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This article is a survey of our recent work on the connections between Koba–Nielsen amplitudes and local zeta functions (in the sense of Gel’fand, Weil, Igusa, Sato, Bernstein, Denef, Loeser, etc.). Our research program is motivated by the fact that the p-adic strings seem to be related in some interesting ways with ordinary strings. p-Adic string amplitudes share desired characteristics with their Archimedean counterparts, such as crossing symmetry and invariance under Möbius transformations. A direct connection between p-adic amplitudes and the Archimedean ones is through the limit p→1. Gerasimov and Shatashvili studied the limit p→1 of the p-adic effective action introduced by Brekke, Freund, Olson and Witten. They showed that this limit gives rise to a boundary string field theory, which was previously proposed by Witten in the context of background independent string theory. Explicit computations in the cases of 4 and 5 points show that the Feynman amplitudes at the tree level of the Gerasimov–Shatashvili Lagrangian are related to the limit p→1 of the p-adic Koba–Nielsen amplitudes. At a mathematical level, this phenomenon is deeply connected with the topological zeta functions introduced by Denef and Loeser. A Koba–Nielsen amplitude is just a new type of local zeta function, which can be studied using embedded resolution of singularities. In this way, one shows the existence of a meromorphic continuations for the Koba–Nielsen amplitudes as functions of the kinematic parameters. The Koba–Nielsen local zeta functions are algebraic-geometric integrals that can be defined over arbitrary local fields (for instance R, C, Qp, Fp((T))), and it is completely natural to expect connections between these objects. The limit p tends to one of the Koba–Nielsen amplitudes give rise to new amplitudes which we have called Denef–Loeser amplitudes. Throughout the article, we have emphasized the explicit calculations in the cases of 4 and 5 points.

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Zúñiga-Galindo, W. A. "Local zeta functions and fundamental solutions for pseudo-differential operators over p-adic fields." P-Adic Numbers, Ultrametric Analysis, and Applications 3, no. 4 (November 19, 2011): 344–58. http://dx.doi.org/10.1134/s207004661104008x.

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Sakellaridis, Yiannis. "On the unramified spectrum of spherical varieties over p-adic fields." Compositio Mathematica 144, no. 4 (July 2008): 978–1016. http://dx.doi.org/10.1112/s0010437x08003485.

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AbstractThe description of irreducible representations of a group G can be seen as a problem in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G×G by left and right multiplication. For a split p-adic reductive group G over a local non-archimedean field, unramified irreducible smooth representations are in bijection with semisimple conjugacy classes in the ‘Langlands dual’ group. We generalize this description to an arbitrary spherical variety X of G as follows. Irreducible unramified quotients of the space $C_c^\infty (X)$ are in natural ‘almost bijection’ with a number of copies of AX*/WX, the quotient of a complex torus by the ‘little Weyl group’ of X. This leads to a description of the Hecke module of unramified vectors (a weak analog of geometric results of Gaitsgory and Nadler), and an understanding of the phenomenon that representations ‘distinguished’ by certain subgroups are functorial lifts. In the course of the proof, rationality properties of spherical varieties are examined and a new interpretation is given for the action, defined by Knop, of the Weyl group on the set of Borel orbits.

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QIU, HUA, and WEIYI SU. "THE CONNECTION BETWEEN THE ORDERS OF p-ADIC CALCULUS AND THE DIMENSIONS OF THE WEIERSTRASS TYPE FUNCTION IN LOCAL FIELDS." Fractals 15, no. 03 (September 2007): 279–87. http://dx.doi.org/10.1142/s0218348x07003599.

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This paper investigates the Weierstrass type function in local fields whose graph is a chaotic repelling set of a discrete dynamical system, and proves that their exists a linear connection between the orders of its p-adic calculus and the dimensions of the corresponding graphs.

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ANGLÈS, BRUNO, and TATIANA BELIAEVA. "ON WEIL NUMBERS IN CYCLOTOMIC FIELDS." International Journal of Number Theory 05, no. 05 (August 2009): 871–84. http://dx.doi.org/10.1142/s1793042109002432.

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In this paper, we study the p-adic behavior of Weil numbers in the cyclotomic ℤp-extension of the pth cyclotomic field. We determine the characteristic ideal of the quotient of semi-local units by Weil numbers in terms of the characteristic ideals of some classical modules that appear in the Iwasawa theory. In a recent preprint [9] by Nguyen Quang Do and Nicolas, a generalization of this result to a semi-simple case was obtained.

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Rojas, J. Maurice, and Yuyu Zhu. "A complexity chasm for solving sparse polynomial equations over p -adic fields." ACM Communications in Computer Algebra 54, no. 3 (September 2020): 86–90. http://dx.doi.org/10.1145/3457341.3457343.

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The applications of solving systems of polynomial equations are legion: The real case permeates all of non-linear optimization as well as numerous problems in engineering. The p -adic case leads to many classical questions in number theory, and is close to many applications in cryptography, coding theory, and computational number theory. As such, it is important to understand the complexity of solving systems of polynomial equations over local fields. Furthermore, the complexity of solving structured systems --- such as those with a fixed number of monomial terms or invariance with respect to a group action --- arises naturally in many computational geometric applications and is closely related to a deeper understanding of circuit complexity (see, e.g., [8]). Clearly, if we are to fully understand the complexity of solving sparse polynomial systems, then we should at least be able to settle the univariate case, e.g., classify when it is possible to separate and approximate roots in deterministic time polynomial in the input size.

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Dixon, John D. "Computing subfields in algebraic number fields." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 49, no. 3 (December 1990): 434–48. http://dx.doi.org/10.1017/s1446788700032432.

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AbstractLet K:= Q(α) be an algebraic number field which is given by specifying the minimal polynomial f(X) for α over Q. We describe a procedure for finding the subfields L of K by constructing pairs (w(X), g(X)) of polynomials over Q such that L= Q(w(α)) and g(X) is the minimal polynomial for w(α). The construction uses local information obtained from the Frobenius-Chebotarev theorem about the Galois group Gal(f), and computations over p-adic extensions of Q.

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Colliot-Thélène, Jean-Louis, Raman Parimala, and Venapally Suresh. "Patching and local-global principles for homogeneous spaces over function fields of p-adic curves." Commentarii Mathematici Helvetici 87, no. 4 (2012): 1011–33. http://dx.doi.org/10.4171/cmh/276.

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Beuzart-Plessis, R. "Endoscopie et conjecture locale raffinée de Gan–Gross–Prasad pour les groupes unitaires." Compositio Mathematica 151, no. 7 (February 18, 2015): 1309–71. http://dx.doi.org/10.1112/s0010437x14007891.

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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.

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Haines, Thomas J., and Timo Richarz. "The test function conjecture for local models of Weil-restricted groups." Compositio Mathematica 156, no. 7 (July 2020): 1348–404. http://dx.doi.org/10.1112/s0010437x20007162.

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We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. Our result covers the (modified) local models attached to all connected reductive groups over $p$-adic local fields with $p\geqslant 5$. In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme.

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Freitas, Nuno, Bartosz Naskręcki, and Michael Stoll. "The generalized Fermat equation with exponents 2, 3,." Compositio Mathematica 156, no. 1 (November 26, 2019): 77–113. http://dx.doi.org/10.1112/s0010437x19007693.

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We study the generalized Fermat equation $x^{2}+y^{3}=z^{p}$, to be solved in coprime integers, where $p\geqslant 7$ is prime. Modularity and level-lowering techniques reduce the problem to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve $X(p)$. We develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti-symplectically isomorphic $p$-torsion modules. Using these criteria we produce the minimal list of twists of $X(p)$ that have to be considered, based on local information at 2 and 3; this list depends on $p\hspace{0.2em}{\rm mod}\hspace{0.2em}24$. We solve the equation completely when $p=11$, which previously was the smallest unresolved $p$. One new ingredient is the use of the ‘Selmer group Chabauty’ method introduced by the third author, applied in an elliptic curve Chabauty context, to determine relevant points on $X_{0}(11)$ defined over certain number fields of degree 12. This result is conditional on the generalized Riemann hypothesis, which is needed to show correctness of the computation of the class groups of five specific number fields of degree 36. We also give some partial results for the case $p=13$. The source code for the various computations is supplied as supplementary material with the online version of this article.

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FEHM, ARNO, and SEBASTIAN PETERSEN. "ON THE RANK OF ABELIAN VARIETIES OVER AMPLE FIELDS." International Journal of Number Theory 06, no. 03 (May 2010): 579–86. http://dx.doi.org/10.1142/s1793042110003071.

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A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the ℤ(p)-module A(K) ⊗ ℤ(p) is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.

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Matringe, Nadir, and Omer Offen. "Gamma Factors, Root Numbers, and Distinction." Canadian Journal of Mathematics 70, no. 3 (June 1, 2018): 683–701. http://dx.doi.org/10.4153/cjm-2017-011-6.

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AbstractWe 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.

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Hoshi, Yuichiro. "A note on the geometricity of open homomorphisms between the absolute Galois groups of p-adic local fields." Kodai Mathematical Journal 36, no. 2 (June 2013): 284–98. http://dx.doi.org/10.2996/kmj/1372337519.

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Kai, Wataru. "A higher-dimensional generalization of Lichtenbaum duality in terms of the Albanese map." Compositio Mathematica 152, no. 9 (July 14, 2016): 1915–34. http://dx.doi.org/10.1112/s0010437x16007600.

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In this article, we present a conjectural formula describing the cokernel of the Albanese map of zero-cycles of smooth projective varieties $X$ over $p$-adic fields in terms of the Néron–Severi group and provide a proof under additional assumptions on an integral model of $X$. The proof depends on a non-degeneracy result of Brauer–Manin pairing due to Saito–Sato and on Gabber–de Jong’s comparison result of cohomological and Azumaya–Brauer groups. We will also mention the local–global problem for the Albanese cokernel; the abelian group on the ‘local side’ turns out to be a finite group.

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Büyükboduk, Kâzım. "Stark units and the main conjectures for totally real fields." Compositio Mathematica 145, no. 5 (September 2009): 1163–95. http://dx.doi.org/10.1112/s0010437x09004163.

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AbstractThe main theorem of the author’s thesis suggests that it should be possible to lift the Kolyvagin systems of Stark units, constructed by the author in an earlier paper, to a Kolyvagin system over the cyclotomic Iwasawa algebra. In this paper, we verify that this is indeed the case. This construction of Kolyvagin systems over the cyclotomic Iwasawa algebra from Stark units provides the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank (in the sense of Mazur and Rubin) is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement in the context of local Iwasawa theory, assuming the truth of the Rubin–Stark conjecture and Leopoldt’s conjecture. This statement in the local Iwasawa theory context turns out to be interesting in its own right, as it suggests a relation between the solutions to p-adic and complex Stark conjectures.

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Česnavičius, Kęstutis, and Naoki Imai. "The remaining cases of the Kramer–Tunnell conjecture." Compositio Mathematica 152, no. 11 (July 29, 2016): 2255–68. http://dx.doi.org/10.1112/s0010437x16007624.

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For an elliptic curve $E$ over a local field $K$ and a separable quadratic extension of $K$, motivated by connections to the Birch and Swinnerton-Dyer conjecture, Kramer and Tunnell have conjectured a formula for computing the local root number of the base change of $E$ to the quadratic extension in terms of a certain norm index. The formula is known in all cases except some where $K$ is of characteristic $2$, and we complete its proof by reducing the positive characteristic case to characteristic $0$. For this reduction, we exploit the principle that local fields of characteristic $p$ can be approximated by finite extensions of $\mathbb{Q}_{p}$: we find an elliptic curve $E^{\prime }$ defined over a $p$-adic field such that all the terms in the Kramer–Tunnell formula for $E^{\prime }$ are equal to those for $E$.

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Cunningham, Clifton, and David Roe. "FROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF -ADIC TORI." Journal of the Institute of Mathematics of Jussieu 17, no. 1 (October 13, 2015): 1–37. http://dx.doi.org/10.1017/s1474748015000286.

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We consider the rigid monoidal category of character sheaves on a smooth commutative group scheme $G$ over a finite field $k$, and expand the scope of the function-sheaf dictionary from connected commutative algebraic groups to this setting. We find the group of isomorphism classes of character sheaves on $G$, and show that it is an extension of the group of characters of $G(k)$ by a cohomology group determined by the component group scheme of $G$. We also classify all morphisms in the category character sheaves on $G$. As an application, we study character sheaves on Greenberg transforms of locally finite type Néron models of algebraic tori over local fields. This provides a geometrization of quasicharacters of $p$-adic tori.

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Mochizuki, Shinichi. "Galois Sections in Absolute Anabelian Geometry." Nagoya Mathematical Journal 179 (2005): 17–45. http://dx.doi.org/10.1017/s0027763000025599.

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AbstractWe show that isomorphisms between arithmetic fundamental groups of hyperbolic curves over p-adic local fields preserve the decomposition groups of all closed points (respectively, closed points arising from torsion points of the underlying elliptic curve), whenever the hyperbolic curves in question are isogenous to hyperbolic curves of genus zero defined over a number field (respectively, are once-punctured elliptic curves [which are not necessarily defined over a number field]). We also show that, under certain conditions, such isomorphisms preserve certain canonical “integral structures” at the cusps [i.e., points at infinity] of the hyperbolic curve.

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Elder, G. Griffith, and Manohar L. Madan. "Galois Module Structure of the Integers in Wildly Ramified Cp × Cp Extensions." Canadian Journal of Mathematics 49, no. 4 (August 1, 1997): 722–35. http://dx.doi.org/10.4153/cjm-1997-035-2.

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AbstractLet L/K be a finite Galois extension of local fields which are finite extensions of ℚp, the field of p-adic numbers. Let Gal(L/K) = G, and 𝔒L and ℤp be the rings of integers in L and ℚp, respectively. And let 𝔓L denote the maximal ideal of 𝔒L. We determine, explicitly in terms of specific indecomposable ℤp[G]-modules, the ℤp[G]-module structure of 𝔒L and 𝔓L, for L, a composite of two arithmetically disjoint, ramified cyclic extensions of K, one of which is only weakly ramified in the sense of Erez [6].

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Goldberg, David, and Freydoon Shahidi. "On the Tempered Spectrum of Quasi-Split Classical Groups II." Canadian Journal of Mathematics 53, no. 2 (April 1, 2001): 244–77. http://dx.doi.org/10.4153/cjm-2001-011-7.

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AbstractWe determine the poles of the standard intertwining operators for a maximal parabolic subgroup of the quasi-split unitary group defined by a quadratic extension E/F of p-adic fields of characteristic zero. We study the case where the Levi component M ≃ GLn(E) × Um(F), with n ≡ m (mod 2). This, along with earlier work, determines the poles of the local Rankin-Selberg product L-function L(s, t′ × τ), with t′ an irreducible unitary supercuspidal representation of GLn(E) and τ a generic irreducible unitary supercuspidal representation of Um(F). The results are interpreted using the theory of twisted endoscopy.

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Langlands, R. P., and D. Shelstad. "Orbital Integrals on Forms of SL(3), II." Canadian Journal of Mathematics 41, no. 3 (June 1, 1989): 480–507. http://dx.doi.org/10.4153/cjm-1989-022-0.

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In the paper [6] we described in a precise fashion the notion of transfer of orbital integrals from a reductive group over a local field to an endoscopic group. We did not, however, prove the existence of the transfer. This remains, indeed, an unsolved problem, although in [7] we have reduced it to a local problem at the identity.In the present paper we solve this local problem for two special cases, the group SL(3), which is not so interesting, and the group SU(3), and then conclude that transfer exists for any group of type A2.The methods are those of [4], and are based on techniques of Igusa for the study of the asymptotic behavior of integrals on p-adic manifolds. (As observed in [7], the existence of the transfer over archimedean fields is a result of earlier work by Shelstad.)

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Beuzart-Plessis, Raphaël. "Expression d'un facteur epsilon de paire par une formule intégrale." Canadian Journal of Mathematics 66, no. 5 (October 1, 2014): 993–1049. http://dx.doi.org/10.4153/cjm-2013-042-4.

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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.

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Hattori, Shin. "Ramification theory and perfectoid spaces." Compositio Mathematica 150, no. 5 (April 3, 2014): 798–834. http://dx.doi.org/10.1112/s0010437x1300763x.

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AbstractLet $K_1$ and $K_2$ be complete discrete valuation fields of residue characteristic $p>0$. Let $\pi _{K_1}$ and $\pi _{K_2}$ be their uniformizers. Let $L_1/K_1$ and $L_2/K_2$ be finite extensions with compatible isomorphisms of rings $\mathcal{O}_{K_1}/(\pi _{K_1}^m)\, {\simeq }\, \mathcal{O}_{K_2}/(\pi _{K_2}^m)$ and $\mathcal{O}_{L_1}/(\pi _{K_1}^m)\, {\simeq }\, \mathcal{O}_{L_2}/(\pi _{K_2}^m)$ for some positive integer $m$ which is no more than the absolute ramification indices of $K_1$ and $K_2$. Let $j\leq m$ be a positive rational number. In this paper, we prove that the ramification of $L_1/K_1$ is bounded by $j$ if and only if the ramification of $L_2/K_2$ is bounded by $j$. As an application, we prove that the categories of finite separable extensions of $K_1$ and $K_2$ whose ramifications are bounded by $j$ are equivalent to each other, which generalizes a theorem of Deligne to the case of imperfect residue fields. We also show the compatibility of Scholl’s theory of higher fields of norms with the ramification theory of Abbes–Saito, and the integrality of small Artin and Swan conductors of $p$-adic representations with finite local monodromy.

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FLICKER, YUVAL Z., and DMITRII ZINOVIEV. "COMPUTATION OF A TWISTED CHARACTER OF A SMALL REPRESENTATION OF GL(3, E)." International Journal of Number Theory 08, no. 05 (July 6, 2012): 1153–230. http://dx.doi.org/10.1142/s1793042112500704.

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Let E/F be a quadratic extension of p-adic fields, p ≠ 2. Let [Formula: see text] be the involution of E over F. The representation π of GL (3, E) normalizedly induced from the trivial representation of the maximal parabolic subgroup is invariant under the involution [Formula: see text]. We compute — by purely local means — the σ-twisted character [Formula: see text] of π. We show that it is σ-unstable, namely its value at one σ-regular-elliptic conjugacy class within a stable such class is equal to negative its value at the other such conjugacy class within the stable class, or zero when the σ-regular-elliptic stable conjugacy class consists of a single such conjugacy class. Further, we relate this twisted character to the twisted endoscopic lifting from the trivial representation of the "unstable" twisted endoscopic group U (2, E/F) of GL (3, E). In particular π is σ-elliptic, that is, [Formula: see text] is not identically zero on the σ-elliptic set.

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Stoltenberg-Hansen, V., and J. V. Tucker. "Complete local rings as domains." Journal of Symbolic Logic 53, no. 2 (June 1988): 603–24. http://dx.doi.org/10.1017/s0022481200028498.

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Contents: Introduction. §1: Computable rings and modules. §2: Ideal membership relation. §3: Effective structured domains. §4: Completion of a local ring as a domain. §5: The recursive completion. Epilogue. References.Introduction. Completion is an important general mathematical device. Often, but not always, a completion takes the following form. Let A be a topological algebraic structure whose topology is derived from a metric. For A, a topological algebra Â and an embedding i: A → Â are constructed such that Â is a complete metric space in which A is densely embedded by i. The long list of structures for which such completions exist begins with Cantor's construction of the real number field and includes objects like the p-adic integers, Baire space, and Boolean algebras. In Bourbaki [6] a careful and thorough account of completions for arbitrary topological groups and fields is given, for which it is important to note that the topological structures need not be metrizable, but must possess a uniformity.The effectiveness of the completion process of a computable structure A cannot be readily studied using the tools of computable algebra, simply because the resulting structure Â is almost invariably uncountable. However, in particular cases, it has been possible to define and study the substructure Ak of computable elements of Â; this has been done for the structures mentioned above, starting with the field of recursive real numbers.In this paper we analyse the effectivity of the completion of a local ring R. We do this using structured Scott-Ershov domains. Our study may be considered as a prototype containing methods applicable to a broad class of completions, including all the examples mentioned above, except for the real number field, which needs a generalisation of the domain concept.A Scott-Ershov domain D formalises how a set Dt of possibly “infinite” elements, called total elements, is constructed from a set Dc of “finite” elements, called compact elements. This is achieved by means of an approximation ordering which determines a topology on D and, in particular, on Dt. Our methodology is to associate to a given topological algebra A a structured domain D(A) such that the total elements D(A)t form a topological algebra topologically isomorphic to A. In such circumstances A is said to be domain definable by D(A). The theory of computability for domains is now applied to study the effectivity of the topological algebra A.

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CHAN, PING-SHUN, and YUVAL Z. FLICKER. "CYCLIC ODD DEGREE BASE CHANGE LIFTING FOR UNITARY GROUPS IN THREE VARIABLES." International Journal of Number Theory 05, no. 07 (November 2009): 1247–309. http://dx.doi.org/10.1142/s1793042109002687.

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Let F be a number field or a p-adic field of odd residual characteristic. Let E be a quadratic extension of F, and F' an odd degree cyclic field extension of F. We establish a base-change functorial lifting of automorphic (respectively, admissible) representations from the unitary group U (3, E/F) to the unitary group U (3, F' E/F'). As a consequence, we classify, up to certain restrictions, the packets of U (3, F' E/F') which contain irreducible automorphic (respectively, admissible) representations invariant under the action of the Galois group Gal (F' E/E). We also determine the invariance of individual representations. This work is the first study of base change into an algebraic group whose packets are not all singletons, and which does not satisfy the rigidity, or "strong multiplicity one", theorem. Novel phenomena are encountered: e.g. there are invariant packets where not every irreducible automorphic (respectively, admissible) member is Galois-invariant. The restriction that the residual characteristic of the local fields be odd may be removed once the multiplicity one theorem for U(3) is proved to hold unconditionally without restriction on the dyadic places.

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Chan, Charlotte. "The cohomology of semi-infinite Deligne–Lusztig varieties." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 768 (November 1, 2020): 93–147. http://dx.doi.org/10.1515/crelle-2019-0039.

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AbstractWe prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne–Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes {X_{h}}. Boyarchenko’s two conjectures are on the maximality of {X_{h}} and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invariant {1/n} in the case {h=2} (the “lowest level”) by the work of Boyarchenko–Weinstein on the cohomology of a special affinoid in the Lubin–Tate tower. We prove that the number of rational points of {X_{h}} attains its Weil–Deligne bound, so that the cohomology of {X_{h}} is pure in a very strong sense. We prove that the torus-eigenspaces of the cohomology group {H_{c}^{i}(X_{h})} are irreducible representations and are supported in exactly one cohomological degree. Finally, we give a complete description of the homology groups of the semi-infinite Deligne–Lusztig varieties attached to any division algebra, thus giving a geometric realization of a large class of supercuspidal representations of these groups. Moreover, the correspondence {\theta\mapsto H_{c}^{i}(X_{h})[\theta]} agrees with local Langlands and Jacquet–Langlands correspondences. The techniques developed in this paper should be useful in studying these constructions for p-adic groups in general.

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Gomez, Raul, Dmitry Gourevitch, and Siddhartha Sahi. "Generalized and degenerate Whittaker models." Compositio Mathematica 153, no. 2 (February 2017): 223–56. http://dx.doi.org/10.1112/s0010437x16007788.

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We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction of epimorphisms from the generalized Whittaker model corresponding to a nilpotent orbit to any degenerate Whittaker model corresponding to the same orbit, and to certain degenerate Whittaker models corresponding to bigger orbits. We also give choice-free definitions of generalized and degenerate Whittaker models. Finally, we explain how our methods imply analogous results for Whittaker–Fourier coefficients of automorphic representations. For $\text{GL}_{n}(\mathbb{F})$ this implies that a smooth admissible representation $\unicode[STIX]{x1D70B}$ has a generalized Whittaker model ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ corresponding to a nilpotent coadjoint orbit ${\mathcal{O}}$ if and only if ${\mathcal{O}}$ lies in the (closure of) the wave-front set $\operatorname{WF}(\unicode[STIX]{x1D70B})$. Previously this was only known to hold for $\mathbb{F}$ non-archimedean and ${\mathcal{O}}$ maximal in $\operatorname{WF}(\unicode[STIX]{x1D70B})$, see Moeglin and Waldspurger [Modeles de Whittaker degeneres pour des groupes p-adiques, Math. Z. 196 (1987), 427–452]. We also express ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ as an iteration of a version of the Bernstein–Zelevinsky derivatives [Bernstein and Zelevinsky, Induced representations of reductive p-adic groups. I., Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), 441–472; Aizenbud et al., Derivatives for representations of$\text{GL}(n,\mathbb{R})$and$\text{GL}(n,\mathbb{C})$, Israel J. Math. 206 (2015), 1–38]. This enables us to extend to $\text{GL}_{n}(\mathbb{R})$ and $\text{GL}_{n}(\mathbb{C})$ several further results by Moeglin and Waldspurger on the dimension of ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ and on the exactness of the generalized Whittaker functor.

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AWTREY, CHAD. "DODECIC 3-ADIC FIELDS." International Journal of Number Theory 08, no. 04 (May 16, 2012): 933–44. http://dx.doi.org/10.1142/s1793042112500558.

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Let n be an integer and p be a prime number. An important problem in number theory is to classify the degree n extensions of the p-adic numbers through their arithmetic invariants. The most difficult cases arise when p divides n and n is composite. In this paper, we consider the case n = 12 and p = 3; the degrees n < 12 having previously been determined.

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Zaharescu, Alexandru. "INTEGRAL BASES OVER p-ADIC FIELDS." Bulletin of the Korean Mathematical Society 40, no. 3 (August 1, 2003): 509–20. http://dx.doi.org/10.4134/bkms.2003.40.3.509.

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ZAHARESCU, ALEXANDRU. "LIPSCHITZIAN ELEMENTS OVER p-ADIC FIELDS." Glasgow Mathematical Journal 47, no. 2 (July 27, 2005): 363–72. http://dx.doi.org/10.1017/s0017089505002594.

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Ormsby, Kyle M. "Motivic invariants of p-adic fields." Journal of K-theory 7, no. 3 (May 19, 2011): 597–618. http://dx.doi.org/10.1017/is011004017jkt153.

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AbstractWe provide a complete analysis of the motivic Adams spectral sequences converging to the bigraded coefficients of the 2-complete algebraic Johnson-Wilson spectra BPGL〈n〉 over p-adic fields. These spectra interpolate between integral motivic cohomology (n = 0), a connective version of algebraic K-theory (n = 1), and the algebraic Brown-Peterson spectrum (n = ∞). We deduce that, over p-adic fields, the 2-complete BPGL〈n〉 splits over 2-complete BPGL〈0〉, implying that the slice spectral sequence for BPGL collapses.This is the first in a series of two papers investigating motivic invariants of p-adic fields, and it lays the groundwork for an understanding of the motivic Adams-Novikov spectral sequence over such base fields.

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Ruelle, Ph, E. Thiran, D. Verstegen, and J. Weyers. "Quantum mechanics on p‐adic fields." Journal of Mathematical Physics 30, no. 12 (December 1989): 2854–74. http://dx.doi.org/10.1063/1.528468.

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Dubhashi, D. P. "Quantifier Elimination in p-adic Fields." Computer Journal 36, no. 5 (May 1, 1993): 419–26. http://dx.doi.org/10.1093/comjnl/36.5.419.

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Alacoque, C., P. Ruelle, E. Thiran, D. Verstegen, and J. Weyers. "Quantum amplitudes on p-adic fields." Physics Letters B 211, no. 1-2 (August 1988): 59–62. http://dx.doi.org/10.1016/0370-2693(88)90807-6.

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Guilloux, Antonin. "Yet another $p$-adic hyperbolic disc: Hilbert distance for $p$-adic fields." Groups, Geometry, and Dynamics 10, no. 1 (2016): 9–43. http://dx.doi.org/10.4171/ggd/341.

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Nekovář, Jan, and Wiesława Nizioł. "Syntomic cohomology and p-adic regulators for varieties over p-adic fields." Algebra & Number Theory 10, no. 8 (October 7, 2016): 1695–790. http://dx.doi.org/10.2140/ant.2016.10.1695.

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Albeverio, Sergio, and Sergei V. Kozyrev. "Pseudodifferential p-adic vector fields and pseudodifferentiation of a composite p-adic function." P-Adic Numbers, Ultrametric Analysis, and Applications 2, no. 1 (January 2010): 21–34. http://dx.doi.org/10.1134/s2070046610010024.

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Brink, D., H. Godinho, and P. H. A. Rodrigues. "Simultaneous diagonal equations over p-adic fields." Acta Arithmetica 132, no. 4 (2008): 393–99. http://dx.doi.org/10.4064/aa132-4-8.

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Scowcroft, Philip. "More on imaginaries in p-adic fields." Journal of Symbolic Logic 62, no. 1 (March 1997): 1–13. http://dx.doi.org/10.2307/2275728.

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According to [4, p. 1154], a complete L-theory T eliminates imaginaries just in case for every L-formula φ(x1,… , xm, y1, …, yn), every model M of T, and every ā Є Mn, there is a subset A of M's domain with the following property: if N ≽ M and f is an automorphism of N, thenif and only ifAmong the several equivalent conditions discussed in [4, p. 1155], one may single out the following: if T is a complete theory in which two distinct objects are definable, T eliminates imaginaries just in case every T-definable n-ary equivalence relation may be defined by a formulawhere g is a T-definable n-ary function taking k-tuples as values (for some natural number k).Say that an L-structure M eliminates imaginaries just in case Th(M) does. If L is the language of rings with unit, [4, p. 1158] shows that any algebraically closed field eliminates imaginaries, and [2, p. 629] points out that any real-closed field eliminates imaginaries.

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Journal articles on the topic 'Local and $p$-adic fields'

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