Recent zbMATH articles in MSC 11Shttps://zbmath.org/atom/cc/11S2022-07-25T18:03:43.254055ZWerkzeugOn \(\lambda \)-Changhee-Hermite polynomialshttps://zbmath.org/1487.110292022-07-25T18:03:43.254055Z"Pathan, M. A."https://zbmath.org/authors/?q=ai:pathan.mahmood-ahmad"Khan, Waseem A."https://zbmath.org/authors/?q=ai:khan.waseem-ahmadSummary: In this paper, we introduce a new class of \(\lambda \)-analogues of the Changhee-Hermite polynomials and generalized Gould-Hopper-Appell type \(\lambda \)-Changhee polynomials, and present some properties and identities of these polynomials. A new class of polynomials generalizing different classes of Hermite polynomials such as the real Gould-Hopper, as well as the 1D and 2D holomorphic, ternary and polyanalytic complex Hermite polynomials and their relationship to the Appell type \(\lambda \)-Changhee polynomials are also discussed.On the local doubling \(\gamma \)-factor for classical groups over function fieldshttps://zbmath.org/1487.110502022-07-25T18:03:43.254055Z"Kakuhama, Hirotaka"https://zbmath.org/authors/?q=ai:kakuhama.hirotakaLet $F$ be a local field and let $G$ be a classical group over $F$. The author of the paper under review uses the doubling method of Piatetski-Shapiro, Rallis [\textit{S. Gelbart} et al., Explicit constructions of automorphic \(L\)-functions. Springer, Cham (1987; Zbl 0612.10022); \textit{E. M. Lapid} and \textit{S. Rallis}, Ohio State Univ. Math. Res. Inst. Publ. 11, 309--359 (2005; Zbl 1188.11023); \textit{I. I. Piatetski-Shapiro} and \textit{S. Rallis}, Proc. Natl. Acad. Sci. USA 83, 4589--4593 (1986; Zbl 0599.12012)] to irreducible representations of $G(F)\times F^\times$ in the case $ch(F)$ is an odd prime to give give a precise definition of a $\gamma$-factor $\gamma^{V}(s,\pi\boxtimes\omega,\psi)$ for an irreducible representation of $G(F)$, a character $\omega$ of $F^\times$, and a nontrivial additive character $\psi$ of $F$. Here $\boxtimes$ is the free multiplicative convolution of two non-commutative variables. Beside of the author approach for above definition he proves and uses some fundamental properties that are sufficient to define it uniquely. The fact that shows the definition is ``precise''. The process for precise definition in the case $ch(F) =0$ already is established [\textit{W. T. Gan}, Nagoya Math. J. 208, 67--95 (2012; Zbl 1280.11028); \textit{H. Kakuhama}, Manuscr. Math. 163, No. 1--2, 57--86 (2020; Zbl 1470.11129); \textit{E. M. Lapid} and \textit{S. Rallis}, Ohio State Univ. Math. Res. Inst. Publ. 11, 309--359 (2005; Zbl 1188.11023); \textit{S. Yamana}, Invent. Math. 196, No. 3, 651--732 (2014; Zbl 1303.11054)]. The paper under review is a well-written one with strong results and it is a good resource for interested researchers in the field.
Reviewer: Manouchehr Misaghian (Prairie View)Dirichlet series expansions of \(p\)-adic \(\mathrm{L}\)-functionshttps://zbmath.org/1487.111012022-07-25T18:03:43.254055Z"Knospe, Heiko"https://zbmath.org/authors/?q=ai:knospe.heiko"Washington, Lawrence C."https://zbmath.org/authors/?q=ai:washington.lawrence-cLet \(\chi\) be a Dirichlet character of conductor \(f\). It is well known that the associated complex \(L\)-function \(L(s,\chi)\) is given by an expansion
\[
L(s,\chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}
\]
if the real part of \(s\) is greater than \(1\). Recall that in general an expansion of the form \(\sum_{n=1}^{\infty} a_n/n^s\) with \(a_n \in \mathbb{C}\) is called a Dirichlet series. \par
Now assume that \(\chi\) is even and let \(p\) be prime, which we assume to be odd for simplicity. Then there is also a \(p\)-adic \(L\)-function \(L_p(s,\chi)\), which is a \(p\)-adic meromorphic function that interpolates values of complex \(L\)-series at non-positive integers. Starting from a formula for \(L_p(s,\chi)\) in the book of the second named author (Theorem 5.11 in [Introduction to cyclotomic fields. 2nd ed. New York, NY: Springer (1997; Zbl 0966.11047)]), the authors derive at a Dirichlet series expansion for \(L_p(s,\chi)\). More precisely, for any integer \(c>1\), which is coprime to \(pf\), one has the formula
\[
-(1 - \chi(c)\langle c \rangle^{1-s}) \cdot L_p(s,\chi) = \lim_{n \rightarrow \infty} \sum_{\genfrac{}{}{0pt}{}{a=1}{p \nmid a} }^{fp^n} \chi \omega^{-1}(a) \frac{\varepsilon_{a,c,fp^n}}{\langle a \rangle^s},
\]
where \(\omega\) denotes the Teichmüller character and the \(\varepsilon_{a,c,fp^n}\) are certain explicit (half-) integers. A similar formula is obtained for \(S\)-truncated \(p\)-adic \(L\)-series. \par
It is also shown that the numbers \(\varepsilon_{a,c,fp^n}\) are related to the Bernoulli distribution \(E_1\) on \(X := \mathbb{Z}/ f\mathbb{Z} \times \mathbb{Z}_p\) as follows. One first replaces \(E_1\) with a regularization \(E_{1,c}\) which is actually a measure. Then one has
\[
\varepsilon_{a,c,fp^n} = E_{1,c}(a + fp^n X).
\]
In a final section, the authors specialize their formulas to different regularization parameters \(c\). In particular, the case \(c=2\) is considered. In this case the obtained results are similar to those of \textit{D. Delbourgo} [J. Aust. Math. Soc. 81, No. 2, 215--224 (2006; Zbl 1207.11112); Proc. Edinb. Math. Soc., II. Ser. 52, No. 3, 583--606 (2009; Zbl 1243.11104)], where the method of proof is different. Similar expansions for slightly different \(p\)-adic \(L\)-functions are due to \textit{M. S. Kim} and \textit{S. Hu} [J. Number Theory 132, No. 12, 2977--3015 (2012; Zbl 1272.11130)].
Reviewer: Andreas Nickel (Essen)Extensions of absolute values on two subfieldshttps://zbmath.org/1487.120032022-07-25T18:03:43.254055Z"Ding, Zhiguo"https://zbmath.org/authors/?q=ai:ding.zhiguo"Zieve, Michael E."https://zbmath.org/authors/?q=ai:zieve.michael-eThe theory of absolute values is large subdomain of modern mathematics. One of the classical topics in this theory is the study of the set of absolute values on a field \(F\) which extend a prescribed absolute value on a subfield \(K\).
In this paper, the authors study the absolute values on a field which simultaneously extend absolute values on two subfields. They apply their results to count points on the fibered product of two curve morphisms \(C\to X\) and \(D\to X\) which lie over prescribed points on \(C\) and \(D\). As the authors explain, this situation arises in many areas of mathematics, including complex dynamics, complex analysis, algebraic topology, number theory, functional equations, and coding theory. They give a general and abstract description of these absolute values, which appear to be useful in applications.
In the preliminary section, the authors recall some standard facts about absolute values which will be used in this article. In the main section, the authors provide a common generalization of most variants of Abhyankar's lemma, which is a valuable tool with many applications.
Reviewer: Mouad Moutaoukil (Fès)On triangulable tensor products of \(B\)-pairs and trianguline representationshttps://zbmath.org/1487.140522022-07-25T18:03:43.254055Z"Berger, Laurent"https://zbmath.org/authors/?q=ai:berger.laurent"Di Matteo, Giovanni"https://zbmath.org/authors/?q=ai:di-matteo.giovanniTrianguline representations (introduced by \textit{P. Colmez} [Astérisque 319, 213--258 (2008; Zbl 1168.11022)] are a certain class of \(p\)-adic representations of \(\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_p)\), including crystalline, and semistable representations of Fontaine. They are defined using their associated \((\varphi,\Gamma)\)-modules. In the article under review, the authors prove the following theorem: if \(V\) and \(V'\) are representations of \(\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_p)\), \(V \otimes V'\) is trianguline, then \(V\) and \(V'\) are potentially trianguline.
Reviewer: Dingxin Zhang (Beijing)