This note answers, and generalizes, a question of Kaisa Matomaki. We show that give two cuspidal automorphic representations $pi_1$ and $pi_2$ of $GL_n$ over a number field $F$ of respective conductor $N_1,$ $N_2,$ every character $chi$ such that $pi_1otimeschisimeqpi_2$ of conductor $Q,$ satisfies the bound: $Q^nmid N_1N_2.$ If at every finite place $v,$ $pi_{1,v}$ is a discrete series whenever it is ramified, then $Q^n$ divides the least common multiple $[N_1, N_2].$
{"title":"A constraint for twist equivalence of cusp forms on GL$(n)$","authors":"D. Ramakrishnan, Liyang Yang","doi":"10.7169/FACM/1913","DOIUrl":"https://doi.org/10.7169/FACM/1913","url":null,"abstract":"This note answers, and generalizes, a question of Kaisa Matomaki. We show that give two cuspidal automorphic representations $pi_1$ and $pi_2$ of $GL_n$ over a number field $F$ of respective conductor $N_1,$ $N_2,$ every character $chi$ such that $pi_1otimeschisimeqpi_2$ of conductor $Q,$ satisfies the bound: $Q^nmid N_1N_2.$ If at every finite place $v,$ $pi_{1,v}$ is a discrete series whenever it is ramified, then $Q^n$ divides the least common multiple $[N_1, N_2].$","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44514141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $A$ be a set of $N$ vectors in ${mathbb Z}^n$ and let $v$ be a vector in ${mathbb C}^N$ that has minimal negative support for $A$. Such a vector $v$ gives rise to a formal series solution of the $A$-hypergeometric system with parameter $beta=Av$. If $v$ lies in ${mathbb Q}^n$, then this series has rational coefficients. Let $p$ be a prime number. We characterize those $v$ whose coordinates are rational, $p$-integral, and lie in the closed interval $[-1,0]$ for which the corresponding normalized series solution has $p$-integral coefficients. From this we deduce further integrality results for hypergeometric series.
设$A$是${mathbb Z}^ N$中$N$向量的集合，设$v$是${mathbb C}^N$中对$A$具有最小负支持的向量。这样的向量$v$给出了参数$beta=Av$的$ a $-超几何系统的形式级数解。如果$v$在${mathbb Q}^n$中，则该级数具有有理系数。设p是质数。我们刻画了那些坐标是有理的，p$-积分的，并且在闭合区间$[-1,0]$中，对应的归一化级数解具有p$-积分系数的$v$。由此进一步导出了超几何级数的完整性结果。
{"title":"On integrality properties of hypergeometric series","authors":"A. Adolphson, S. Sperber","doi":"10.7169/FACM/1843","DOIUrl":"https://doi.org/10.7169/FACM/1843","url":null,"abstract":"Let $A$ be a set of $N$ vectors in ${mathbb Z}^n$ and let $v$ be a vector in ${mathbb C}^N$ that has minimal negative support for $A$. Such a vector $v$ gives rise to a formal series solution of the $A$-hypergeometric system with parameter $beta=Av$. If $v$ lies in ${mathbb Q}^n$, then this series has rational coefficients. Let $p$ be a prime number. We characterize those $v$ whose coordinates are rational, $p$-integral, and lie in the closed interval $[-1,0]$ for which the corresponding normalized series solution has $p$-integral coefficients. From this we deduce further integrality results for hypergeometric series.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46561057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a degree 1 function $Finmathcal{S}^{sharp}$ and a real number $alpha$, we consider the linear twist $F(s,alpha)$, proving that it satisfies a functional equation reflecting $s$ into $1-s$, which can be seen as a Hurwitz-Lerch type of functional equation. We also derive some results on the distribution of the zeros of the linear twist.
给定一个1次函数$Finmathcal{S}^{sharp}$和一个实数$alpha$，我们考虑线性扭曲$F(S，alpha)$，证明它满足一个反映$ S $为$1- S $的泛函方程，这可以看作是一个Hurwitz-Lerch型泛函方程。我们还得到了关于线性扭转的零点分布的一些结果。
{"title":"On the linear twist of degree $1$ functions in the extended Selberg class","authors":"Giamila Zaghloul","doi":"10.7169/facm/1801","DOIUrl":"https://doi.org/10.7169/facm/1801","url":null,"abstract":"Given a degree 1 function $Finmathcal{S}^{sharp}$ and a real number $alpha$, we consider the linear twist $F(s,alpha)$, proving that it satisfies a functional equation reflecting $s$ into $1-s$, which can be seen as a Hurwitz-Lerch type of functional equation. We also derive some results on the distribution of the zeros of the linear twist.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46591765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Products of consecutive values of some quartic polynomials","authors":"A. Dubickas","doi":"10.7169/FACM/1733","DOIUrl":"https://doi.org/10.7169/FACM/1733","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.7169/FACM/1733","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46694784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Lucas non-Wieferich primes in arithmetic progressions","authors":"S. S. Rout","doi":"10.7169/facm/1709","DOIUrl":"https://doi.org/10.7169/facm/1709","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.7169/facm/1709","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42696359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that for most primes $p$, the set of Fibonomials forms an additive basis of order $8$ for the group of residue classes modulo $p$.
我们证明了对于大多数素数$p$，对于模$p$的残差类组，纤维组元形成$8$阶的加性基。
{"title":"A note on fibonomial coefficients","authors":"V. C. García, F. Luca","doi":"10.7169/FACM/1697","DOIUrl":"https://doi.org/10.7169/FACM/1697","url":null,"abstract":"We show that for most primes $p$, the set of Fibonomials forms an additive basis of order $8$ for the group of residue classes modulo $p$.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.7169/FACM/1697","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41879877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Explicit expression of a Barban & Vehov Theorem","authors":"Mohamed Haye Betah","doi":"10.7169/facm/1712","DOIUrl":"https://doi.org/10.7169/facm/1712","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.7169/facm/1712","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44387935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a recent paper of Dokchitser--Dokchitser--Maistret--Morgan, the authors introduced the concept of a cluster picture associated to a hyperelliptic curve from which they are able to recover numerous invariants, including the inertia representation on the first etale cohomology group of the curve. The purpose of this paper is to explore the functionality of these cluster pictures and prove that the inertia representation of a hyperelliptic curve is a function of its cluster picture.
{"title":"Clusters, inertia, and root numbers","authors":"Matthew Bisatt","doi":"10.7169/facm/1973","DOIUrl":"https://doi.org/10.7169/facm/1973","url":null,"abstract":"In a recent paper of Dokchitser--Dokchitser--Maistret--Morgan, the authors introduced the concept of a cluster picture associated to a hyperelliptic curve from which they are able to recover numerous invariants, including the inertia representation on the first etale cohomology group of the curve. The purpose of this paper is to explore the functionality of these cluster pictures and prove that the inertia representation of a hyperelliptic curve is a function of its cluster picture.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43739262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We bound Kloosterman-like sums of the shape [ sum_{n=1}^N exp(2pi i (x lfloor f(n)rfloor+ y lfloor f(n)rfloor^{-1})/p), ] with integers parts of a real-valued, twice-differentiable function $f$ is satisfying a certain limit condition on $f''$, and $lfloor f(n)rfloor^{-1}$ is meaning inversion modulo~$p$. As an immediate application, we obtain results concerning the distribution of modular inverses inverses $lfloor f(n)rfloor^{-1} pmod{p}$. The results apply, in particular, to Piatetski-Shapiro sequences $ lfloor t^crfloor$ with $cin(1,frac{4}{3})$. The proof is an adaptation of an argument used by Banks and the first named author in a series of papers from 2006 to 2009.
{"title":"Kloosterman sums with twice-differentiable functions","authors":"I. Shparlinski, Marc Technau","doi":"10.7169/facm/1845","DOIUrl":"https://doi.org/10.7169/facm/1845","url":null,"abstract":"We bound Kloosterman-like sums of the shape [ sum_{n=1}^N exp(2pi i (x lfloor f(n)rfloor+ y lfloor f(n)rfloor^{-1})/p), ] with integers parts of a real-valued, twice-differentiable function $f$ is satisfying a certain limit condition on $f''$, and $lfloor f(n)rfloor^{-1}$ is meaning inversion modulo~$p$. As an immediate application, we obtain results concerning the distribution of modular inverses inverses $lfloor f(n)rfloor^{-1} pmod{p}$. The results apply, in particular, to Piatetski-Shapiro sequences $ lfloor t^crfloor$ with $cin(1,frac{4}{3})$. The proof is an adaptation of an argument used by Banks and the first named author in a series of papers from 2006 to 2009.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41520636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent $s d$ case, fixing all pairs of distnaces leads to an overdetermined system, so $q^{binom{k+1}{2}}$ is no longer the correct number of congruence classes. We determine the correct number, and prove that $|E|gtrsim q^s$ still determines a positive proportion of all congruence classes, for the same $s$ as in the $kleq d$ case.
{"title":"Congruence classes of large configurations in vector spaces over finite fields","authors":"Alex McDonald","doi":"10.7169/facm/1814","DOIUrl":"https://doi.org/10.7169/facm/1814","url":null,"abstract":"Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent $s d$ case, fixing all pairs of distnaces leads to an overdetermined system, so $q^{binom{k+1}{2}}$ is no longer the correct number of congruence classes. We determine the correct number, and prove that $|E|gtrsim q^s$ still determines a positive proportion of all congruence classes, for the same $s$ as in the $kleq d$ case.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47520377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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