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1. CJM Online first

Levin, Aaron; Wang, Julie Tzu-Yueh
On non-Archimedean curves omitting few components and their arithmetic analogues
Let $\mathbf{k}$ be an algebraically closed field complete with respect to a non-Archimedean absolute value of arbitrary characteristic. Let $D_1,\dots, D_n$ be effective nef divisors intersecting transversally in an $n$-dimensional nonsingular projective variety $X$. We study the degeneracy of non-Archimedean analytic maps from $\mathbf{k}$ into $X\setminus \cup_{i=1}^nD_i$ under various geometric conditions. When $X$ is a rational ruled surface and $D_1$ and $D_2$ are ample, we obtain a necessary and sufficient condition such that there is no non-Archimedean analytic map from $\mathbf{k}$ into $X\setminus D_1 \cup D_2$. Using the dictionary between non-Archimedean Nevanlinna theory and Diophantine approximation that originated in earlier work with T. T. H. An, % we also study arithmetic analogues of these problems, establishing results on integral points on these varieties over $\mathbb{Z}$ or the ring of integers of an imaginary quadratic field.

Keywords:non-Archimedean Picard theorem, non-Archimedean analytic curves, integral points
Categories:11J97, 32P05, 32H25

2. CJM Online first

Garibaldi, Skip; Nakano, Daniel K.
Bilinear and quadratic forms on rational modules of split reductive groups
The representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the question of whether a given complex representation is symplectic or orthogonal has been solved since at least the 1950s. Similar results for Weyl modules of split reductive groups over fields of characteristic different from 2 hold by using similar proofs. This paper considers analogues of these results for simple, induced and tilting modules of split reductive groups over fields of prime characteristic as well as a complete answer for Weyl modules over fields of characteristic 2.

Keywords:orthogonal representations, symmetric tensors, alternating forms, characteristic 2, split reductive groups
Categories:20G05, 11E39, 11E88, 15A63, 20G15

3. CJM Online first

Cojocaru, Alina Carmen; Shulman, Andrew Michael
The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank
Let $\psi$ be a generic Drinfeld module of rank $r \geq 2$. We study the first elementary divisor $d_{1, \wp}(\psi)$ of the reduction of $\psi$ modulo a prime $\wp$, as $\wp$ varies. In particular, we prove the existence of the density of the primes $\wp$ for which $d_{1, \wp} (\psi)$ is fixed. For $r = 2$, we also study the second elementary divisor (the exponent) of the reduction of $\psi$ modulo $\wp$ and prove that, on average, it has a large norm. Our work is motivated by the study of J.-P. Serre of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M.R. Murty.

Keywords:Drinfeld modules, density theorems
Categories:11R45, 11G09, 11R58

4. CJM 2015 (vol 67 pp. 1046)

Dubickas, Arturas; Sha, Min; Shparlinski, Igor
Explicit Form of Cassels' $p$-adic Embedding Theorem for Number Fields
In this paper, we mainly give a general explicit form of Cassels' $p$-adic embedding theorem for number fields. We also give its refined form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over $\mathbb{Z}$, we give a general unconditional upper bound for the smallest prime number $p$ such that $f$ has a simple root modulo $p$.

Keywords:number field, $p$-adic embedding, height, polynomial, cyclotomic field
Categories:11R04, 11S85, 11G50, 11R09, 11R18

5. CJM Online first

Fité, Francesc; González, Josep; Lario, Joan Carles
Frobenius distribution for quotients of Fermat curves of prime exponent
Let $\mathcal{C}$ denote the Fermat curve over $\mathbb{Q}$ of prime exponent $\ell$. The Jacobian $\operatorname{Jac}(\mathcal{C})$ of~$\mathcal{C}$ splits over $\mathbb{Q}$ as the product of Jacobians $\operatorname{Jac}(\mathcal{C}_k)$, $1\leq k\leq \ell-2$, where $\mathcal{C}_k$ are curves obtained as quotients of $\mathcal{C}$ by certain subgroups of automorphisms of $\mathcal{C}$. It is well known that $\operatorname{Jac}(\mathcal{C}_k)$ is the power of an absolutely simple abelian variety $B_k$ with complex multiplication. We call degenerate those pairs $(\ell,k)$ for which $B_k$ has degenerate CM type. For a non-degenerate pair $(\ell,k)$, we compute the Sato-Tate group of $\operatorname{Jac}(\mathcal{C}_k)$, prove the generalized Sato-Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of $(\ell,k)$ being degenerate or not, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the $\ell$-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

Keywords:Sato-Tate group, Fermat curve, Frobenius distribution
Categories:11D41, 11M50, 11G10, 14G10

6. CJM Online first

Gras, Georges
Les $\theta$-régulateurs locaux d'un nombre algébrique -- Conjectures $p$-adiques
Let $K/\mathbb{Q}$ be Galois and let $\eta\in K^\times$ be such that $\operatorname{Reg}_\infty (\eta) \ne 0$. We define the local $\theta$-regulators $\Delta_p^\theta(\eta) \in \mathbb{F}_p$ for the $\mathbb{Q}_p\,$-irreducible characters $\theta$ of $G=\operatorname{Gal}(K/\mathbb{Q})$. A linear representation ${\mathcal L}^\theta\simeq \delta \, V_\theta$ is associated with $\Delta_p^\theta (\eta)$ whose nullity is equivalent to $\delta \geq 1$. Each $\Delta_p^\theta (\eta)$ yields $\operatorname{Reg}_p^\theta (\eta)$ modulo $p$ in the factorization $\prod_{\theta}(\operatorname{Reg}_p^\theta (\eta))^{\varphi(1)}$ of $\operatorname{Reg}_p^G (\eta) := \frac{ \operatorname{Reg}_p(\eta)}{p^{[K : \mathbb{Q}\,]} }$ (normalized $p$-adic regulator). From $\operatorname{Prob}\big (\Delta_p^\theta(\eta) = 0 \ \& \ {\mathcal L}^\theta \simeq \delta \, V_\theta\big ) \leq p^{- f \delta^2}$ ($f \geq 1$ is a residue degree) and the Borel-Cantelli heuristic, we conjecture that, for $p$ large enough, $\operatorname{Reg}_p^G (\eta)$ is a $p$-adic unit or that $p^{\varphi(1)} \parallel \operatorname{Reg}_p^G (\eta)$ (a single $\theta$ with $f=\delta=1$); this obstruction may be lifted assuming the existence of a binomial probability law confirmed through numerical studies (groups $C_3$, $C_5$, $D_6$). This 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.

Keywords:$p$-adic regulators, Leopoldt-Jaulent conjecture, Frobenius group determinants, characters, Fermat quotient, Abelian $p$-ramification, probabilistic number theory
Categories:11F85, 11R04, 20C15, 11C20, 11R37, 11R27, 11Y40

7. CJM 2015 (vol 67 pp. 654)

Lim, Meng Fai; Murty, V. Kumar
Growth of Selmer groups of CM Abelian varieties
Let $p$ be an odd prime. We study the variation of the $p$-rank of the Selmer group of Abelian varieties with complex multiplication in certain towers of number fields.

Keywords:Selmer group, Abelian variety with complex multiplication, $\mathbb{Z}_p$-extension, $p$-Hilbert class tower
Categories:11G15, 11G10, 11R23, 11R34

8. CJM Online first

Bonfanti, Matteo Alfonso; van Geemen,
Abelian Surfaces with an Automorphism and Quaternionic Multiplication
We construct one dimensional families of Abelian surfaces with quaternionic multiplication which also have an automorphism of order three or four. Using Barth's description of the moduli space of $(2,4)$-polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space which parametrizes Abelian surfaces with real multiplication by $\mathbf{Z}[\sqrt{2}]$.

Keywords:abelian surfaces, moduli, shimura curves
Categories:14K10, 11G10, 14K20

9. CJM Online first

Chandee, Vorrapan; David, Chantal; Koukoulopoulos, Dimitris; Smith, Ethan
The frequency of elliptic curve groups over prime finite fields
Letting $p$ vary over all primes and $E$ vary over all elliptic curves over the finite field $\mathbb{F}_p$, we study the frequency to which a given group $G$ arises as a group of points $E(\mathbb{F}_p)$. It is well-known that the only permissible groups are of the form $G_{m,k}:=\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/mk\mathbb{Z}$. Given such a candidate group, we let $M(G_{m,k})$ be the frequency to which the group $G_{m,k}$ arises in this way. Previously, the second and fourth named authors determined an asymptotic formula for $M(G_{m,k})$ assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for $M(G_{m,k})$, pointwise and on average. In particular, we show that $M(G_{m,k})$ is bounded above by a constant multiple of the expected quantity when $m\le k^A$ and that the conjectured asymptotic for $M(G_{m,k})$ holds for almost all groups $G_{m,k}$ when $m\le k^{1/4-\epsilon}$. We also apply our methods to study the frequency to which a given integer $N$ arises as the group order $\#E(\mathbb{F}_p)$.

Keywords:average order, elliptic curves, primes in short intervals
Categories:11G07, 11N45, 11N13, 11N36

10. CJM 2015 (vol 67 pp. 597)

Drappeau, Sary
Sommes friables d'exponentielles et applications
An integer is said to be $y$-friable if its greatest prime factor is less than $y$. In this paper, we obtain estimates for exponential sums over $y$-friable numbers up to $x$ which are non-trivial when $y \geq \exp\{c \sqrt{\log x} \log \log x\}$. As a consequence, we obtain an asymptotic formula for the number of $y$-friable solutions to the equation $a+b=c$ which is valid unconditionnally under the same assumption. We use a contour integration argument based on the saddle point method, as developped in the context of friable numbers by Hildebrand and Tenenbaum, and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers.

Keywords:théorie analytique des nombres, entiers friables, méthode du col
Categories:12N25, 11L07

11. CJM Online first

Stange, Katherine E.
Integral Points on Elliptic Curves and Explicit Valuations of Division Polynomials
Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ and non-torsion point $P \in E(\mathbb{Q})$, there is at most one integral multiple $[n]P$ such that $n \gt C$. The proof is a modification of a proof of Ingram giving an unconditional but not uniform bound. The new ingredient is a collection of explicit formulae for the sequence $v(\Psi_n)$ of valuations of the division polynomials. For $P$ of non-singular reduction, such sequences are already well described in most cases, but for $P$ of singular reduction, we are led to define a new class of sequences called \emph{elliptic troublemaker sequences}, which measure the failure of the Néron local height to be quadratic. As a corollary in the spirit of a conjecture of Lang and Hall, we obtain a uniform upper bound on $\widehat{h}(P)/h(E)$ for integer points having two large integral multiples.

Keywords:elliptic divisibility sequence, Lang's conjecture, height functions
Categories:11G05, 11G07, 11D25, 11B37, 11B39, 11Y55, 11G50, 11H52

12. CJM Online first

Takeda, Shuichiro
Metaplectic Tensor Products for Automorphic Representation of $\widetilde{GL}(r)$
Let $M=\operatorname{GL}_{r_1}\times\cdots\times\operatorname{GL}_{r_k}\subseteq\operatorname{GL}_r$ be a Levi subgroup of $\operatorname{GL}_r$, where $r=r_1+\cdots+r_k$, and $\widetilde{M}$ its metaplectic preimage in the $n$-fold metaplectic cover $\widetilde{\operatorname{GL}}_r$ of $\operatorname{GL}_r$. For automorphic representations $\pi_1,\dots,\pi_k$ of $\widetilde{\operatorname{GL}}_{r_1}(\mathbb{A}),\dots,\widetilde{\operatorname{GL}}_{r_k}(\mathbb{A})$, we construct (under a certain technical assumption, which is always satisfied when $n=2$) an automorphic representation $\pi$ of $\widetilde{M}(\mathbb{A})$ which can be considered as the ``tensor product'' of the representations $\pi_1,\dots,\pi_k$. This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place $v$, $\pi_v$ is equivalent to the local metaplectic tensor product of $\pi_{1,v},\dots,\pi_{k,v}$ defined by Mezo. Then we show that if all of $\pi_i$ are cuspidal (resp. square-integrable modulo center), then the metaplectic tensor product is cuspidal (resp. square-integrable modulo center). We also show that (both locally and globally) the metaplectic tensor product behaves in the expected way under the action of a Weyl group element, and show the compatibility with parabolic inductions.

Keywords:automorphic forms, representations of covering groups

13. CJM 2014 (vol 67 pp. 893)

Mok, Chung Pang; Tan, Fucheng
Overconvergent Families of Siegel-Hilbert Modular Forms
We construct one-parameter families of overconvergent Siegel-Hilbert modular forms. This result has applications to construction of Galois representations for automorphic forms of non-cohomological weights.

Keywords:p-adic automorphic form, rigid analytic geometry
Categories:11F46, 14G22

14. CJM 2014 (vol 67 pp. 848)

Köck, Bernhard; Tait, Joseph
Faithfulness of Actions on Riemann-Roch Spaces
Given a faithful action of a finite group $G$ on an algebraic curve~$X$ of genus $g_X\geq 2$, we give explicit criteria for the induced action of~$G$ on the Riemann-Roch space~$H^0(X,\mathcal{O}_X(D))$ to be faithful, where $D$ is a $G$-invariant divisor on $X$ of degree at least~$2g_X-2$. This leads to a concise answer to the question when the action of~$G$ on the space~$H^0(X, \Omega_X^{\otimes m})$ of global holomorphic polydifferentials of order $m$ is faithful. If $X$ is hyperelliptic, we furthermore provide an explicit basis of~$H^0(X, \Omega_X^{\otimes m})$. Finally, we give applications in deformation theory and in coding theory and we discuss the analogous problem for the action of~$G$ on the first homology $H_1(X, \mathbb{Z}/m\mathbb{Z})$ if $X$ is a Riemann surface.

Keywords:faithful action, Riemann-Roch space, polydifferential, hyperelliptic curve, equivariant deformation theory, Goppa code, homology
Categories:14H30, 30F30, 14L30, 14D15, 11R32

15. CJM 2014 (vol 66 pp. 993)

Beuzart-Plessis, Raphaël
Expression d'un facteur epsilon de paire par une formule intégrale
Let $E/F$ be a quadratic extension of $p$-adic fields and let $d$, $m$ be nonnegative integers of distinct parities. Fix admissible irreducible tempered representations $\pi$ and $\sigma$ of $GL_d(E)$ and $GL_m(E)$ respectively. We assume that $\pi$ and $\sigma$ are conjugate-dual. That is to say $\pi\simeq \pi^{\vee,c}$ and $\sigma\simeq \sigma^{\vee,c}$ where $c$ is the non trivial $F$-automorphism of $E$. This implies, we can extend $\pi$ to an unitary representation $\tilde{\pi}$ of a nonconnected group $GL_d(E)\rtimes \{1,\theta\}$. Define $\tilde{\sigma}$ the same way. We state and prove an integral formula for $\epsilon(1/2,\pi\times \sigma,\psi_E)$ involving the characters of $\tilde{\pi}$ and $\tilde{\sigma}$. This formula is related to the local Gan-Gross-Prasad conjecture for unitary groups.

Keywords:epsilon factor, twisted groups
Categories:22E50, 11F85

16. CJM 2014 (vol 67 pp. 795)

Di Nasso, Mauro; Goldbring, Isaac; Jin, Renling; Leth, Steven; Lupini, Martino; Mahlburg, Karl
On a Sumset Conjecture of Erdős
Erdős conjectured that for any set $A\subseteq \mathbb{N}$ with positive lower asymptotic density, there are infinite sets $B,C\subseteq \mathbb{N}$ such that $B+C\subseteq A$. We verify Erdős' conjecture in the case that $A$ has Banach density exceeding $\frac{1}{2}$. As a consequence, we prove that, for $A\subseteq \mathbb{N}$ with positive Banach density (a much weaker assumption than positive lower density), we can find infinite $B,C\subseteq \mathbb{N}$ such that $B+C$ is contained in the union of $A$ and a translate of $A$. Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős' conjecture for subsets of the natural numbers that are pseudorandom.

Keywords:sumsets of integers, asymptotic density, amenable groups, nonstandard analysis
Categories:11B05, 11B13, 11P70, 28D15, 37A45

17. CJM 2014 (vol 67 pp. 424)

Samart, Detchat
Mahler Measures as Linear Combinations of $L$-values of Multiple Modular Forms
We study the Mahler measures of certain families of Laurent polynomials in two and three variables. Each of the known Mahler measure formulas for these families involves $L$-values of at most one newform and/or at most one quadratic character. In this paper, we show, either rigorously or numerically, that the Mahler measures of some polynomials are related to $L$-values of multiple newforms and quadratic characters simultaneously. The results suggest that the number of modular $L$-values appearing in the formulas significantly depends on the shape of the algebraic value of the parameter chosen for each polynomial. As a consequence, we also obtain new formulas relating special values of hypergeometric series evaluated at algebraic numbers to special values of $L$-functions.

Keywords:Mahler measures, Eisenstein-Kronecker series, $L$-functions, hypergeometric series
Categories:11F67, 33C20

18. CJM 2014 (vol 67 pp. 507)

Borwein, Peter; Choi, Stephen; Ferguson, Ron; Jankauskas, Jonas
On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk
We investigate the numbers of complex zeros of Littlewood polynomials $p(z)$ (polynomials with coefficients $\{-1, 1\}$) inside or on the unit circle $|z|=1$, denoted by $N(p)$ and $U(p)$, respectively. Two types of Littlewood polynomials are considered: Littlewood polynomials with one sign change in the sequence of coefficients and Littlewood polynomials with one negative coefficient. We obtain explicit formulas for $N(p)$, $U(p)$ for polynomials $p(z)$ of these types. We show that, if $n+1$ is a prime number, then for each integer $k$, $0 \leq k \leq n-1$, there exists a Littlewood polynomial $p(z)$ of degree $n$ with $N(p)=k$ and $U(p)=0$. Furthermore, we describe some cases when the ratios $N(p)/n$ and $U(p)/n$ have limits as $n \to \infty$ and find the corresponding limit values.

Keywords:Littlewood polynomials, zeros, complex roots
Categories:11R06, 11R09, 11C08

19. CJM 2014 (vol 67 pp. 198)

Murty, V. Kumar; Patankar, Vijay M.
Tate Cycles on Abelian Varieties with Complex Multiplication
We consider Tate cycles on an Abelian variety $A$ defined over a sufficiently large number field $K$ and having complex multiplication. We show that there is an effective bound $C = C(A,K)$ so that to check whether a given cohomology class is a Tate class on $A$, it suffices to check the action of Frobenius elements at primes $v$ of norm $ \leq C$. We also show that for a set of primes $v$ of $K$ of density $1$, the space of Tate cycles on the special fibre $A_v$ of the Néron model of $A$ is isomorphic to the space of Tate cycles on $A$ itself.

Keywords:Abelian varieties, complex multiplication, Tate cycles
Categories:11G10, 14K22

20. CJM 2014 (vol 66 pp. 1078)

Lanphier, Dominic; Skogman, Howard
Values of Twisted Tensor $L$-functions of Automorphic Forms Over Imaginary Quadratic Fields
Let $K$ be a complex quadratic extension of $\mathbb{Q}$ and let $\mathbb{A}_K$ denote the adeles of $K$. We find special values at all of the critical points of twisted tensor $L$-functions attached to cohomological cuspforms on $GL_2(\mathbb{A}_K)$, and establish Galois equivariance of the values. To investigate the values, we determine the archimedean factors of a class of integral representations of these $L$-functions, thus proving a conjecture due to Ghate. We also investigate analytic properties of these $L$-functions, such as their functional equations.

Keywords:twisted tensor $L$-function, cuspform, hypergeometric series
Categories:11F67, 11F37

21. CJM 2014 (vol 67 pp. 286)

Bell, Jason P.; Lagarias, Jeffrey C.
A Skolem-Mahler-Lech Theorem for Iterated Automorphisms of $K$-algebras
This paper proves a commutative algebraic extension of a generalized Skolem-Mahler-Lech theorem due to the first author. Let $A$ be a finitely generated commutative $K$-algebra over a field of characteristic $0$, and let $\sigma$ be a $K$-algebra automorphism of $A$. Given ideals $I$ and $J$ of $A$, we show that the set $S$ of integers $m$ such that $\sigma^m(I) \supseteq J$ is a finite union of complete doubly infinite arithmetic progressions in $m$, up to the addition of a finite set. Alternatively, this result states that for an affine scheme $X$ of finite type over $K$, an automorphism $\sigma \in \operatorname{Aut}_K(X)$, and $Y$ and $Z$ any two closed subschemes of $X$, the set of integers $m$ with $\sigma^m(Z ) \subseteq Y$ is as above. The paper presents examples showing that this result may fail to hold if the affine scheme $X$ is not of finite type, or if $X$ is of finite type but the field $K$ has positive characteristic.

Keywords:automorphisms, endomorphisms, affine space, commutative algebras, Skolem-Mahler-Lech theorem
Categories:11D45, 14R10, 11Y55, 11D88

22. CJM 2013 (vol 66 pp. 826)

Kim, Byoung Du
Signed-Selmer Groups over the $\mathbb{Z}_p^2$-extension of an Imaginary Quadratic Field
Let $E$ be an elliptic curve over $\mathbb Q$ which has good supersingular reduction at $p\gt 3$. We construct what we call the $\pm/\pm$-Selmer groups of $E$ over the $\mathbb Z_p^2$-extension of an imaginary quadratic field $K$ when the prime $p$ splits completely over $K/\mathbb Q$, and prove they enjoy a property analogous to Mazur's control theorem. Furthermore, we propose a conjectural connection between the $\pm/\pm$-Selmer groups and Loeffler's two-variable $\pm/\pm$-$p$-adic $L$-functions of elliptic curves.

Keywords:elliptic curves, Iwasawa theory

23. CJM 2013 (vol 66 pp. 1305)

Koskivirta, Jean-Stefan
Congruence Relations for Shimura Varieties Associated with $GU(n-1,1)$
We prove the congruence relation for the mod-$p$ reduction of Shimura varieties associated to a unitary similitude group $GU(n-1,1)$ over $\mathbb{Q}$, when $p$ is inert and $n$ odd. The case when $n$ is even was obtained by T. Wedhorn and O. B?ltel, as a special case of a result of B. Moonen, when the $\mu$-ordinary locus of the $p$-isogeny space is dense. This condition fails in our case. We show that every supersingular irreducible component of the special fiber of $p\textrm{-}\mathscr{I}sog$ is annihilated by a degree one polynomial in the Frobenius element $F$, which implies the congruence relation.

Keywords:Shimura varieties, congruence relation
Categories:11G18, 14G35, 14K10

24. CJM 2013 (vol 67 pp. 214)

Szpruch, Dani
Symmetric Genuine Spherical Whittaker Functions on $\overline{GSp_{2n}(F)}$
Let $F$ be a p-adic field of odd residual characteristic. Let $\overline{GSp_{2n}(F)}$ and $\overline{Sp_{2n}(F)}$ be the metaplectic double covers of the general symplectic group and the symplectic group attached to the $2n$ dimensional symplectic space over $F$. Let $\sigma$ be a genuine, possibly reducible, unramified principal series representation of $\overline{GSp_{2n}(F)}$. In these notes we give an explicit formulas for a spanning set for the space of Spherical Whittaker functions attached to $\sigma$. For odd $n$, and generically for even $n$, this spanning set is a basis. The significant property of this set is that each of its elements is unchanged under the action of the Weyl group of $\overline{Sp_{2n}(F)}$. If $n$ is odd then each element in the set has an equivariant property that generalizes a uniqueness result of Gelbart, Howe and Piatetski-Shapiro. Using this symmetric set, we construct a family of reducible genuine unramified principal series representations which have more then one generic constituent. This family contains all the reducible genuine unramified principal series representations induced from a unitary data and exists only for $n$ even.

Keywords:metaplectic group, Casselman Shalika Formula

25. CJM 2013 (vol 65 pp. 1320)

Taniguchi, Takashi; Thorne, Frank
Orbital $L$-functions for the Space of Binary Cubic Forms
We introduce the notion of orbital $L$-functions for the space of binary cubic forms and investigate their analytic properties. We study their functional equations and residue formulas in some detail. Aside from their intrinsic interest, the results from this paper are used to prove the existence of secondary terms in counting functions for cubic fields. This is worked out in a companion paper.

Keywords:binary cubic forms, prehomogeneous vector spaces, Shintani zeta functions, $L$-functions, cubic rings and fields
Categories:11M41, 11E76
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