Let $N/K$ be a finite Galois extension of $p$-adic number fields and let $ρ^\mathrm{nr} : G_K \to \mathrm{Gl}_r(\mathbb Z_p)$ be an $r$-dimensional unramified representation of the absolute Galois group $G_K$ which is the restriction of an unramified representation $ρ^\mathrm{nr}_{\mathbb Q_p} : G_{\mathbb Q_p} \to \mathrm{Gl}_r(\mathbb Z_p)$. In this paper we consider the $\mathrm{Gal}(N/K)$-equivariant local $ε$-conjecture for the $p$-adic representation $T = \mathbb Z_p^r(1)(ρ^\mathrm{nr})$. For example, if $A$ is an abelian variety of dimension $r$ defined over $\mathbb Q_p$ with good ordinary reduction, then the Tate module $T = T_p\hat A$ associated to the formal group $\hat A$ of $A$ is a $p$-adic representation of this form. We prove the conjecture for all tame extensions $N/K$ and a certain family of weakly and wildly ramified extensions $N/K$. This generalizes previous work of Izychev and Venjakob in the tame case and of the authors in the weakly and wildly ramified case.
Let G be the unramified unitary group in three variables defined over a p-adic field F of odd resudual characteristic. Gelbart, Piatetski-Shapiro and Baruch attached zeta integrals of Rankin-Selberg type to irreducible generic representations of G. In this paper, we formulate a conjecture on L- and epsilon-factors defined through zeta integrals in terms of local newforms for G, which is an analogue of the result by Casselman and Deligne for GL(2). We prove our conjecture for the generic supercuspidal representations of G.
Let G be the unramified unitary group in three variables defined over a p-adic field of odd residual characteristic. In this paper, we establish a theory of newforms for the Rankin-Selberg integral for G introduced by Gelbart and Piatetski-Shapiro. We describe L and epsilon-factors defined through zeta integrals in terms of newforms. We show that zeta integrals of newforms for generic representations attain L-factors. As a corollary, we get an explicit formula for epsilon-factors of generic representations.
We study the equivariant local epsilon constant conjecture, denoted by $C_{EP}^{na}(N/K,V)$, as formulated in various forms by Kato, Benois and Berger, Fukaya and Kato and others, for certain 1-dimensional twists $T=\mathbb{Z}_p(χ^{nr})(1)$ of $\mathbb{Z}_p(1)$. Following ideas of recent work of Izychev and Venjakob we prove that for $T=\mathbb{Z}_p(1)$ a conjecture of Breuning is equivalent to $C_{EP}^{na}(N/K,V)$. As our main result we show the validity of $C_{EP}^{na}(N/K,V)$ for certain wildly and weakly ramified abelian extensions $N/K$. A crucial step in the proof is the construction of …
We study the local epsilon constant conjecture as formulated by Breuning. This conjecture fits into the general framework of the equivariant Tamagawa number conjecture (ETNC) and should be interpreted as a consequence of the expected compatibility of the ETNC with the functional equation of Artin-L-functions. Let K / Q_p be unramified. Under some mild technical assumption we prove Breuning's conjecture for weakly ramified abelian extensions N / K with cyclic ramification group. As a consequence of Breuning's local-global principle we obtain the validity of the global epsilon constant conjectur…
In this article, we investigate the variance of local $\varepsilon$-factor for a modular form with arbitrary nebentypus with respect to twisting by a quadratic character. We detect the type of the supercuspidal representation from that. For modular forms with trivial nebentypus, similar results are proved by Pacetti. Our method however is completely different from that of Pacetti and we use representation theory crucially. For ramified principal series (with $\infdiv{p}{N}$ and $p$ odd) and unramified supercuspidal representations of level zero, we relate these numbers with the Morita's $p$-ad…
In this paper we formulate a conjecture on the relationship between the equivariant ε-constants (associated to a local p-adic representation V and a finite extension of local fields L/K) and local Galois cohomology groups of a Galois stable \mathbb{Z}_{p}-lattice T of V. We prove the conjecture for L/K being an unramified extension of degree prime to p and T being a p-adic Tate module of a one-dimensional Lubin-Tate group defined over \mathbb{Z}_{p} by extending the ideas of \cite{Breu} from the case of the multiplicative group \mathbb{G}_{m} to arbitrary one-dimensional Lubin-Tate groups. For…
Let F be a finite extension of Qp and G be GL(2,F). When V is the tensor product of three infinite dimensional, irreducible, admissible representations of G, the space of G-invariant linear forms has dimension 0 or 1. When a non-zero linear form exists, one wants to find an element of V which is not in its kernel : this is a test vector. Gross and Prasad found explicit test vectors when the three representations are unramified principal series, and when the three representations are unramified twists of the Steinberg representation. In this paper, we find an explicit test vector when two of th…
This paper contains a complete proof of Fukaya's and Kato's epsilon$-isomorphism conjecture in [23] for invertible Λ-modules (the case of V = V_0(r) where V_0 is unramified of dimension 1). Our results rely heavily on Kato's unpublished proof of (commutative) epsilon-isomorphisms for one dimensional representations of G_{Q_p} in [27], but apart from fixing some sign-ambiguities in (loc.\ cit.) we use the theory of (ϕ,Γ)-modules instead of syntomic cohomology. Also, for the convenience of the reader we give a slight modification or rather reformulation of it in the language of [23] and extend i…
Let $q$ be a prime with $q \equiv 7 \mod 8$, and let $K=\mathbb{Q}(\sqrt{-q})$. Then $2$ splits in $K$, and we write $\mathfrak{p}$ for either of the primes $K$ above $2$. Let $K_\infty$ be the unique $\mathbb{Z}_2$-extension of $K$ unramified outside $\mathfrak{p}$ with $n$-th layer $K_n$. For certain quadratic extensions $F/K$, we prove a simple exact formula for the $λ$-invariant of the Galois group of the maximal abelian 2-extension unramified outside $\mathfrak{p}$ of the field $F_\infty = FK_\infty$. Equivalently, our result determines the exact $\mathbb{Z}_2$-corank of certain Selmer gr…
Let U(N) be the quasi-split unitary group in N variables for a quadratic unramified extension of p-adic fields. We compute the characters of simple supercuspidal representations of twisted GL(N) and U(N). Comparing them by the endoscopic character relation, we determine the liftings of simple supercuspidal representations of U(N) to GL(N), under the assumption that p is not equal to 2.
In this article, we study the sum of additively twisted Fourier coefficients of an irreducible cuspidal automorphic representation of $\mathrm{GL}_2$ or $\mathrm{GL}_3$ over an arbitrary number field. When the representation is unramified at all non-archimedean places, we prove the Wilton type bound for $\mathrm{GL}_2$ and the Miller type bound for $\mathrm{GL}_3$ which are uniform in terms of the additive character.
We prove a twist formula for the epsilon factor of a constructible sheaf on a projective smooth variety over a finite field in terms of characteristic class of the sheaf. This formula is a modified version of the formula conjectured by Kato and Saito in [Ann. Math., 168 (2008):33-96, Conjecture 4.3.11]. We give two applications of the twist formula. Firstly, we prove that the characteristic classes of constructible étale sheaves on projective smooth varieties over a finite field are compatible with proper push-forward. Secondly, we show that the two Swan classes in the literature are the same …
We show that for a large class of finite groups G, the number of Galois extensions E/Q of group G and discriminant $|d_E|\leq y$ grows like a power of $y$ (for some specified exponent). The groups G are the regular Galois groups over Q and the extensions E/Q that we count are obtained by specialization from a given regular Galois extension F/Q(T). The extensions E/Q can further be prescribed any unramified local behavior at each suitably large prime $p\leq \log (y)/δ$ for some $δ\geq 1$. This result is a step toward the Malle conjecture on the number of Galois extensions of given group and bou…
Let $L/K$ be a finite Galois extension of $p$-adic fields and let $L_{\infty}$ be the unramified $\mathbb Z_p$-extension of $L$. Then $L_{\infty}/K$ is a one-dimensional $p$-adic Lie extension. In the spirit of the main conjectures of equivariant Iwasawa theory, we formulate a conjecture which relates the equivariant local epsilon constants attached to the finite Galois intermediate extensions $M/K$ of $L_{\infty}/K$ to a natural arithmetic invariant arising from the étale cohomology of the constant sheaf $\mathbb Q_p/\mathbb Z_p$ on the spectrum of $L_{\infty}$. We give strong evidence of the…
Given two pure representations of the absolute Galois group of an $\ell$-adic number field with coefficients in $\overline{\mathbb{Q}}_p$ (with $\ell\neq p$), we show that the Frobenius-semisimplifications of the associated Weil--Deligne representations are twists of each other by an integral power of certain unramified character if they have equal normalized traces. This is an analogue of a recent result of Patankar and Rajan in the context of local Galois representations.
Let $π$ traverse a sequence of cuspidal automorphic representations of GL(2) with large prime level, unramified central character and bounded infinity type. For G either of the groups GL(1) or PGL(2), let H(G) denote the assertion that subconvexity holds for G-twists of the adjoint $L$-function of $π$, with polynomial dependence upon the conductor of the twist. We show that H(GL(1)) implies H(PGL(2)). In geometric terms, H(PGL(2)) corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term, H(GL(1)) to the special case in which the relevant…
Consider an irreducible, admissible representation $π$ of GL(2,$F$) whose restriction to GL(2,$F)^+$ breaks up as a sum of two irreducible representations $π_+ + π_-$. If $π=r_θ$, the Weil representation of GL(2,$F$) attached to a character $θ$ of $K^*$ which does not factor through the norm map from $K$ to $F$, then $χ\in \hat{K^*}$ with $(χ>. θ^{-1})|_{F^{*}}=ω_{K/F}$ occurs in ${r_θ}_+$ if and only if $ε(θχ^{-1},ψ_0)=ε(\bar θχ^{-1},ψ_0)=1$ and in ${r_θ}_-$ if and only if both the epsilon factors are -1. But given a conductor $n$, can we say precisely how many such $χ$ will appear in $π$? We…
The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $Π$ of $\mathrm{GL}(2n)$ over a totally real field $F$ admitting a Shalika model. We use a modular symbol approach, along the global lines of the work of Ash and Ginzburg, but our results are more definitive since we draw heavily upon the methods used in the recent and separate works of all the three authors. By construction our $p$-adic $L$-functions are distributions on the Galois group of the maximal abelian extension of $F$ unramified outside $p\infty$. More…
We study the relationship between potential equivalence and character theory; we observe that potential equivalence of a representation $ρ$ is determined by an equality of an $m$-power character $g\mapsto Tr(ρ(g^m))$ for some natural number $m$. Using this, we extend Faltings' finiteness criteria to determine the equivalence of two $\ell$-adic, semisimple representations of the absolute Galois group of a number field, to the context of potential equivalence. We also discuss finiteness results for twist unramified representations.
Let pi be an automorphic representation on GL(r, A_Q) for r=1, 2, or 3. Let d be a fundamental discriminant and chi_d the corresponding quadratic Dirichlet character. We consider the question of the least d, relative to the data (level, weight or eigenvalue) of pi, such that the central value of the twisted L-series is nonzero, i.e. L(1/2, pi x chi_d) != 0. For example, let N be the level of pi. Using multiple Dirichlet series, we prove the nonvanishing of a central twisted $L$-value with d << N^{1/2} for GL(1), d << N^1 for GL(2), and d << N^2 for GL(3), the last case assuming that a certain …
Let $G$ be the group of rational points of a general linear group over a non-archimedean local field $F$. We show that certain representations of open, compact-mod-centre subgroups of $G$, (the maximal simple types of Bushnell and Kutzko) can be realized as concrete spaces. In the level zero case our result is essentially due to Gelfand. This allows us, for a supercuspidal representation $π$ of $G$, to compute a distinguished matrix coefficient of $π$. By integrating, we obtain an explicit Whittaker function for $π$. We use this to compute the epsilon factor of pairs, for supercuspidal represe…
We compute by a purely local method the (elliptic) twisted by transpose-inverse character χ_{π_Y} of the representation π_Y=I_{(3,1)}(1_3xχ_Y) of G=GL(4,F), where F is a p-adic field, p not 2, and Y is an unramified quadratic extension of F; χ_Y is the nontrivial character of F^\x/N_{Y/F}Y^x. The representation π_Y is normalizedly induced from \pmatrix m_3&\ast 0&m_1\endpmatrix \mapstoχ_Y(m_1), m_i in GL(i,F), on the maximal parabolic subgroup of type (3,1). We show that the twisted character χ_{π_Y} of π_Y is an unstable function: its value at a twisted regular elliptic conjugacy class with n…
Let $F$ be a non-Archimedean local field with the residual characteristic $p$. We construct a "good" number of smooth irreducible $\bar{\mathbf{F}}_p$-representations of $GL_2(F)$, which are supersingular in the sense of Barthel and Livné. If $F=\mathbf{Q}_p$ then results of Breuil imply that our construction gives all the supersingular representations up to the twist by an unramified quasi-character. We conjecture this is true for arbitrary $F$.
Let $E/F$ be a quadratic extension of non archimedean local fields of odd residual characteristic. We prove a conjecture of Prasad and Takloo-Bighash, in the case of cuspidal representations of depth zero of $\mathrm{GL}(2m,F)$. This conjecture characterizes distinction for the pair $(\mathrm{GL}(2m,F),\mathrm{GL}(m,E))$ with respect to a character $μ\circ \mathrm{det}$ of $\mathrm{GL}(m,E)$, in terms of certain conditions on Langlands paremeters, including an epsilon value. We also compute the multiplicity of the involved equivariant linear forms when $E/F$ is unramified, and also when $μ$ is…
In this work we develop an integral representation for the partial $L$-function of a pair $π\timesτ$ of genuine irreducible cuspidal automorphic representations, $π$ of the $m$-fold covering of Matsumoto of the symplectic group $Sp_{2n}$, and $τ$ of a certain covering group of $GL_k$, with arbitrary $m$, $n$ and $k$. Our construction is based on the recent extension by Cai, Friedberg, Ginzburg and the author, of the classical doubling method of Piatetski-Shapiro and Rallis, from rank-$1$ twists to arbitrary rank twists. We prove a basic global identity for the integral and compute the local in…
Let $\mathbf{F}$ be a number field and $\mathfrak{q},\mathfrak{l}$ two coprime integral ideals with $\mathfrak{q}$ squarefree and $π_1,π_2$ two fixed unitary automorphic representations of $\mathrm{PGL}_2(\mathbb{A}_{\mathbf{F}})$ unramified at all finite places. In this paper, we use regularized integrals to obtain a formula that links the first moment of $L(π\otimesπ_1\otimesπ_2,\tfrac{1}{2})$ twisted by the Hecke eigenvalues $λ_π(\mathfrak{l})$, where $π$ runs through unitary automorphic representations of $\mathrm{PGL}_2(\mathbb{A}_{\mathbf{F}})$ with conductor dividing $\mathfrak{q}$, wit…
Let $G_{n}=\operatorname{GL}_{n}(F)$, where $F$ is a non-archimedean local field with residue characteristic $p$ and where $n=2k$ is even. In this article, we investigate a question occurring in the decomposition of the category of $\ell$-modular smooth representations of $G_n$ into Bernstein blocks (where $\ell\neq p$). The easiest block not investigated in \cite{guiraud} is the one defined by the standard parabolic subgroup with Levi factor $M=\GL_k(F) \times \GL_k(F)$ and by an $M$-representation of the form $π_0 \boxtimes π_0$ with $π_0$ a supercuspidal $\GL_k(F)$-representation. This bloc…
In this paper we construct explicitly a square integrable residual automorphic representation of the special orthogonal group $SO_{2n}$, through Eisenstein series. We show that this representation comes from an elliptic Arthur parameter $ψ$ and appears in the space $L^2(SO_{2n}(\mathbb{Q})\backslash SO_{2n}(\mathbb{A}_{\mathbb{Q}}))$ with multiplicity one. Next, we consider parameters whose Hecke matrices, at the unramified places, have eigenvalues bigger (in absolute value), than those of the parameter constructed before. The main result is, that these parameters cannot be cuspidal. We establ…
Iwasawa theory of modular forms over anticyclotomic $\mathbb{Z}_p$-extensions of imaginary quadratic fields has been studied by several authors, starting from the works of Bertolini-Darmon and Iovita-Spiess, under the crucial assumption that the prime $p$ is unramified in $K$. We start in this article the systematic study of anticyclotomic $p$-adic $L$-functions when $p$ is ramified in $K$. In particular, when $f$ is a weight $2$ modular form attached to an elliptic curve $E/\mathbb{Q}$ having multiplicative reduction at $p$, and $p$ is ramified in $K$, we show an analogue of the exceptional z…
Let S be a finite set of primes, p in S, and Q_S a maximal algebraic extension of Q unramified outside S and infinity. Assume that |S|>=2. We show that the natural maps Gal(Q_p^bar/Q_p) --> Gal(Q_S/Q) are injective. Much of the paper is devoted to the problem of constructing selfdual automorphic cuspidal representations of GL(2n,A_Q) with prescribed properties at all places, that we study via the twisted trace formula of J. Arthur. The techniques we develop shed also some lights on the orthogonal/symplectic alternative for selfdual representations of GL(2n).
Let $K$ be a field, let $G$ be a finite group, and let $\bar X\rightarrow \bar Y$ be a $G$-Galois branched cover of varieties over $K^{sep}$. Given a mere cover model $X\rightarrow Y$ of this cover over $K$, in Part I of this paper I observe that there is a unique minimal field $E$ over which $X\rightarrow Y$ becomes Galois, and I prove that $E/K$ is Galois with group a subgroup of $Aut(G)$. In Part II of this paper, by making the additional assumption that $K$ is a field of definition (i.e., that there exists \it some \rm Galois model over $K$), I am able to give an explicit description of th…
