We introduce an explicit family of representations of the double affine Hecke algebra $\mathbb {H}$ acting on spaces of quasi-polynomials, defined in terms of truncated Demazure-Lusztig type operators. We show that these quasi-polynomial representations provide concrete realisations of a natural family of cyclic Y-parabolically induced $\mathbb {H}$-representations. We recover Cherednik’s well-known polynomial representation as a special case.

The quasi-polynomial representation gives rise to a family of commuting operators acting on spaces of quasi-polynomials. These generalize the Cherednik operators, which are fundamental in the study of Macdonald polynomials. We provide a detailed study of their joint eigenfunctions, which may be regarded as quasi-polynomial, multi-parametric generalisations of nonsymmetric Macdonald polynomials. We also introduce generalizations of symmetric Macdonald polynomials, which are invariant under a multi-parametric generalization of the standard Weyl group action.

We connect our results to the representation theory of metaplectic covers of reductive groups over non-archimedean local fields. We introduce root system generalizations of the metaplectic polynomials from our previous work by taking a suitable restriction and reparametrization of the quasi-polynomial generalizations of Macdonald polynomials. We show that metaplectic Iwahori-Whittaker functions can be recovered by taking the Whittaker limit of these metaplectic polynomials.

Inspired by Nakamura’s work [36] on $\epsilon $-isomorphisms for $(\varphi ,\Gamma )$-modules over (relative) Robba rings with respect to the cyclotomic theory, we formulate an analogous conjecture for L-analytic Lubin-Tate $(\varphi _L,\Gamma _L)$-modules over (relative) Robba rings for any finite extension L of $\mathbb {Q}_p.$ In contrast to Kato’s and Nakamura’s setting, our conjecture involves L-analytic cohomology instead of continuous cohomology within the generalized Herr complex. Similarly, we restrict to the identity components of $D_{cris}$ and $D_{dR},$ respectively. For rank one modules of the above type or slightly more generally for trianguline ones, we construct $\epsilon $-isomorphisms for their Lubin-Tate deformations satisfying the desired interpolation property.

Let F be a non-archimedean locally compact field of residual characteristic p, let $G=\operatorname {GL}_{r}(F)$ and let $\widetilde {G}$ be an n-fold metaplectic cover of G with $\operatorname {gcd}(n,p)=1$. We study the category $\operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$ of complex smooth representations of $\widetilde {G}$ having inertial equivalence class $\mathfrak {s}=(\widetilde {M},\mathcal {O})$, which is a block of the category $\operatorname {Rep}(\widetilde {G})$, following the ‘type theoretical’ strategy of Bushnell-Kutzko.

Precisely, first we construct a ‘maximal simple type’ $(\widetilde {J_{M}},\widetilde {\lambda }_{M})$ of $\widetilde {M}$ as an $\mathfrak {s}_{M}$-type, where $\mathfrak {s}_{M}=(\widetilde {M},\mathcal {O})$ is the related cuspidal inertial equivalence class of $\widetilde {M}$. Along the way, we prove the folklore conjecture that every cuspidal representation of $\widetilde {M}$ could be constructed explicitly by a compact induction. Secondly, we construct ‘simple types’ $(\widetilde {J},\widetilde {\lambda })$ of $\widetilde {G}$ and prove that each of them is an $\mathfrak {s}$-type of a certain block $\operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$. When $\widetilde {G}$ is either a Kazhdan-Patterson cover or Savin’s cover, the corresponding blocks turn out to be those containing discrete series representations of $\widetilde {G}$. Finally, for a simple type $(\widetilde {J},\widetilde {\lambda })$ of $\widetilde {G}$, we describe the related Hecke algebra $\mathcal {H}(\widetilde {G},\widetilde {\lambda })$, which turns out to be not far from an affine Hecke algebra of type A, and is exactly so if $\widetilde {G}$ is one of the two special covers mentioned above.

We leave the construction of a ‘semi-simple type’ related to a general block $\operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$ to a future phase of the work.

In this paper, we complete the proof of the conjecture of Gross and Zagier concerning algebraicity of higher Green functions at a single CM point on the product of modular curves. The new ingredient is an analogue of the incoherent Eisenstein series over a real quadratic field, which is constructed as the Doi-Naganuma theta lift of a deformed theta integral on hyperbolic 1-space.

Arthur packets have been defined for pure real forms of symplectic and special orthogonal groups following two different approaches. The first approach, due to Arthur, Moeglin, and Renard uses harmonic analysis. The second approach, due to Adams, Barbasch, and Vogan uses microlocal geometry. We prove that the two approaches produce essentially equivalent Arthur packets. This extends previous work of the authors and J. Adams for the quasisplit real forms.

We consider integral models of Hilbert modular varieties with Iwahori level structure at primes over p, first proving a Kodaira–Spencer isomorphism that gives a concise description of their dualizing sheaves. We then analyze fibres of the degeneracy maps to Hilbert modular varieties of level prime to p and deduce the vanishing of higher direct images of structure and dualizing sheaves, generalizing prior work with Kassaei and Sasaki (for p unramified in the totally real field F). We apply the vanishing results to prove flatness of the finite morphisms in the resulting Stein factorizations, and combine them with the Kodaira–Spencer isomorphism to simplify and generalize the construction of Hecke operators at primes over p on Hilbert modular forms (integrally and mod p).

Let $K={\mathbb {Q}}(\sqrt {-7})$ and $\mathcal {O}$ the ring of integers in $K$. The prime $2$ splits in $K$, say $2{\mathcal {O}}={\mathfrak {p}}\cdot {\mathfrak {p}}^*$. Let $A$ be an elliptic curve defined over $K$ with complex multiplication by $\mathcal {O}$. Assume that $A$ has good ordinary reduction at both $\mathfrak {p}$ and ${\mathfrak {p}}^*$. Write $K_\infty$ for the field generated by the $2^\infty$–division points of $A$ over $K$ and let ${\mathcal {G}}={\mathrm {Gal}}(K_\infty /K)$. In this paper, by adopting a congruence formula of Yager and De Shalit, we construct the two-variable $2$-adic $L$-function on $\mathcal {G}$. Then by generalizing De Shalit’s local structure theorem to the two-variable setting, we prove a two-variable elliptic analogue of Iwasawa’s theorem on cyclotomic fields. As an application, we prove that every branch of the two-variable measure has Iwasawa $\mu$ invariant zero.

We determine the asymptotic behavior of the coefficients of Hecke polynomials. In particular, this allows us to determine signs of these coefficients when the level or the weight is sufficiently large. In all but finitely many cases, this also verifies a conjecture on the nanvanishing of the coefficients of Hecke polynomials.

We prove the Ramanujan and Sato–Tate conjectures for Bianchi modular forms of weight at least $2$. More generally, we prove these conjectures for all regular algebraic cuspidal automorphic representations of $\operatorname {\mathrm {GL}}_2(\mathbf {A}_F)$ of parallel weight, where F is any CM field. We deduce these theorems from a new potential automorphy theorem for the symmetric powers of $2$-dimensional compatible systems of Galois representations of parallel weight.

Let ${ F}/{ F}_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq 2$ with Galois automorphism $\sigma $, and let R be an algebraically closed field of characteristic $\ell \notin \{0,p\}$. We reduce the classification of $\operatorname {GL}_n({ F}_0)$-distinguished cuspidal R-representations of $\operatorname {GL}_n({ F})$ to the level $0$ setting. Moreover, under a parity condition, we give necessary conditions for a $\sigma $-self-dual cuspidal R-representation to be distinguished. Finally, we classify the distinguished cuspidal ${\overline {\mathbb {F}}_{\ell }}$-representations of $\operatorname {GL}_n({ F})$ having a distinguished cuspidal lift to ${\overline {\mathbb {Q}}_\ell }$.

We study deformation theory of mod p Galois representations of p-adic fields with values in generalised tori, such as L-groups of (possibly non-split) tori. We show that the corresponding deformation rings are formally smooth over a group algebra of a finite abelian p-group. We compute their dimension and the set of irreducible components.

In this work, we develop an integral representation for the partial L-function of a pair $\pi \times \tau $ of genuine irreducible cuspidal automorphic representations, $\pi $ of the m-fold covering of Matsumoto of the symplectic group $\operatorname {\mathrm {Sp}}_{2n}$ and $\tau $ of a certain covering group of $\operatorname {\mathrm {GL}}_k$, 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 integrals with unramified data. Our global results are subject to certain conjectures, but when $k=1$ they are unconditional for all m. One possible future application of this work will be a Shimura-type lift of representations from covering groups to general linear groups. In a recent work, we used the present results in order to provide an analytic definition of local factors for representations of the m-fold covering of $\operatorname {\mathrm {Sp}}_{2n}$.

Let X be a smooth, projective and geometrically connected curve defined over a finite field ${\mathbb {F}}_q$ of characteristic p different from $2$ and $S\subseteq X$ a subset of closed points. Let $\overline {X}$ and $\overline {S}$ be their base changes to an algebraic closure of ${\mathbb {F}}_q$. We study the number of $\ell $-adic local systems $(\ell \neq p)$ in rank $2$ over $\overline {X}-\overline {S}$ with all possible prescribed tame local monodromies fixed by k-fold iterated action of Frobenius endomorphism for every $k\geq 1$. In all cases, we confirm conjectures of Deligne predicting that these numbers behave as if they were obtained from a Lefschetz fixed point formula. In fact, our counting results are expressed in terms of the numbers of some Higgs bundles.

H. H. Chan, K. S. Chua and P. Solé [‘Quadratic iterations to $\pi $ associated to elliptic functions to the cubic and septic base’, Trans. Amer. Math. Soc. 355(4) (2002), 1505–1520] found that, for each positive integer d, there are theta series $A_d, B_d$ and $C_d$ of weight one that satisfy the Pythagoras-like relationship $A_d^2=B_d^2+C_d^2$. In this article, we show that there are two collections of theta series $A_{b,d}, B_{b,d}$ and $C_{b,d}$ of weight one that satisfy $A_{b,d}^2=B_{b,d}^2+C_{b,d}^2,$ where b and d are certain integers.

In this paper, we study the twisted Ruelle zeta function associated with the geodesic flow of a compact, hyperbolic, odd-dimensional manifold X. The twisted Ruelle zeta function is associated with an acyclic representation $\chi \colon \pi _{1}(X) \rightarrow \operatorname {\mathrm {GL}}_{n}(\mathbb {C})$, which is close enough to an acyclic, unitary representation. In this case, the twisted Ruelle zeta function is regular at zero and equals the square of the refined analytic torsion, as it is introduced by Braverman and Kappeler in [6], multiplied by an exponential, which involves the eta invariant of the even part of the odd-signature operator, associated with $\chi $.

We introduce the L-series of weakly holomorphic modular forms using Laplace transforms and give their functional equations. We then determine converse theorems for vector-valued harmonic weak Maass forms, Jacobi forms, and elliptic modular forms of half-integral weight in Kohnen plus space.

In his “lost notebook,” Ramanujan used iterated derivatives of two theta functions to define sequences of q-series $\{U_{2t}(q)\}$ and $\{V_{2t}(q)\}$ that he claimed to be quasimodular. We give the first explicit proof of this claim by expressing them in terms of “partition Eisenstein series,” extensions of the classical Eisenstein series $E_{2k}(q),$ defined by

$$ \begin{align*}\lambda=(1^{m_1}, 2^{m_2},\dots, n^{m_n}) \vdash n \ \ \ \longmapsto \ \ \ E_{\lambda}(q):= E_2(q)^{m_1} E_4(q)^{m_2}\cdots E_{2n}(q)^{m_n}. \end{align*} $$

For functions $\phi : \mathcal {P}\mapsto {\mathbb C}$ on partitions, the weight $2n$ partition Eisenstein trace is

$$ \begin{align*}\operatorname{\mathrm{Tr}}_n(\phi;q):=\sum_{\lambda \vdash n} \phi(\lambda)E_{\lambda}(q). \end{align*} $$

For all t, we prove that $U_{2t}(q)=\operatorname {\mathrm {Tr}}_t(\phi _U;q)$ and $V_{2t}(q)=\operatorname {\mathrm {Tr}}_t(\phi _V;q),$ where $\phi _U$ and $\phi _V$ are natural partition weights, giving the first explicit quasimodular formulas for these series.

We prove an exact control theorem, in the sense of Hida theory, for the ordinary part of the middle degree étale cohomology of certain Hilbert modular varieties, after localizing at a suitable maximal ideal of the Hecke algebra. Our method of proof builds upon the techniques introduced by Loeffler–Rockwood–Zerbes (2023, Spherical varieties and p-adic families of cohomology classes); another important ingredient in our proof is the recent work of Caraiani–Tamiozzo (2023, Compositio Mathematica 159, 2279–2325) on the vanishing of the étale cohomology of Hilbert modular varieties with torsion coefficients outside the middle degree. This work will be used in forthcoming work of the author to show that the Asai–Flach Euler system corresponding to a quadratic Hilbert modular form varies in Hida families.

The Eichler–Selberg trace formula expresses the trace of Hecke operators on spaces of cusp forms as weighted sums of Hurwitz–Kronecker class numbers. We extend this formula to a natural class of relations for traces of singular moduli, where one views class numbers as traces of the constant function $j_0(\tau )=1$. More generally, we consider the singular moduli for the Hecke system of modular functions

For each $\nu \geq 0$ and $m\geq 1$, we obtain an Eichler–Selberg relation. For $\nu =0$ and $m\in \{1, 2\},$ these relations are Kaneko’s celebrated singular moduli formulas for the coefficients of $j(\tau ).$ For each $\nu \geq 1$ and $m\geq 1,$ we obtain a new Eichler–Selberg trace formula for the Hecke action on the space of weight $2 \nu +2$ cusp forms, where the traces of $j_m(\tau )$ singular moduli replace Hurwitz–Kronecker class numbers. These formulas involve a new term that is assembled from values of symmetrized shifted convolution L-functions.

We give sharp point-wise bounds in the weight-aspect on fourth moments of modular forms on arithmetic hyperbolic surfaces associated to Eichler orders. Thereby, we strengthen a result of Xia and extend it to co-compact lattices. We realize this fourth moment by constructing a holomorphic theta kernel on $\mathbf {G} \times \mathbf {G} \times \mathbf {SL}_{2}$, for $\mathbf {G}$ an indefinite inner form of $\mathbf {SL}_2$ over $\mathbb {Q}$, based on the Bergman kernel, and considering its $L^2$-norm in the Weil variable. The constructed theta kernel further gives rise to new elementary theta series for integral quadratic forms of signature $(2,2)$.