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 $(2,2)$.

In this paper, we study the universal lifting spaces of local Galois representations valued in arbitrary reductive group schemes when $\ell \neq p$. In particular, under certain technical conditions applicable to any root datum, we construct a canonical smooth component in such spaces, generalizing the minimally ramified deformation condition previously studied for classical groups. Our methods involve extending the notion of isotypic decomposition for a $\operatorname {\mathrm {GL}}_n$-valued representation to general reductive group schemes. To deal with certain scheme-theoretic issues coming from this notion, we are led to a detailed study of certain families of disconnected reductive groups, which we call weakly reductive group schemes. Our work can be used to produce geometric lifts for global Galois representations, and we illustrate this for $\mathrm {G}_2$-valued representations.

Let G be a split connected reductive group defined over $\mathbb {Z}$. Let F and $F'$ be two non-Archimedean m-close local fields, where m is a positive integer. D. Kazhdan gave an isomorphism between the Hecke algebras $\mathrm {Kaz}_m^F :\mathcal {H}\big (G(F),K_F\big ) \rightarrow \mathcal {H}\big (G(F'),K_{F'}\big )$, where $K_F$ and $K_{F'}$ are the mth usual congruence subgroups of $G(F)$ and $G(F')$, respectively. On the other hand, if $\sigma $ is an automorphism of G of prime order l, then we have Brauer homomorphism $\mathrm {Br}:\mathcal {H}(G(F),U(F))\rightarrow \mathcal {H}(G^\sigma (F),U^\sigma (F))$, where $U(F)$ and $U^\sigma (F)$ are compact open subgroups of $G(F)$ and $G^\sigma (F),$ respectively. In this article, we study the compatibility between these two maps in the local base change setting. Further, an application of this compatibility is given in the context of linkage – which is the representation theoretic version of Brauer homomorphism.

We prove the existence of a vector-valued cusp form for the full modular group for which the nth derivative of its L-function does not vanish under certain conditions. As an application, we generalize our result to Kohnen’s plus space and prove an analogous result for Jacobi forms.

Let f and g be two distinct normalized primitive holomorphic cusp forms of even integral weight $k_{1}$ and $k_{2}$ for the full modular group $SL(2,\mathbb {Z})$, respectively. Suppose that $\lambda _{f\times f\times f}(n)$ and $\lambda _{g\times g\times g}(n)$ are the n-th Dirichlet coefficient of the triple product L-functions $L(s,f\times f\times f)$ and $L(s,g\times g\times g)$. In this paper, we consider the sign changes of the sequence $\{\lambda _{f\times f\times f}(n)\}_{n\geq 1}$ and $\{\lambda _{f\times f\times f}(n)\lambda _{g\times g\times g}(n)\}_{n\geq 1}$ in short intervals and establish quantitative results for the number of sign changes for $n \leq x$, which improve the previous results.

We prove a comparison theorem between Greenberg–Benois $\mathcal {L}$-invariants and Fontaine–Mazur $\mathcal {L}$-invariants. Such a comparison theorem supplies an affirmative answer to a speculation of Besser–de Shalit.

Let $F$ be a totally real field in which $p$ is unramified and let $B$ be a quaternion algebra over $F$ which splits at at most one infinite place. Let $\overline {r}:\operatorname {{\mathrm {Gal}}}(\overline {F}/F)\rightarrow \mathrm {GL}_2(\overline {\mathbb {F}}_p)$ be a modular Galois representation which satisfies the Taylor–Wiles hypotheses. Assume that for some fixed place $v|p$, $B$ ramifies at $v$ and $F_v$ is isomorphic to $\mathbb {Q}_p$ and $\overline {r}$ is generic at $v$. We prove that the admissible smooth representations of the quaternion algebra over $\mathbb {Q}_p$ coming from mod $p$ cohomology of Shimura varieties associated to $B$ have Gelfand–Kirillov dimension $1$. As an application we prove that the degree-two Scholze's functor (which is defined by Scholze [On the $p$-adic cohomology of the Lubin–Tate tower, Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), 811–863]) vanishes on generic supersingular representations of $\mathrm {GL}_2(\mathbb {Q}_p)$. We also prove some finer structure theorems about the image of Scholze's functor in the reducible case.

We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain lower bounds for the sizes of Galois orbits of points in a generalised Hecke orbit in terms of this height function, assuming the ‘weakly adelic Mumford–Tate hypothesis’ and prove the generalised André–Pink–Zannier conjecture under this assumption, using Pila–Zannier strategy.

Compared with algebraic varieties the local monodromy of Drinfeld modules appears to be hopelessly complex: the image of the wild inertia subgroup under Tate module representations is infinite save for the case of potential good reduction. Nonetheless, we show that Tate modules of Drinfeld modules are ramified in a limited way: the image of a sufficiently deep ramification subgroup is trivial. This leads to a new invariant, the local conductor of a Drinfeld module. We establish an upper bound on the conductor in terms of the volume of the period lattice. As an intermediate step we develop a theory of normed lattices in function field arithmetic including the notion of volume. We relate normed lattices to vector bundles on projective curves. With the aid of Castelnuovo–Mumford regularity this implies a volume bound on norms of lattice generators, and the conductor inequality follows. Last but not least we describe the image of inertia for Drinfeld modules with period lattices of rank $1$. Just as in the theory of local $\ell$-adic Galois representations this image is commensurable with a commutative unipotent algebraic subgroup. However, in the case of Drinfeld modules such a subgroup can be a product of several copies of $\mathbf {G}_a$.

We investigate the discrepancy between the distributions of the random variable $\log L (\sigma , f \times f, X)$ and that of $\log L(\sigma +it, f \times f)$, that is,

$$ \begin{align*} D_{\sigma} (T) := \sup_{\mathcal{R}} |\mathbb{P}_T(\log L(\sigma+it, f \times f) \in \mathcal{R}) - \mathbb{P}(\log L(\sigma, f \times f, X) \in \mathcal{R})|, \end{align*} $$

where the supremum is taken over rectangles $\mathcal {R}$ with sides parallel to the coordinate axes. For fixed $T>3$ and $2/3 <\sigma _0 < \sigma < 1$, we prove that

$$ \begin{align*} D_{\sigma} (T) \ll \frac{1}{(\log T)^{\sigma}}. \end{align*} $$