We show the Harris–Viehmann conjecture under some Hodge–Newton reducibility condition for a generalisation of the diamond of a non-basic Rapoport–Zink space at infinite level, which appears as a cover of the non-semi-stable locus in the Hecke stack. We show also that the cohomology of the non-semi-stable locus with coefficients coming from a cuspidal Langlands parameter vanishes. As an application, we show the Hecke eigensheaf property in Fargues’ conjecture for cuspidal Langlands parameters in the $ {\mathrm {GL}}_2$-case.

Let f(z) be the normalized primitive holomorphic Hecke eigenforms of even integral weight k for the full modular group $SL(2,\mathbb{Z})$ and denote $L(s,\mathrm{sym}^{2}f)$ be the symmetric square L-function attached to f(z). Suppose that $\lambda_{\mathrm{sym}^{2}f}(n)$ be the $\mathrm{Fourier}$ coefficient of $L(s,\mathrm{sym}^{2}f)$. In this paper, we investigate the sum $\sum\limits_{n\leqslant x}\lambda^{j}_{\mathrm{sym}^{2}f }(n) $ for $j\geqslant 3$ and obtain some new results which improve on previous error estimates. We also consider the sum $\sum\limits_{n\leqslant x}\lambda^{j}_{f }(n^{2})$ and get some similar results.

We study the local theta correspondence for dual pairs of the form $\mathrm {Aut}(C)\times F_{4}$ over a p-adic field, where C is a composition algebra of dimension $2$ or $4$, by restricting the minimal representation of a group of type E. We investigate this restriction through the computation of maximal parabolic Jacquet modules and the Fourier–Jacobi functor.

As a consequence of our results, we prove a multiplicity one result for the $\mathrm {Spin}(9)$-invariant linear functionals of irreducible representations of $F_{4}$ and classify the $\mathrm {Spin}(9)$-distinguished representations.

In p-adic Hodge theory and the p-adic Langlands program, Banach spaces with $\mathbf {Q}_p$-coefficients and p-adic Lie group actions are central. Studying the subrepresentation of G-locally analytic vectors, $W^{\mathrm {la}}$, is useful because $W^{\mathrm {la}}$ can be studied via the Lie algebra $\mathrm {Lie}(G)$, which simplifies the action of G. Additionally, $W^{\mathrm {la}}$ often behaves as a decompletion of W, making it closer to an algebraic or geometric object.

This article introduces a notion of locally analytic vectors for W in a mixed characteristic setting, specifically for $\mathbf {Z}_p$-Tate algebras. This generalization encompasses the classical definition and also specializes to super-Hölder vectors in characteristic p. Using binomial expansions instead of Taylor series, this new definition bridges locally analytic vectors in characteristic $0$ and characteristic p.

Our main theorem shows that under certain conditions, the map $W \mapsto W^{\mathrm {la}}$ acts as a descent, and the derived locally analytic vectors $\mathrm {R}_{\mathrm {la}}^i(W)$ vanish for $i \geq 1$. This result extends Theorem C of [Por24], providing new tools for propagating information about locally analytic vectors from characteristic $0$ to characteristic p.

We provide three applications: a new proof of Berger-Rozensztajn’s main result using characteristic $0$ methods, the introduction of an integral multivariable ring $\widetilde {\mathbf {A}}_{\mathrm {LT}}^{\dagger ,\mathrm {la}}$ in the Lubin-Tate setting, and a novel interpretation of the classical Cohen ring $\mathbf {{A}}_{\mathbf {Q}_p}$ from the theory of $(\varphi ,\Gamma )$-modules in terms of locally analytic vectors.

Rigid meromorphic cocycles are defined in the setting of orthogonal groups of arbitrary real signature and constructed in some instances via a p-adic analogue of Borcherds’ singular theta lift. The values of rigid meromorphic cocycles at special points of an associated p-adic symmetric space are then conjectured to belong to class fields of suitable global reflex fields, suggesting an eventual framework for explicit class field theory beyond the setting of CM fields explored in the treatise of Shimura and Taniyama.

Let $\Gamma$ be a Schottky subgroup of $\mathrm{SL}_2(\mathbb{Z})$ and let $X=\Gamma \backslash {\mathbb{H}}^2$ be the associated hyperbolic surface. We consider the family of Hecke congruence coverings of $X$, which we denote as usual by $ X_0(q) = \Gamma _0(q)\backslash {\mathbb{H}}^2$. Conditional on the Lindelöf Hypothesis for quadratic L-functions, we establish a uniform and explicit spectral gap for the Laplacian on $ X_0(q)$ for “almost” all prime levels $q$. Assuming the generalized Riemann hypothesis for quadratic $L$-functions, we obtain an even larger spectral gap.

We construct a mod $\ell $ congruence between a Klingen Eisenstein series (associated with a classical newform $\phi $ of weight k) and a Siegel cusp form f with irreducible Galois representation. We use this congruence to show non-vanishing of the Bloch–Kato Selmer group $H^1_f(\mathbf {Q}, \operatorname {\mathrm {ad}}^0\rho _{\phi }(2-k)\otimes \mathbf {Q}_{\ell }/\mathbf {Z}_{\ell })$ under certain assumptions and provide an example. We then prove an $R=dvr$ theorem for the Fontaine–Laffaille universal deformation ring of ${\overline {\rho }}_f$ under some assumptions, in particular, that the residual Selmer group $H^1_f(\mathbf {Q}, \operatorname {\mathrm {ad}}^0{\overline {\rho }}_{\phi }(k-2))$ is cyclic. For this, we prove a result about extensions of Fontaine–Laffaille modules. We end by formulating conditions for when $H^1_f(\mathbf {Q}, \operatorname {\mathrm {ad}}^0{\overline {\rho }}_{\phi }(k-2))$ is non-cyclic and the Eisenstein ideal is non-principal.

We give a new proof of Heath-Brown’s full asymptotic expansion for the second moment of Dirichlet L-functions and we obtain a corresponding asymptotic expansion for a twisted first moment of Hecke–Maass L-functions.

Let E be an elliptic curve over the rationals which does not have complex multiplication. Serre showed that the adelic representation attached to $E/\mathbb {Q}$ has open image, and in particular, there is a minimal natural number $C_E$ such that the mod $\ell $ representation ${\bar {\rho }}_{E,\ell }$ is surjective for any prime $\ell> C_E$. Assuming the Generalized Riemann Hypothesis, Mayle–Wang gave explicit bounds for $C_E$ which are logarithmic in the conductor of E and have explicit constants. The method is based on using effective forms of the Chebotarev Density Theorem together with the Faltings–Serre method, in particular, using the “deviation group” of the $2$-adic representations attached to two elliptic curves.

By considering quotients of the deviation group and a characterization of the images of the $2$-adic representation $\rho _{E,2}$ by Rouse and Zureick–Brown, we show in this article how to further reduce the constants in Mayle–Wang’s results. Another result of independent interest are improved effective isogeny theorems for elliptic curves over the rationals.

Let $f $ be a normalized Hecke eigenform of even weight $k \geq 2$ for $SL_2(\mathbb {Z})$. In this article, we establish an asymptotic formula for the shifted convolution sum of a general divisor function, where the sum involves the Fourier coefficients of a multi-folded L-function weighted with a kernel function.

We determine explicit generators for the ring of modular forms associated with the moduli spaces of K3 surfaces with automorphism group $(\mathbb {Z}/2\mathbb {Z})^2$ and of Picard rank 13 and higher. The K3 surfaces in question carry a canonical Jacobian elliptic fibration and the modular form generators appear as coefficients in the Weierstrass-type equations describing these fibrations.

Pursuing ideas in [6], we determine the isometry classes of unimodular lattices of rank $28$, as well as the isometry classes of unimodular lattices of rank $29$ without nonzero vectors of norm $\leq 2$. We also provide some invariant that allows to distinguish these lattices and to independently check with a computer that our lists are complete.

We extend the work of N. Zubrilina on murmuration of modular forms to the case when prime-indexed coefficients are replaced by squares of primes. Our key observation is that the shape of the murmuration density is the same.

Let $a(n)$ be the nth Dirichlet coefficient of the automorphic L-function or the Rankin–Selberg L-function. We investigate the cancellation of $a(n)$ over sequences linked to the Waring–Goldbach problem, by establishing a non-trivial bound for the additive twisted sums over primes on ${\mathrm {GL}}_m$. The bound does not depend on the generalized Ramanujan conjecture or the non-existence of Landau–Siegel zeros. Furthermore, we present an application associated with the Sato–Tate conjecture and propose a conjecture about the Goldbach conjecture on average bound.

Using an explicit Eichler–Shimura–Harder isomorphism, we establish the analog of Manin’s rationality theorem for Bianchi periods and hence special values of L-functions of Bianchi cusp forms. This gives a new short proof of a result of Hida in the case of Euclidean imaginary quadratic fields. In particular, we give an explicit proof using the space of Bianchi period polynomials constructed by Karabulut and describe the action of Hecke operators.

We formulate Guo–Jacquet type fundamental lemma conjectures and arithmetic transfer conjectures for inner forms of $GL_{2n}$. Our main results confirm these conjectures for division algebras of invariant $1/4$ and $3/4$.

For an even positive integer n, we study rank-one Eisenstein cohomology of the split orthogonal group $\mathrm {O}(2n+2)$ over a totally real number field $F.$ This is used to prove a rationality result for the ratios of successive critical values of degree-$2n$ Langlands L-functions associated to the group $\mathrm {GL}_1 \times \mathrm {O}(2n)$ over F. The case $n=2$ specializes to classical results of Shimura on the special values of Rankin–Selberg L-functions attached to a pair of Hilbert modular forms.

We construct an fpqc gerbe $\mathcal {E}_{\dot {V}}$ over a global function field F such that for a connected reductive group G over F with finite central subgroup Z, the set of $G_{\mathcal {E}_{\dot {V}}}$-torsors contains a subset $H^{1}(\mathcal {E}_{\dot {V}}, Z \to G)$ which allows one to define a global notion of (Z-)rigid inner forms. There is a localization map $H^{1}(\mathcal {E}_{\dot {V}}, Z \to G) \to H^{1}(\mathcal {E}_{v}, Z \to G)$, where the latter parametrizes local rigid inner forms (cf. [8, 6]) which allows us to organize local rigid inner forms across all places v into coherent families. Doing so enables a construction of (conjectural) global L-packets and a conjectural formula for the multiplicity of an automorphic representation $\pi $ in the discrete spectrum of G in terms of these L-packets. We also show that, for a connected reductive group G over a global function field F, the adelic transfer factor $\Delta _{\mathbb {A}}$ for the ring of adeles $\mathbb {A}$ of F serving an endoscopic datum for G decomposes as the product of the normalized local transfer factors from [6].

We give a short proof of the anticyclotomic analogue of the “strong” main conjecture of Kurihara on Fitting ideals of Selmer groups for elliptic curves with good ordinary reduction under mild hypotheses. More precisely, we completely determine the initial Fitting ideal of Selmer groups over finite subextensions of an imaginary quadratic field in its anticyclotomic $\mathbb {Z}_p$-extension in terms of Bertolini–Darmon’s theta elements.

In the early 2000s, Ramakrishna asked the question: for the elliptic curve

\[E\;:\; y^2 = x^3 - x,\]

$a_p(E)$

$E/\mathbb{Q}$