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

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

$a_p(E)$

$E/\mathbb{Q}$

We construct explicit generating series of arithmetic extensions of Kudla’s special divisors on integral models of unitary Shimura varieties over CM fields with arbitrary split levels and prove that they are modular forms valued in the arithmetic Chow groups. This provides a partial solution to Kudla’s modularity problem. The main ingredient in our construction is S. Zhang’s theory of admissible arithmetic divisors. The main ingredient in the proof is an arithmetic mixed Siegel-Weil formula.

The cyclicity and Koblitz conjectures ask about the distribution of primes of cyclic and prime-order reduction, respectively, for elliptic curves over $\mathbb {Q}$. In 1976, Serre gave a conditional proof of the cyclicity conjecture, but the Koblitz conjecture (refined by Zywina in 2011) remains open. The conjectures are now known unconditionally “on average” due to work of Banks–Shparlinski and Balog–Cojocaru–David. Recently, there has been a growing interest in the cyclicity conjecture for primes in arithmetic progressions (AP), with relevant work by Akbal–Güloğlu and Wong. In this article, we adapt Zywina’s method to formulate the Koblitz conjecture for AP and refine a theorem of Jones to establish results on the moments of the constants in both the cyclicity and Koblitz conjectures for AP. In doing so, we uncover a somewhat counterintuitive phenomenon: On average, these two constants are oppositely biased over congruence classes. Finally, in an accompanying repository, we give Magma code for computing the constants discussed in this article.

Let p be an odd prime, and suppose that $E_1$ and $E_2$ are two elliptic curves which are congruent modulo p. Fix an Artin representation $\tau\,{:}\,G_{F}\rightarrow \mathrm{GL}_2(\mathbb{C})$ over a totally real field F, induced from a Hecke character over a CM-extension $K/F$. Assuming $E_1$ and $E_2$ are ordinary at p, we compute the variation in the $\mu$- and $\lambda$-invariants for the $\tau$-part of the Iwasawa Main Conjecture, as one switches from $E_1$ to $E_2$. Provided an Euler system exists, it will follow directly that IMC$(E_1,\tau)$ is true if and only if IMC$(E_2,\tau)$ is true.

This article presents new rationality results for the ratios of critical values of Rankin–Selberg L-functions of $\mathrm {GL}(n) \times \mathrm {GL}(n')$ over a totally imaginary field $F.$ The proof is based on a cohomological interpretation of Langlands’s contant term theorem via rank-one Eisenstein cohomology for the group $\mathrm {GL}(N)/F,$ where $N = n+n'.$ The internal structure of the totally imaginary base field has a delicate effect on the Galois equivariance properties of the critical values.

We state and prove an extension of the global Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture to the case of some Eisenstein series on unitary groups $U_n\times U_{n+1}$. Our theorems are based on a comparison of the Jacquet-Rallis trace formulas. A new point is the expression of some interesting spectral contributions in these formulas in terms of integrals of relative characters. As an application of our main theorems, we prove the global Gan-Gross-Prasad and the Ichino-Ikeda conjecture for Bessel periods of unitary groups.

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 local field of characteristic not equal to 2. In this article, we prove the local converse theorem for quasi-split $\mathrm {O}_{2n}(F)$ and $\mathrm {SO}_{2n}(F)$, via the description of the local theta correspondence between $\mathrm {O}_{2n}(F)$ and $\mathrm {Sp}_{2n}(F)$. More precisely, as a main step, we explicitly describe the precise behavior of the $\gamma $-factors under the correspondence. Furthermore, we apply our results to prove the weak rigidity theorems for irreducible generic cuspidal automorphic representations of $\mathrm {O}_{2n}(\mathbb {A})$ and $\mathrm {SO}_{2n}(\mathbb {A})$, respectively, where $\mathbb {A}$ is a ring of adele of a global number field L.

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.