A classical construction of Katz gives a purely algebraic construction of Eisenstein–Kronecker series using the Gauß–Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and $p$ -adic properties of real-analytic Eisenstein series. In the first part of this paper we provide an alternative algebraic construction of Eisenstein–Kronecker series via the Poincaré bundle. Building on this, we give in the second part a new conceptional construction of Katz’ two-variable $p$ -adic Eisenstein measure through $p$ -adic theta functions of the Poincaré bundle.

Let $d_{3}(n)$ be the divisor function of order three. Let $g$ be a Hecke–Maass form for $\unicode[STIX]{x1D6E4}$ with $\unicode[STIX]{x1D6E5}g=(1/4+t^{2})g$ . Suppose that $\unicode[STIX]{x1D706}_{g}(n)$ is the $n$ th Hecke eigenvalue of  $g$ . Using the Voronoi summation formula for $\unicode[STIX]{x1D706}_{g}(n)$ and the Kuznetsov trace formula, we estimate a shifted convolution sum of $d_{3}(n)$ and $\unicode[STIX]{x1D706}_{g}(n)$ and show that

$$\begin{eqnarray}\mathop{\sum }_{n\leq x}d_{3}(n)\unicode[STIX]{x1D706}_{g}(n-1)\ll _{t,\unicode[STIX]{x1D700}}x^{8/9+\unicode[STIX]{x1D700}}.\end{eqnarray}$$

$d_{3}$

Let $F$ be a totally real field and let $p$ be an odd prime which is totally split in $F$ . We define and study one-dimensional ‘partial’ eigenvarieties interpolating Hilbert modular forms over $F$ with weight varying only at a single place $v$ above $p$ . For these eigenvarieties, we show that methods developed by Liu, Wan and Xiao apply and deduce that, over a boundary annulus in weight space of sufficiently small radius, the partial eigenvarieties decompose as a disjoint union of components which are finite over weight space. We apply this result to prove the parity version of the Bloch–Kato conjecture for finite slope Hilbert modular forms with trivial central character (with a technical assumption if $[F:\mathbb{Q}]$ is odd), by reducing to the case of parallel weight $2$ . As another consequence of our results on partial eigenvarieties, we show, still under the assumption that $p$ is totally split in $F$ , that the ‘full’ (dimension $1+[F:\mathbb{Q}]$ ) cuspidal Hilbert modular eigenvariety has the property that many (all, if $[F:\mathbb{Q}]$ is even) irreducible components contain a classical point with noncritical slopes and parallel weight $2$ (with some character at $p$ whose conductor can be explicitly bounded), or any other algebraic weight.

In this paper we construct indefinite theta series for lattices of arbitrary signature $(p,q)$ as ‘incomplete’ theta integrals, that is, by integrating the theta forms constructed by the second author with J. Millson over certain singular $q$ -chains in the associated symmetric space $D$ . These chains typically do not descend to homology classes in arithmetic quotients of $D$ , and consequently the theta integrals do not give rise to holomorphic modular forms, but rather to the non-holomorphic completions of certain mock modular forms. In this way we provide a general geometric framework for the indefinite theta series constructed by Zwegers and more recently by Alexandrov, Banerjee, Manschot, and Pioline, Nazaroglu, and Raum. In particular, the coefficients of the mock modular forms are identified with intersection numbers.

We develop a general procedure to study the combinatorial structure of Arthur packets for $p$ -adic quasisplit $\mathit{Sp}(N)$ and $O(N)$ following the works of Mœglin. This will allow us to answer many delicate questions concerning the Arthur packets of these groups, for example the size of the packets.

We give a $q$ -analogue of the following congruence: for any odd prime $p$ ,

$$\begin{eqnarray}\mathop{\sum }_{k=0}^{(p-1)/2}(-1)^{k}(6k+1)\frac{(\frac{1}{2})_{k}^{3}}{k!^{3}8^{k}}\mathop{\sum }_{j=1}^{k}\biggl(\frac{1}{(2j-1)^{2}}-\frac{1}{16j^{2}}\biggr)\equiv 0\;(\text{mod}\;p),\end{eqnarray}$$

$q$

Given a property of representations satisfying a basic stability condition, Ramakrishna developed a variant of Mazur’s Galois deformation theory for representations with that property. We introduce an axiomatic definition of pseudorepresentations with such a property. Among other things, we show that pseudorepresentations with a property enjoy a good deformation theory, generalizing Ramakrishna’s theory to pseudorepresentations.

In this article we construct a p-adic three-dimensional eigenvariety for the group $U$ (2,1)( $E$ ), where $E$ is a quadratic imaginary field and $p$ is inert in $E$ . The eigenvariety parametrizes Hecke eigensystems on the space of overconvergent, locally analytic, cuspidal Picard modular forms of finite slope. The method generalized the one developed in Andreatta, Iovita and Stevens [ $p$ -adic families of Siegel modular cuspforms Ann. of Math. (2) 181, (2015), 623–697] by interpolating the coherent automorphic sheaves when the ordinary locus is empty. As an application of this construction, we reprove a particular case of the Bloch–Kato conjecture for some Galois characters of $E$ , extending the results of Bellaiche and Chenevier to the case of a positive sign.

In this article we obtain an explicit formula for certain Rankin–Selberg type Dirichlet series associated to certain Siegel cusp forms of half-integral weight. Here these Siegel cusp forms of half-integral weight are obtained from the composition of the Ikeda lift and the Eichler–Zagier–Ibukiyama correspondence. The integral weight version of the main theorem was obtained by Katsurada and Kawamura. The result of the integral weight case is a product of an $L$ -function and Riemann zeta functions, while the half-integral weight case is an infinite summation over negative fundamental discriminants with certain infinite products. To calculate an explicit formula for such Rankin–Selberg type Dirichlet series, we use a generalized Maass relation and adjoint maps of index-shift maps of Jacobi forms.

In this paper, we study some aspects of the local fields generated by division values of formal Drinfeld modules.

We prove that the complete $L$ -function associated to any cuspidal automorphic representation of $\operatorname{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$ has infinitely many simple zeros.

We answer a challenge posed in Booker [ $L$ -functions as distributions. Math. Ann. 363(1–2) (2015), 423–454, §1.3] by proving a version of Weil’s converse theorem [Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149–156] that assumes a functional equation for character twists but allows their root numbers to vary arbitrarily.

Let $p$ be a prime, let $K$ be a complete discrete valuation field of characteristic $0$ with a perfect residue field of characteristic $p$ , and let $G_{K}$ be the Galois group. Let $\unicode[STIX]{x1D70B}$ be a fixed uniformizer of $K$ , let $K_{\infty }$ be the extension by adjoining to $K$ a system of compatible $p^{n}$ th roots of $\unicode[STIX]{x1D70B}$ for all $n$ , and let $L$ be the Galois closure of $K_{\infty }$ . Using these field extensions, Caruso constructs the $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$ -modules, which classify $p$ -adic Galois representations of $G_{K}$ . In this paper, we study locally analytic vectors in some period rings with respect to the $p$ -adic Lie group $\operatorname{Gal}(L/K)$ , in the spirit of the work by Berger and Colmez. Using these locally analytic vectors, and using the classical overconvergent $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules, we can establish the overconvergence property of the $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$ -modules.

We study $\text{Sp}_{2n}(F)$ -distinction for representations of the quasi-split unitary group $U_{2n}(E/F)$ in $2n$ variables with respect to a quadratic extension $E/F$ of $p$ -adic fields. A conjecture of Dijols and Prasad predicts that no tempered representation is distinguished. We verify this for a large family of representations in terms of the Mœglin–Tadić classification of the discrete series. We further study distinction for some families of non-tempered representations. In particular, we exhibit $L$ -packets with no distinguished members that transfer under base change to $\text{Sp}_{2n}(E)$ -distinguished representations of $\text{GL}_{2n}(E)$ .

In this paper various analytic techniques are combined in order to study the average of a product of a Hecke $L$ -function and a symmetric square $L$ -function at the central point in the weight aspect. The evaluation of the second main term relies on the theory of Maaß forms of half-integral weight and the Rankin–Selberg method. The error terms are bounded using the Liouville–Green approximation.

We show that the Galois cohomology groups of $p$ -adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ can be computed via the generalization of Herr’s complex to multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules. Using Tate duality and a pairing for multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules we extend this to analogues of the Iwasawa cohomology. We show that all $p$ -adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ are overconvergent and, moreover, passing to overconvergent multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules is an equivalence of categories. Finally, we prove that the overconvergent Herr complex also computes the Galois cohomology groups.

We study the variation of $\unicode[STIX]{x1D707}$ -invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the $p$ -adic zeta function. This lower bound forces these $\unicode[STIX]{x1D707}$ -invariants to be unbounded along the family, and we conjecture that this lower bound is an equality. When $U_{p}-1$ generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the $p$ -adic $L$ -function is simply a power of $p$ up to a unit (i.e.  $\unicode[STIX]{x1D706}=0$ ). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.

We consider families of Siegel eigenforms of genus $2$ and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call the Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zero locus defined by the Galois level of the family admits an automorphic description: it is the locus of points that arise from overconvergent eigenforms for $\operatorname{GL}_{2}$ , via a $p$ -adic Langlands lift attached to the symmetric cube representation. Our proof goes via the comparison of the Galois level with a ‘fortuitous’ congruence ideal. Some of the $p$ -adic lifts are interpolated by a morphism of rigid analytic spaces from an eigencurve for $\operatorname{GL}_{2}$ to an eigenvariety for $\operatorname{GSp}_{4}$ , while the remainder appear as isolated points on the eigenvariety.

We discuss the generalizations of the concept of Chebyshev’s bias from two perspectives. First, we give a general framework for the study of prime number races and Chebyshev’s bias attached to general L-functions satisfying natural analytic hypotheses. This extends the cases previously considered by several authors and involving, among others, Dirichlet L-functions and Hasse–Weil L-functions of elliptic curves over Q. This also applies to new Chebyshev’s bias phenomena that were beyond the reach of the previously known cases. In addition, we weaken the required hypotheses such as GRH or linear independence properties of zeros of L-functions. In particular, we establish the existence of the logarithmic density of the set $ \{x \ge 2:\sum\nolimits_{p \le x} {\lambda _f}(p) \ge 0\}$ for coefficients (λf(p)) of general L-functions conditionally on a much weaker hypothesis than was previously known.

Motivated by Ramanujan’s continued fraction and the work of Richmond and Szekeres [‘The Taylor coefficients of certain infinite products’, Acta Sci. Math. (Szeged)40(3–4) (1978), 347–369], we investigate vanishing coefficients along arithmetic progressions in four quotients of infinite product expansions and obtain similar results. For example, $a_{1}(5n+4)=0$ , where $a_{1}(n)$ is defined by

$$\begin{eqnarray}\displaystyle {\displaystyle \frac{(q,q^{4};q^{5})_{\infty }^{3}}{(q^{2},q^{3};q^{5})_{\infty }^{2}}}=\mathop{\sum }_{n=0}^{\infty }a_{1}(n)q^{n}. & & \displaystyle \nonumber\end{eqnarray}$$