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 $F$ 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.

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 construct the $\unicode[STIX]{x1D6EC}$-adic crystalline and Dieudonné analogues of Hida’s ordinary $\unicode[STIX]{x1D6EC}$-adic étale cohomology, and employ integral $p$-adic Hodge theory to prove $\unicode[STIX]{x1D6EC}$-adic comparison isomorphisms between these cohomologies and the $\unicode[STIX]{x1D6EC}$-adic de Rham cohomology studied in Cais [The geometry of Hida families I:$\unicode[STIX]{x1D6EC}$-adic de Rham cohomology, Math. Ann. (2017), doi:10.1007/s00208-017-1608-1] as well as Hida’s $\unicode[STIX]{x1D6EC}$-adic étale cohomology. As applications of our work, we provide a ‘cohomological’ construction of the family of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules attached to Hida’s ordinary $\unicode[STIX]{x1D6EC}$-adic étale cohomology by Dee [$\unicode[STIX]{x1D6F7}$–$\unicode[STIX]{x1D6E4}$modules for families of Galois representations, J. Algebra 235 (2001), 636–664], and we give a new and purely geometric proof of Hida’s finiteness and control theorems. We also prove suitable $\unicode[STIX]{x1D6EC}$-adic duality theorems for each of the cohomologies we construct.

We prove a level raising mod $\ell =2$ theorem for elliptic curves over $\mathbb{Q}$. It generalizes theorems of Ribet and Diamond–Taylor and also explains different sign phenomena compared to odd $\ell$. We use it to study the 2-Selmer groups of modular abelian varieties with common mod 2 Galois representation. As an application, we show that the 2-Selmer rank can be arbitrary in level raising families.

Let $p>2$ be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call pseudo-Barsotti–Tate representations, over arbitrary finite extensions of $\mathbb{Q}_{p}$. As a consequence, we establish (under the usual Taylor–Wiles hypothesis) the weight part of Serre’s conjecture for $\text{GL}(2)$ over arbitrary totally real fields.

Generalizing previous results of Deligne–Serre and Taylor, Galois representations are attached to cuspidal automorphic representations of unitary groups whose Archimedean component is a holomorphic limit of discrete series. The main ingredient is a construction of congruences, using the Hasse invariant, that is independent of $q$-expansions.

Let π(f) be a nearly ordinary automorphic representation of the multiplicative group of an indefinite quaternion algebra B over a totally real field F with associated Galois representation ρf. Let K be a totally complex quadratic extension of F embedding in B. Using families of CM points on towers of Shimura curves attached to B and K, we construct an Euler system for ρf. We prove that it extends to p-adic families of Galois representations coming from Hida theory and dihedral ℤdp-extensions. When this Euler system is non-trivial, we prove divisibilities of characteristic ideals for the main conjecture in dihedral and modular Iwasawa theory.

Let χ be the primitive Dirichlet character of conductor 49 defined by χ(3)=ζ for ζ a primitive 42nd root of unity. We explicitly compute the slopes of the U7 operator acting on the space of overconvergent modular forms on X1(49) with weight k and character χ7k−6 or χ8−7k, depending on the embedding of ℚ(ζ) into ℂ7. By applying results of Coleman and of Cohen and Oesterlé, we are then able to deduce the slopes of U7 acting on all classical Hecke newforms of the same weight and character.

We study the possible weights of an irreducible two-dimensional mod p representation of which is modular in the sense that it comes from an automorphic form on a definite quaternion algebra with centre F which is ramified at all places dividing p, where F is a totally real field. In most cases we determine the precise list of possible weights; in the remaining cases we determine the possible weights up to a short and explicit list of exceptions.

We present a level-raising result for families of p-adic automorphic forms for a definite quaternion algebra D over ℚ. The main theorem is an analogue of a theorem for classical automorphic forms due to Diamond and Taylor. We show that certain families of forms old at a prime l intersect with families of l-new forms (at a non-classical point). One of the ingredients in the proof of Diamond and Taylor’s theorem (which also played a role in earlier work of Taylor) is the definition of a suitable pairing on the space of automorphic forms. In our situation one cannot define such a pairing on the infinite dimensional space of p-adic automorphic forms, so instead we introduce a space defined with respect to a dual coefficient system and work with a pairing between the usual forms and the dual space. A key ingredient is an analogue of Ihara’s lemma which shows an interesting asymmetry between the usual and the dual spaces.

Let $V$ be a complete discrete valuation ring with residue field $k$ of characteristic $p>0$ and fraction field $K$ of characteristic zero. Let ${\cal S}$ be a formal scheme over $V$ and let $\mathfrak{X}\to {\cal S}$ be a locally projective formal abelian scheme. In this paper we prove that, under suitable natural conditions on the Hasse–Witt matrix of $\mathfrak{X}\otimes_V V/\mathit{pV}$, the kernel of the Frobenius morphism on $\mathfrak{X}_k$ can be canonically lifted to a finite and flat subgroup scheme of $\mathfrak{X}$ over an admissible blow-up of ${\cal S}$, called the ‘canonical subgroup of $\mathfrak{X}$’. This is done by a careful study of torsors under group schemes of order $p$ over $\mathfrak{X}$. We also present a filtration on ${\rm H}^1(\mathfrak{X},\mu_p)$ in the spirit of the Hodge–Tate decomposition.

Here, a classical observation of Siegel is generalized by determining all the linear relations among the initial Fourier coefficients of a modular form on $\SL_2(\ZZ)$. As a consequence, spaces $M_k$ are identified, in which there are universal $p$-divisibility properties for certain $p$-power coefficients. As a corollary, let $f(z)=\sum_{n=1}^{\infty}a_f(n)q^n \in S_k\cap O_{L}[[q]]$ be a normalized Hecke eigenform (note that $q:=e^{2\pi i z}$), and let $k\equiv \delta(k)\pmod{12}$, where $\delta(k)\in \{4, 6, 8, 10, 14\}$. Reproducing earlier results of Hatada and Hida, if $p$ is a prime for which $k\equiv \delta(k)\pmod{p-1}$, and $\mathfrak{p}\subset O_L$ is a prime ideal above $p$, a proof is given to show that $a_f(p)\equiv 0\pmod{\mathfrak{p}}$.

Let $\tau$ be a point in the upper half-plane such that the elliptic curve corresponding to $\tau$ can be defined over $\mathbb{Q}$, and let f be a modular form on the full modular group with rational Fourier coefficients. By applying the Ramanujan differential operator D to f, we obtain a family of modular forms Dlf. In this paper we study the behavior of $D^l(f)(\tau)$ modulo the powers of a prime p > 3. We show that for $p \equiv 1 \bmod 3$ the quantities $D^l(f)(\tau)$, suitably normalized, satisfy Kummer-type congruences, and that for $p \equiv 2 \bmod 3$ the p-adic valuations of $D^l(f)(\tau)$ grow arbitrarily large. We prove these congruences by making a connection with a certain elliptic curve whose reduction modulo p is ordinary if $p \equiv 1 \bmod 3$ and supersingular otherwise.

It is shown that, given continuous, absolutely irreducible representations of ${\rm Gal}(\mathbb{Q}^{\rm ac}/\mathbb{Q})$ with values in ${\rm GL}_2(\mathbb{F}_2)$ and ${\rm GL}_2(\mathbb{F}_3)$, and having cyclotomic determinant, there is a weight 2 newform of some level whose mod $2$ and mod $3$ representations are equivalent to those given.