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We present a comprehensive study of the geometry of Hilbert $p$-adic eigenvarieties at parallel weight one intersection points of their cuspidal and Eisenstein loci. The Galois theoretic approach presents genuine difficulties due to the lack of good deformation theory for pseudo-characters irregular at $p$ and reflects the rich local geometry at such points. We believe that our geometric results lead to deeper insight into the arithmetic of Hilbert automorphic forms and we produce several applications in Iwasawa theory to support that., Comment: 42 pages

The aim of this paper is to investigate the trivial zeros of the Katz $p$-adic $L$-functions by the CM congruence. We prove the existence of trivial zeros of the Katz $p$-adic $L$-functions for general CM fields and establish a first derivative formula of the cyclotomic $p$-adic $L$-functions at trivial zeros under some Leopoldt hypothesis. The crucial ingredients in our proof are a special case of $p$-adic Kronecker limit formula for CM fields and a leading term formula of anticyclotomic $p$-adic $L$-functions at trivial zeros via the explicit congruences between CM and non-CM Hida families of Hilbert cusp forms., Comment: 55 pages. Minor typos are fixed

This article extends our study of the geometry of the $p$-adic eigencurve at a point defined by a weight $1$ cuspform $f$ irregular at $p$ and having complex multiplication, and the implications in Iwasawa and in Hida theories. The novel results include the determination of the Fourier coefficients of certain non-classical $p$-adic modular forms belonging to the generalized eigenspace of $f$, in terms of $p$-adic logarithms of algebraic numbers. We also compute the "mysterious" cross-ratios of the $p$-ordinary filtrations of the Hida families containing $f$., Comment: to appear in J. Th. Nombres Bordeaux (Iwasawa 2019 special issue)

Let $f_{\mathrm{new}}$ be a classical newform of weight $\geq 2$ and prime to $p$ level. We study the arithmetic of $f_{\mathrm{new}}$ and its unique $p$-stabilisation $f$ when $f_{\mathrm{new}}$ is $p$-irregular, that is, when its Hecke polynomial at $p$ admits a single repeated root. In particular, we study $p$-adic weight families through $f$ and its base-change to an imaginary quadratic field $F$ where $p$ splits, and prove that the respective eigencurves are both Gorenstein at $f$. We use this to construct a two-variable $p$-adic $L$-function over a Coleman family through $f$, and a three-variable $p$-adic $L$-function over the base-change of this family to $F$. We relate the two- and three-variable $p$-adic $L$-functions via $p$-adic Artin formalism. These results are used in work of Xin Wan to prove the Iwasawa Main Conjecture in this case. In an appendix, we prove results towards Hida duality for modular symbols, constructing a pairing between Hecke algebras and families of overconvergent modular symbols and proving that it is non-degenerate locally around any cusp form. This allows us to control the sizes of (classical and Bianchi) Hecke algebras in families., Comment: 28 pages. Changes for v2: Minor corrections and improvements; final version, to appear in Mathematika

The primary goal of this paper is to investigate the geometry of the $p$-adic eigencurve at a point $f$ corresponding to a weight one cuspidal theta series irregular at the prime number $p$. We show that $f$ belongs to exactly three or four irreducible components and study their intersection multiplicities. In particular, we show that the congruence ideal of a CM component has a simple zero at $f$ if and only if a certain anti-cyclotomic $\mathscr{L}$-invariant $\mathscr{L}_-(\varphi)$ does not vanish. Further, using Roy's Strong Six Exponential Theorem we show that at least one amongst $\mathscr{L}_-(\varphi)$ and $\mathscr{L}_-(\varphi^{-1})$ is non-zero. Combined with a divisibility proved by Hida and Tilouine, we deduce that the anti-cyclotomic Katz $p$-adic $L$-function of $\varphi$ has a simple (trivial) zero at $s=0$ if $\mathscr{L}_-(\varphi)$ is non-zero, which can be seen as an anti-cyclotomic analogue of a result of Ferrero and Greenberg. Finally, we propose a formula for the linear term of the two-variable Katz $p$-adic $L$-function of $\varphi$ at $s=0$ extending a conjecture of Gross., Comment: revised version to appear in Advances in Mathematics

We study the geometry of the $p$-adic Siegel eigenvariety $\mathcal{E}$ of paramodular tame level at certain Saito-Kurokawa points having a critical slope. For $k \geq 2$ let $f$ be a cuspidal new eigenform of $\mathrm{S}_{2k-2}(\Gamma_0(N))$ ordinary at a prime $p\nmid N$ with sign $\epsilon_f=-1$ and write $\alpha$ for the $p$-adic unit root of the Hecke polynomial of $f$ at $p$. Let $\pi_\alpha$ be the semi-ordinary $p$-stabilization of the Saito-Kurokawa lift of the cusp form $f$ to $\mathrm{GSp}(4)$ of weight $(k,k)$ and paramodular tame level. Under the assumption that the dimension of the Selmer group $H^1_{f,\mathrm{unr}}(\mathbb{Q},\rho_f(k-1))$ attached to $f$ is at most one and some mild assumptions on the automorphic representation attached to $f$, we show that $\mathcal{E}$ is smooth at the point corresponding to $\pi_\alpha$, and that the irreducible component of $\mathcal{E}$ specializing to $\pi_\alpha$ is not globally endoscopic. Finally we give an application to the Bloch-Kato conjecture, by proving under some mild assumptions that the smoothness failure of $\mathcal{E}$ at $\pi_\alpha$ yields that $\dim H^1_{f,\mathrm{unr}}(\mathbb{Q},\rho_f(k-1))\geq 2$., Comment: 50 pages, to appear in Annales de l'Institut Fourier

Let $F$ be a totally real number field and let $f$ be a classical cuspidal $p$-regular Hilbert modular eigenform over $F$ of parallel weight $1$. Let $x$ be the point on the $p$-adic Hilbert eigenvariety $\mathcal E$ corresponding to an ordinary $p$-stabilization of $f$. We show that if the $p$-adic Schanuel Conjecture is true, then $\mathcal E$ is smooth at $x$ if $f$ has CM. If we additionally assume that $F/\mathbb Q$ is Galois, we show that the weight map is \'etale at $x$ if $f$ has either CM or exotic projective image (which is the case for almost all cuspidal Hilbert modular eigenforms of parallel weight $1$). We prove these results by showing that the completed local ring of the eigenvariety at $x$ is isomorphic to a universal nearly ordinary Galois deformation ring., Comment: The material in the introduction and the final sections was reorganized. The sections on background material were substantially shortened

We prove that the cuspidal eigencurve $C_{\mathrm{cusp}}$ is \'etale over the weight space at any classical weight $1$ Eisenstein point $f$ and meets two Eisenstein components of the eigencurve $C$ transversally at $f$. Further, we prove that the local ring of $C$ at $f$ is Cohen--Macaulay but not Gorenstein and compute the Fourier coefficients of a basis of overconvergent weight $1$ modular forms lying in the same generalised eigenspace as $f$. In addition, we prove an $R=T$ theorem for the local ring at $f$ of the closed subspace of $C$ given by the union of $C_{\mathrm{cusp}}$ and one Eisenstein component and prove unconditionally, via a geometric construction of the residue map, that the corresponding congruence ideal is generated by the Kubota--Leopoldt $p$-adic $L$-function. Finally we obtain a new proof of the Ferrero--Greenberg Theorem and Gross' formula for the derivative of the $p$-adic $L$-function at the trivial zero., Comment: to appear in the American Journal of Mathematics

Let $p\geq 5$ be a prime number. We generalize the results of E. de Shalit about supersingular $j$-invariants in characteristic $p$. We consider supersingular elliptic curves with a basis of $2$-torsion over $\overline{\mathbf{F}}_p$, or equivalently supersingular Legendre $\lambda$-invariants. Let $F_p(X,Y) \in \mathbf{Z}[X,Y]$ be the $p$-th modular polynomial for $\lambda$-invariants. A simple generalization of Kronecker's classical congruence shows that $R(X):=\frac{F_p(X,X^{p})}{p}$ is in $\mathbf{Z}[X]$. We give a formula for $R(\lambda)$ if $\lambda$ is a supersingular. This formula is related to the Manin--Drinfeld pairing used in the $p$-adic uniformization of the modular curve $X(\Gamma_0(p)\cap \Gamma(2))$. This pairing was computed explicitly modulo principal units in a previous work of both authors. Furthermore, if $\lambda$ is supersingular and lives in $\mathbf{F}_p$, then we also express $R(\lambda)$ in terms of a CM lift (which are showed to exist) of the Legendre elliptic curve associated to $\lambda$., Comment: 18 pages

We prove a variant of Oesterl\'e's conjecture about $p$-adic periods of the modular curve $X_0(p)$, with an additional $\Gamma(2)$-structure. We use de Shalit's techniques and $p$-adic uniformization of curves whose reduction is semi-stable., Comment: 22 pages

We show that the Eigenvariety attached to Hilbert modular forms over a totally real field $F$ is smooth at the points corresponding to certain classical weight one theta series and we give a precise criterion for etaleness over the weight space at those points. In the case where the theta series has real multiplication, we construct a non-classical overconvergent generalised eigenform and compute its Fourier coefficients in terms of $p$-adic logarithms of algebraic numbers. Our approach uses deformations and pseudo-deformations of Galois representations., Comment: in French

J.Bella\"iche and M.Dimitrov have shown that the $p$-adic eigencurve is smooth but not etale over the weight space at $p$-regular theta series attached to a character of a real quadratic field $F$ in which $p$ splits. We proof in this paper the existence of an isomorphism between the subring of the completed local ring of the eigencurve at these points fixed by the Atkin-Lehner involution and an universal ring representing a pseudo-deformation problem, and one gives also a precise criterion for which the ramification index is exactly $2$. We finish this paper by proving the smoothness of the nearly ordinary and ordinary Hecke algebras for Hilbert modular forms over $F$ at the cuspidal-overconvergent Eisenstein points which are the base change lift for $\mathrm{GL}(2)_{/F}$ of these theta series.