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\begin{document}

 

 
\title[Overconvergent generalised eigenforms of weight one]
{   Overconvergent generalised eigenforms of weight one \\
and class fields of real quadratic fields
}



\author{Henri Darmon, Alan Lauder and Victor Rotger}



\begin{abstract}
 This article   examines the Fourier expansions of certain 
 non-classical
 $p$-adic modular forms 
 of weight one: the eponymous  {\em generalised eigenforms} 
 of the title, so called because they
 lie in a generalised eigenspace for the  Hecke operators.
 When this generalised eigenspace contains the theta series attached
 to a  character  of a real quadratic field $K$  in
 which the prime $p$ splits,
  the coefficients of the attendant generalised eigenform
 are expressed as $p$-adic  logarithms of algebraic numbers belonging to
  an
 idoneous
 ring class field   of $K$.
  This suggests an approach to ``explicit class field theory" 
 for real quadratic fields  which is simpler than  the one based on Stark's conjecture 
  or  its $p$-adic variants, and is perhaps closer in spirit
  to  the classical theory of singular moduli.
  \end{abstract}
 
 
 
 

\address{H. D.: McGill University, Montreal, Canada}
\email{darmon@math.mcgill.ca}
\address{A. L.: University of Oxford, U. K.}
\email{lauder@maths.ox.ac.uk}
\address{V. R.: Universitat Polit\`{e}cnica de Catalunya, Barcelona, Spain}
\email{victor.rotger@upc.edu}

%\subjclass{11G18, 14G35}


\maketitle

%\tableofcontents



\section{Introduction and statement of the main result}

Fourier coefficients of modular forms
 often describe interesting arithmetic functions. 
 Classical examples are the partition function, the divisor function, and representation numbers of  
 quadratic forms, which are related to Fourier coefficients of the Dedekind eta function, Eisenstein series, 
 and theta series, respectively.
A more modern instance is the appearance
of Frobenius traces of two-dimensional Galois representations
as Fourier coefficients of normalised eigenforms. 
Just as  germane 
to an understanding of  the present work are
the  topological and arithmetic intersection numbers of special cycles arising in the formulae
of 
Hirzebruch-Zagier, Gross-Kohnen-Zagier, and    their  vast generalisations 
growing out of the ``Kudla program". 
   

The main theorem of this paper expresses the Fourier coefficients 
of certain    $p$-adic modular forms of 
weight one as $p$-adic logarithms of algebraic numbers in ring class
 fields of real quadratic fields. It suggests an approach to ``explicit class field theory" 
 for real quadratic fields  which is simpler than  the one based on Stark's (still unproved)
  conjecture \cite{stark}
or  Gross's (more tractable) $p$-adic variant \cite{gross}.
An analogy with    the growing body of work
on  Fourier coefficients of
incoherent Eisenstein series
and
weak harmonic Maass forms suggests
that this approach is perhaps closer 
 in spirit to  the classical theory of singular moduli.
 
 
To set the stage for the main result, let   $K$ be a real quadratic
 field of discriminant $D>0$ and let $\chiK$ denote the
even quadratic Dirichlet character associated to it. Let
$$\psi: G_K:= \Gal(\bar K/K) \lra \C^\times$$
be a ray class character (of order $m$, conductor $\mathfrak{f}_\psi$ and central character
$\chi_\psi$) which is
 of {\em mixed signature}, i.e., which is even at precisely one of the infinite places of $K$ and odd at the other. Hecke's theta series $g:= \theta_\psi$ attached to $\psi$  is a holomorphic newform
 of weight one, level $N$ and nebentype character $\chi$ with Fourier coefficients in $L:=\Q(\mu_m)$,
  where
 $$N= D \cdot {\rm Norm}_{K/\Q} \mathfrak{f}_\psi, \qquad \quad
 \chi = \chiK\chi_\psi.$$
 

Fix a prime $p = \fp \fp'$ which does not divide $N$ and is split in $K/\Q$. 
The eigenform $g$ is said to be {\em regular} at $p$
if the Hecke polynomial
$$ x^2-a_p(g) x + \chi(p) = (x-\psi(\fp'))(x-\psi(\fp)) =: (x-\alpha)(x-\beta)$$
has distinct roots.
 Assume henceforth that this regularity hypothesis holds, and let 
 $$ g_\alpha(z):= g(z) - \beta g(pz), \qquad g_\beta(z) := g(z) -\alpha g(pz)$$
 be the two distinct $p$-stabilisations of $g$, which are eigenvectors for the $U_p$ operator with eigenvalues $\alpha$ and $\beta$ respectively. Note that these stabilisations are both
 ordinary, since $\alpha$ and $\beta$ are roots of unity.
 
 Let $S_k(N,\chi)$ (resp.~$S_k^{(p)}(N,\chi)$) denote the space  of classical (resp.~$p$-adic overconvergent) modular forms of weight $k$, level $N$ and character $\chi$, with coefficients in $\C_p$. 
 The Hecke algebra  $\TT$ of level $Np$ generated over $\Q$ by the operators $T_\ell$
 with $\ell\nmid Np$ and $U_q$ with $q|Np$ acts naturally on the spaces
 $S_k(Np,\chi)$ and $S_k^{(p)}(N,\chi)$. 
 The normalised eigenform $g_\alpha \in S_1(Np,\chi)$ gives rise to an algebra 
 homomorphism $\varphi_{g_\alpha}:\TT\lra L$ satisfying
 $$ \varphi_{g_\alpha}(T_\ell) = a_\ell(g) \mbox{ if } \ell\nmid Np,
 \qquad \varphi_{g_\alpha}(U_\ell) = \left\{
 \begin{array}{ll}  
 a_\ell(g)  & \mbox{ if } \ell | N;\\
\alpha & \mbox{ if } \ell = p,
 \end{array} \right.
 $$
 and $g_\alpha$ generates the
 one-dimensional eigenspace $S_1(Np,\chi)[g_\alpha]$ attached to this system of Hecke eigenvalues. 

 In~\cite{CV},
  Cho and Vatsal make the important observation  that the Coleman-Mazur 
  eigencurve is smooth but {\em not \'etale}
   over weight space at the points     corresponding to $g_\alpha$ and $g_\beta$.
(It is likely that the ramification degree of the weight map
  is always equal to two at these points under the regularity assumption: see \cite{CV}, \cite{BeDi}, 
  and  Adel Bettina's forthcoming PhD thesis 
  \cite{bettina-thesis} for  various results in this direction.)
In particular, letting   $I_{g_\alpha}$ be   the kernel of $\varphi_{g_\alpha}$, 
   %  Assume from now on that the point  on the eigencurve  associated to $g_\alpha$  is a  ramification point  of order $2$ above  weight space.
   the subdmodule of 
   $S_1^{(p)}(N,\chi)$ which is annihilated by $I_{g_\alpha}^2$, denoted
 $S_1^{(p)}(N,\chi)[[g_\alpha]]$, 
 is two-dimensional and contains non-classical forms which do not lie in the 
 image of the natural inclusion
  $$ S_1(Np,\chi)[g_\alpha] \hookrightarrow S_1^{(p)}(N,\chi)[[g_\alpha]].$$ 
 An overconvergent form in  $S_1^{(p)}(N,\chi)[[g_\alpha]]$ which is not a 
 multiple of $g_\alpha$ is called a {\em generalised eigenform} 
 attached to $g_\alpha$, and is said to be 
 {\em normalised} if its first Fourier coefficient is equal to zero.
 Such a normalised generalised eigenform, denoted 
 $$g^\flat_\alpha := \sum_{n=2}^\infty a_n(g^\flat_\alpha) q^n,$$
 is uniquely determined by $g_\alpha$   
 up to scaling, and the Hecke operators act on it by the rule
 \begin{equation}
 \label{eqn:hecke}
 T_\ell g_\alpha^\flat = a_\ell(g_\alpha) g_\alpha^\flat + a_\ell(g_\alpha^\flat) g_\alpha, \qquad
 U_q g_\alpha^\flat = a_q(g_\alpha) g_\alpha^\flat + a_q(g_\alpha^\flat) g_\alpha, 
 \end{equation}
 for all primes $\ell\nmid Np$ and all $q|Np$. 
   Theorem 1 below shows that the Fourier 
   coefficients  $a_n(g^\flat_\alpha)$  are interesting arithmetic quantities
    with a bearing on explicit class field theory for $K$. 
 
 Let $\psi'$ denote the character deduced from $\psi$ by composing it with the involution in $\Gal(K/\Q)$. The ratio
 $\psim :=\psi/\psi'$   is a totally odd ring class character of $K$. 
 Let $H$ denote the ring class field of $K$ which is fixed by the kernel of $\psim$, and set
 $G:= \Gal(H/K)$.
  
 If $\ell\nmid N$ is any rational prime which is inert in $K/\Q$, 
 the corresponding prime $\ell$ of $K$ splits completely in $H/K$, and the set 
 $\Sigma_\ell$ of primes of $H$ above $\ell$
  is   endowed with the structure of a principal
   $G$-set. 
 Given $\lambda\in\Sigma_\ell$,
  let $u(\lambda) \in \cO_H[1/\lambda]^\times\otimes \Q$ be any
  $\lambda$-unit of $H$ satisfying 
 $ \ord_\lambda(u(\lambda)) = 1.$
While $u(\lambda)$ is only defined up to units in $\cO_H^\times$,  
the element
  $$ u(\psim,\lambda) = \sum_{\sigma\in G} \psim^{\!\!-1}(\sigma) \otimes u(\lambda)^\sigma \quad\in\quad
L\otimes   \cO_H[1/\ell]^\times $$
is independent of the choice of generator $u(\lambda)$, 
 since there are no genuine units in $L\otimes \cO_H^\times$ in the 
 eigencomponents for the totally odd character $\psim$. 
 The $\ell$-unit $u(\psim,\lambda)$ does depend on the choice of 
 $\lambda\in \Sigma_\ell$.
Section \ref{sec:proof} below  uses $\psi$ to  define a function 
 $\eta: \Sigma_\ell \lra \mu_m$ for which the element
\begin{equation}
\label{eqn:def-u-l}
 u(\psim,\ell) := \eta(\lambda) \otimes u(\psim,\lambda) \quad \in\quad L\otimes \cO_H[1/\ell]^\times 
 \end{equation}
 depends only on the inert prime $\ell$ and not on 
 the choice of prime $\lambda\in \Sigma_\ell$
 above it.
  
  Fix   embeddings of $L$ and of $H$ into $\bar\Q_p$, and let
  $$\log_p: L\otimes H^\times  \lra \bar\Q_p$$
  be the  resulting $p$-adic logarithm on $H^\times$, extended 
  to $L\otimes H^\times $ by $L$-linearity.
The main result of this paper is

\medskip
\medskip\noindent
{\bf Theorem 1}. 
{\em 
The normalised   generalised eigenform $g^\flat_\alpha$  attached
  to $g_\alpha$ can be scaled in such a way that, for all primes $\ell\nmid N$,
 $$ a_\ell(g^\flat_\alpha) = \left\{ 
 \begin{array}{cl}
 0 & \mbox{ if  } \chi_K(\ell) =+1; \\
 \log_p u(\psim,\ell)    & \mbox{ if  } \chi_K(\ell) = -1.   
 \end{array}
 \right. $$
 More generally, for all $n\ge 2$ with $\gcd(n,N)=1$,
\begin{equation}
\label{eqn:kry}
a_n(g_\alpha^\flat) = \frac{1}{2} \sum_{\ell | n} \log_p \! u(\psim,\ell) \cdot  (\ord_\ell(n)+1)  \cdot a_{n/\ell}(g_\alpha),
\end{equation}
where the sum  runs over primes $\ell$ that are inert in $K$.
}

\medskip\medskip
\noindent
The following two examples illustrate Theorem 1.
 
 \begin{example}
 Let $\psi$ be the quadratic character of $K=\Q(\sqrt{21})$ of 
conductor $\mathfrak{f}_\psi:= (3,\sqrt{21})$ attached to the quadratic
extension $K(\sqrt{3+\sqrt{21}})$ of $K$, 
 so that  $m=2$ and
$L=\Q$.
 The weight one modular form $g$ attached to $\psi$  is of level
$N= 63$ and has for nebentype character  
 the odd quadratic Dirichlet character $\chi_7$ of conductor $7$.
  The ring class character $\psim$ is a genus character
   associated to $K$ and the associated genus field is 
   just the (narrow)  Hilbert class field
$H=\Q(\sqrt{-3},\sqrt{-7})$ of $K$. 

The prime $p=5$ is split in $K$, and the roots of the associated
Hecke polynomial are  $\alpha = 1$ and $\beta=-1$. Hence the generalised eigenspace  of
   $g_\alpha$
 in $S_1^{(5)}(63,\chi_7)$ contains a    normalised generalised 
   eigenform $g_\alpha^\flat$ which is unique up to scaling.
 The fast  algorithms of 
 \cite{lauder-alg} for calculating overconvergent modular forms
  were used to efficiently compute this generalised eigenform numerically with an
 accuracy of $50$ significant $5$-adic digits, producing a modular form whose first 
 non-vanishing Fourier coefficient $a_2(g_\alpha^\flat)$ is equal to $1$.
 
 For primes $\ell < 300$  
that split in $K$ (including $\ell = p$) it was observed that 
$a_\ell(g_\alpha^\flat) = 0$. 
 When $\chiK(\ell) = -1$, 
 it was observed that
\begin{equation}
\label{eqn:inert1}
a_\ell(g_\alpha^\flat) = \frac{\log_p(u(\psim,\ell))}{\log_p(u(\psim,2))},
\end{equation}
where $u(\psim,\ell)$ denotes a suitable fundamental $\ell$-unit of norm $1$ in $H/K$, for all inert $\ell\ge 2$.  (The logarithm of such a unit is unique up to
 sign.)  At the ramified primes
 we observed
\begin{equation}
\label{eqn:ramified1}
a_3(g_\alpha^\flat) =  0, \qquad 
   a_7(g_\alpha^\flat) = \frac{1}{2} \cdot \frac{\log_p(u(\psim,7))}{\log_p(u(\psim,2))}.
   \end{equation}
Here the $2$-unit is $u(\psim,2) := (-3 + \sqrt{-7})/4$.
The numerical values of  the  first few non-zero coefficients  $a_\ell(g_\alpha^\flat)$ for 
$\ell>2$ prime, and the values of $u(\psim,\ell)$ verifying
\eqref{eqn:inert1} and \eqref{eqn:ramified1}, 
are listed in the table below. 
The $p$-adic logarithms were calculated  relative to the 
$5$-adic embedding of  $H$ in which 
$\sqrt{21} \equiv -1 \bmod{5}$. 
 \medskip
 $$ 
\begin{array}{r|c|c}
\ell & a_\ell(g_\alpha^\flat)  \bmod{5^{50}}&  u(\psim, \ell)  \\
\hline 
7   &   20012844832326722144621655295530693   &                     (1 - 4 \sqrt{-3})/7 \\
11 &   9753260368539762436495550803834302     &                     (-3 - 4\sqrt{-7})/11\\
13 &   80089851328257507529397205421800237     &                    (-1 + 15\sqrt{-3})/26 \\
19 &   5387308298676565974776485314728008     &                     (11 + 21\sqrt{-3})/38 \\
23 &   7951947833969991753485495228957006       &                     (9 + 8\sqrt{-7})/23 \\
29 &   59833461154145179184050173388767665       &                    (-27-4 \sqrt{-7} )/29 \\
31 & 8202575240226165174943058721099781        &  (-13 + 35 \sqrt{-3})/62
\end{array}
$$
 \end{example}

\medskip
\medskip



\begin{example}
Let $\chi$ be a  Dirichlet character of conductor $145$ 
with order $4$ at the prime $5$ and order $2$ at the prime $29$.
The space $S_1(145,\chi)$ is one-dimensional and spanned by the modular form
\[ g = q + i q^4 + i q^5 + (-i - 1) q^7 - i q^9 + (-i + 
        1) q^{13} - q^{16} - q^{20} + \cdots .\]
It is the theta series attached to a 
quartic character of $K = \Q(\sqrt{29})$ ramified at
one of the primes above $(5)$.  Level $145$ is the smallest  where one encounters
 weight one theta series  attached to  a character of a real quadratic
 field, but not
to a character of any imaginary quadratic field. 
(There are two non-conjugate such forms, the other of which
appears in \cite[Example 4.1]{DLR}.)

The prime $p = 13$ is split in $K$  and
the roots of the Hecke polynomial for this prime are
$\alpha  = 1$ and $\beta = -i$.
We  view $g_\alpha$ as a $13$-adic modular form using the
 embedding  of  $L=\Q(i)$ into $\Q_{13}$ for which $i \equiv 5 \bmod{13}$.
The coefficients of the normalised eigenform $g_\alpha^\flat$, scaled so that
$a_2(g_\alpha)^\flat = 1$, 
 are given in the second column of the table below 
for the inert primes $\ell = 3, 11, 17$ and $19$ of $K$. 

The ring class field $H$ of conductor $5$ is a cyclic
quartic extension of $K$ given by 
\[ H = K(\sqrt{5},\delta)  \qquad \mbox{ where } \delta^2 = \frac{\sqrt{145} - 15}{32}. \]
Let $\sigma$ be the generator of  $\Gal(H/K) $ defined by
$$\sigma(\sqrt{5}) =  - \sqrt{5}, \qquad 
\sigma(\delta)  =  -\frac{1}{4}(3\sqrt{5} + \sqrt{29}) \delta.$$
We embed $H$ in the quartic unramified extension 
\[ \Q_{13^4} = \Q(\alpha) \mbox{ where } \alpha^4 + 3 \alpha^2 + 12 \alpha + 2 = 0 \]
of $\Q_{13}$, in such a way that
$$\sqrt{29} \equiv 9 ,\quad \sqrt{5} \equiv 8 \alpha^3 + 2 \alpha^2 + 7 \alpha + 10, \quad 
\delta \equiv \alpha^3 + 5\alpha^2 + 6 \alpha + 10 \pmod{13}.$$
For $\ell = 3,11,17$ and $19$,
 it was  verified 
 that
\[ a_\ell(g_\alpha^\flat) = \frac{\log_{13} (u(\psim,\ell))}{\log_{13} (u(\psim,2))} \]
to $20$-digits of $13$-adic precision,
where (denoting the group operation in $L\otimes H^\times$ additively)
\[ u(\psim,\ell) := u_\ell + i \otimes \sigma(u_\ell) - \sigma^2(u_\ell) - i \otimes\sigma^3(u_\ell), \]
for a suitable $\ell$-unit $u_\ell$ of $H$. 
The $2$-unit $u_2$ is given by
\[  u_2 := \frac{1}{2} ( -\sqrt{5} -\sqrt{29} + 6 ) \delta + \frac{1}{8} (\sqrt{29} - 7) \sqrt{5} + \frac{1}{8}(\sqrt{29} + 1),
\]
and the others are listed in the last column of the table below.

\medskip
$$ 
\begin{array}{r|c|l}
\ell & a_\ell(g_\alpha^\flat)  \bmod{13^{20}}  & \qquad \qquad \qquad \qquad u_\ell \\
\hline 
3 &   12915196799386050150007 &
 (\sqrt{5} + \sqrt{29} - 4) \delta + \frac{1}{4} (\sqrt{29} - 4) \sqrt{5} + \frac{1}{4}(2 \sqrt{29} - 13)
 \\
11 &  3524143318627577732842   &  \left( \frac{1}{4} \left( (\sqrt{29} + 1) \sqrt{5} + (-\sqrt{29} + 11) \right) \delta + \frac{1}{4} \left(\sqrt{5} - 1 \right) \right)^4
 \\
17 &  229407992393437964510 &
\left((16\sqrt{29} + 84) \sqrt{5} + (36 \sqrt{29} + 200)\right) \delta    \\
  & & \qquad  \qquad \qquad  + \frac{1}{4} (11 \sqrt{29} + 63) \sqrt{5} + \frac{1}{4}(15 \sqrt{29} + 83)
 \\
19 & 15142834827825079965585 &  
\left( \frac{1}{4} \left((3 \sqrt{29} - 13) \sqrt{5} + (-15 \sqrt{29} + 85)\right) \delta  \right. \\
& & \qquad \qquad \qquad  \left. + \frac{1}{8}(3 \sqrt{29} - 15) \sqrt{5} + \frac{1}{8}(7 \sqrt{29} - 35) \right)^2
\end{array}
$$
\end{example}
 
 
 

  
  \medskip
\medskip
\noindent
\begin{remark}
Theorem 1 was  inspired by the  work of 
Bellaiche and Dimitrov \cite{BeDi}  on the geometry of the
 Coleman-Mazur eigencurve at 
classical weight one points. 
 Theorem 1.1. of  loc.cit.
 asserts that 
  this eigencurve  is  \'etale over weight space 
  at any classical weight one point for which $p$ is regular, 
 {\em unless} $g$ is a theta series of a real quadratic field in which $p$ splits. 
 This explains why  Theorem 1 above focusses on this setting, 
 the only ``regular at $p$" scenario where a non-trivial generalised eigenform
 co-exists with its classical weight one counterpart.
  \end{remark}
  
  \begin{remark}
 Every (totally odd) ring class character of $K$ can be written as $\psi'/\psi$ for a suitable
 ray class character $\psi$ (of mixed signature). (Cf.~ for example Lemma 6.7 of \cite{DR}.)
 Hence all ring class fields of $K$ can be generated by  exponentials of the Fourier coefficients of 
 $g_\alpha^\flat$ for  suitable 
  real dihedral newforms $g$ of weight one attached to $K$, in much the same way that the ring class fields
  of a quadratic imaginary  $K$ can be generated by values of the modular function $j(z)$ at  arguments
   $z\in K$.
  The existence of the generalised eigenforms $g_\alpha^\flat$ of Theorem 1  can therefore be envisioned as 
  an eventual pathway to ``explicit class field theory" for real quadratic fields.
  \end{remark}



\begin{remark}
Comparing \eqref{eqn:kry}
 with the formula for the coefficient denoted $a_n(\phi)$ in 
 Thm.\! \!\!\!1 of \cite{KRY} reveals a strong analogy
 between  
Theorem 1   and  a fundamental result
 of Kudla, Rapoport, and Yang  on
  the Fourier coefficients of central derivatives of incoherent 
 Eisenstein series of weight one.
  When Eisenstein series are replaced by a weight one
   cuspidal newform $g$, 
 the notion of {\em mock modular forms}
 provides  
a fruitful substitute for
Kudla's  incoherent Eisenstein series. 
The  mock modular form attached to $g$,
denoted $g^\sharp$, is   the
 holomorphic part  of a 
{\em  Harmonic weak Maass form} mapping to $g$
 under an appropriate differential operator.
Recent work of   Bill  Duke,  
 Stephan Ehlen,   Yingkun Li,   and  Maryna Viazovska 
relates the  Fourier coefficients of $g^\sharp$ 
to the complex logarithms of  
algebraic numbers belonging to the field cut out by the adjoint $\Ad_g$
 of the two-dimensional 
Artin representation attached to $g$.
The  articles
\cite{Duke-Li}, \cite{ehlen}, \cite{ehlen-thesis}, and \cite{viazovska}
focus largely on the case where 
 $g$ is a theta series of an imaginary quadratic field. The 
 algebraic numbers whose logarithms arise in the Fourier expansion of 
 $g^\sharp$ then belong to 
   abelian extensions of imaginary   rather
 than    real quadratic fields,  and the proofs in
 loc.cit.~rely crucially on   the theory of complex multiplication.
  Such a theory is unavailable for real quadratic fields, and 
  the  techniques exploited in the present work,
 based  on   
  deformations of Galois representations, 
    are thus
  quite different, substantially simpler,   and fundamentally $p$-adic in nature.
  \end{remark}
  
  \begin{remark}
%  Duke and Li  do hint  at an archimedean conjecture which would apply   to arbitrary classical weight one newforms. 
The  concluding section of \cite{Duke-Li} 
makes an experimental study of
  $g^\sharp$ when $g$ is
  an octahedral newform of level $283$, 
in which the Stark conjecture for the Artin $L$-function attached to $\Ad_g$
(as described, for instance, in \S6 of \cite{stark})
 plays an essential role. 
 The case where  $g$ is the
theta series of a character   of a real quadratic field is treated  extensively
in \cite{Li}, where it is explained that   the Fourier 
coefficients of  $g^\sharp$ are expected to be logarithms of algebraic
numbers belonging to $K$ rather that to some non-trivial ring class field of $K$.
This (disappointing, at least for explicit
class field theory) feature of the archimedean context can be traced to the fact that the $L$-function of
the representation induced from $\psim$ does not vanish at $s=0$, and that 
there are no non-trivial Stark
units in the $\psim$-isotypic part of the unit group of $H$. As pointed out in 
\cite{CV} and \cite{BeDi}, it is precisely this phenomenon which leads to the
 existence of the  overconvergent generalised eigenform $g_\alpha^\flat$ 
of Theorem 1.
 \end{remark}
 
 \begin{remark}
 There are other instances where the properties of  weak harmonic Maass forms
 resonate with those of
overconvergent modular forms like $g_\alpha^\flat$. For example,
 Bruinier and Ono  study  the Fourier coefficients of the holomorphic part of
 the weak harmonic Maass form  of weight $1/2$ attached to a classical
 modular form $g$ of weight $3/2$ whose Shimura lift has rational coefficients
 and hence corresponds to an elliptic curve over $\Q$.
 The  main result of \cite{BO} relates these Fourier  coefficients
 to Heegner points on  the elliptic curve  defined over a varying collection of 
 quadratic fields. The  $p$-adic logarithms of the same Heegner points are realised in
 \cite{DT} as the Fourier coefficients of a ``modular form of weight $3/2+\varepsilon$"
arising as an infinitesimal $p$-adic deformation of $g$ over weight space.
  It would be interesting
to flesh out the rather tantalising analogy between
weak harmonic Maass forms and 
$p$-adic deformations of classical eigenforms. To what extent can the latter be
envisaged as non-archimedean counterparts of the former?
\end{remark}

 
 \begin{remark}
 The $\ell$-units $u(\psim,\ell)$   
 are precisely  the Gross-Stark units studied in \cite{DD}, 
 which proposes a {\em conjectural} analytic formula for their $\ell$-adic logarithms 
 refining Gross's $\ell$-adic analogue of the Stark conjectures.
 Theorem 1 above holds unconditionally and concerns the $p$-adic logarithms of the same $\ell$-units, 
 for primes $p\ne \ell$. It therefore bears no direct connection with 
 the Gross-Stark conjecture, even though its  proof, like that 
of the Gross-Stark
 conjecture given in  
 \cite{DDP} and \cite{ventullo}, relies crucially on the deformation theory
 of $p$-adic Galois representations.  
 \end{remark}
 
   
 
 
 \medskip\medskip
We close the introduction with the following corollary of Theorem 1, which
 shows that a naive version of the $q$-expansion 
principle fails for the generalised eigenspace $S_1^{(p)}(N,\chi)[[g_\alpha]]$.

\medskip\medskip
\noindent
{\bf Corollary 2}. 
{\em 
Let $S \subset S_1^{(p)}(N,\chi)[[g_\alpha]]$ be a $\bar\Q$-vector space which is stable under 
all the Hecke operators. Then either $S$ is contained in $S_1(Np,\chi)[g_\alpha]$,  or
it is infinite-dimensional over $\bar\Q$.}

\medskip
\noindent
{\em Proof}.  
If $S$ is not contained in $S_1(Np,\chi)[g_\alpha]$, it contains a 
non-zero (not necessarily normalised) generalised eigenform $h$ of the form 
$g_\alpha^\flat + \lambda g_\alpha$, where $g_\alpha^\flat$ is a normalised 
generalised eigenform and  $\lambda\in \C_p$.
  By \eqref{eqn:hecke} combined with the stability of $S$ under the Hecke operators,
  $$ (T_\ell -a_\ell(g_\alpha)) h = a_\ell(g_\alpha^\flat) g_\alpha$$
  also belongs to $S$, for all primes $\ell\nmid Np$.
    Theorem 1 implies that  the forms
  $ \Omega \log_p(u(\psim,\ell)) g_\alpha$ (for a suitable $\Omega\ne 0$)
  belong to $S$, for all 
  $\ell\nmid Np$ which are inert in $K$. The corollary follows from the linear independence over
  $\bar\Q$ of the $p$-adic logarithms of algebraic numbers.
  



\medskip\noindent
{\small
 \thanks{{\bf Acknowledgements.} The third author was supported by Grant MTM2012-34611.
}}

\medskip
\medskip
  
\section{Proof of Theorem 1}
\label{sec:proof}
The theta series  $g = \theta_\psi$   corresponds to an
 odd, irreducible, two-dimensional Artin representation  
$$ \varrho: G_\Q \lra \GL_2(L),$$
 obtained by inducing $\psi$ from $G_K$ to $G_\Q$.
The two-dimensional $L$-vector space    underlying $\varrho$ 
decomposes as a direct sum of  one-dimensional representations $\psi$ and $\psi'$
when restricted to $G_K$.
Fix an element $\tau_0$ in the complement $G_\Q\setminus G_K$ of $G_K$ in 
$G_\Q$, and 
let  $e_1$ and $e_2$  be eigenvectors for the $G_K$-action attached to 
$\psi$ and $\psi'$ respectively, chosen so that
$e_1 = \varrho(\tau_0) e_2$.  
Relative to this basis,
\begin{equation}
\label{eqn:rho}
\varrho_{|G_K} = \mat{\psi}{0}{0}{\psi'}, 
 \qquad \qquad \varrho_{_{|G_\Q\setminus G_K}} = \mat{0}{\eta'}{\eta}{0}
 \end{equation}
 where $\eta$ and $\eta'$ are $L$-valued functions on $G_\Q\setminus G_K$ 
 given by the rule $\eta(\tau):=\psi(\tau_0 \tau)$ and $\eta'(\tau):=\psi(\tau \tau_0^{-1})$. 
 
 
  

 Let $L_p$ denote a $p$-adic completion of $L$ and let
 $L_p[\varepsilon]$ 
  denote the ring of dual numbers, for which $\varepsilon^2=0$. 
  The theorems of Cho-Vatsal \cite{CV} and Bellaiche-Dimitrov \cite{BeDi}
  show that the tangent space $H^1_{\ord}(\Q,\Ad^0(\varrho))$ of the universal ordinary
  deformation space attached to $\varrho$ with constant determinant, denoted as $t_{\mathcal D'}$ in 
  \cite[Definition 2.1]{BeDi}, is one-dimensional over $L_p$. This means that there is a unique (up to 
  conjugation, and replacing $\varepsilon$ by a non-zero multiple) ordinary lift 
  $$ \tilde\varrho: G_\Q \lra \GL_2(L_p[\varepsilon])$$
of $\varrho$ satisfying 
\begin{equation}\label{ctt}
\det(\tilde\varrho) = \det(\varrho).
\end{equation}
     We begin by observing that   $\tilde\varrho$ 
can be written in  the form
\begin{equation}
\label{eqn:rho-tilde}
\tilde\varrho_{|G_K} = \mat{\psi}{ \psi'\kappa' \cdot \varepsilon}{\psi\kappa \cdot \varepsilon}{\psi'}, 
 \qquad \qquad \tilde\varrho_{_{|G_\Q\setminus G_K}}= \mat{ d_1 \cdot \varepsilon}{\eta'}{\eta}{d_2 \cdot \varepsilon}.
 \end{equation}
The fact that the diagonal entries of $\tilde\varrho_{|G_K}$
 remain ``constant", i.e., belong to $L$ and 
 are equal to those of $\varrho$, follows from \eqref{ctt} and the fact that this is true of the 
 lower right-hand matrix entry, which is unramified at $p$ by the ordinarity of $\tilde\varrho$, 
since there are no non-trivial  homomorphisms from $G_K$ to $L_p$ that are unramified at a prime above $p$.
% \item[b)]
 %the determinant of $\tilde\varrho$ takes values in 
%$L^\times$.
%\end{itemize}
It likewise ensures that the anti-diagonal entries of $\tilde\varrho_{_{|G_\Q\setminus G_K}}$ are equal to those
of $\varrho_{_{|G_\Q\setminus G_K}}$, i.e., are described by the functions $\eta$ and $\eta'$.
\begin{lemma}
The functions $\kappa$ and $\kappa'$ belong to
$H^1(K,L_p(\psim^{\!\!\!-1})) $ and to $H^1(K,L_p(\psim))$
respectively. Their restrictions to $G_H$ are related by the rule
\begin{equation}
\label{eqn:kappa-kappa}
 \kappa'(\sigma) = \frac{\eta'(\tau)}{\eta(\tau)} \kappa(\tau\sigma\tau^{-1}),
 \end{equation}
for any $\tau\in G_\Q\setminus G_K$. 
The class $\kappa$ is unramified at $\mathfrak{p}$, and the class $\kappa'$ is unramified at 
$\mathfrak{p}'$. 
\end{lemma}
\begin{proof}
The first assertion is standard, and follows directly from the fact that $\varrho$ is a homomorphism on $G_K$. The second follows  by a similar argument,
 from a direct calculation of 
the anti-diagonal entries of 
$\tilde\varrho(\tau\sigma\tau^{-1}) = \tilde\varrho(\tau) \tilde\varrho(\sigma) \tilde\varrho(\tau)^{-1}$, 
using the fact that $\psi(\sigma) = \psi'(\sigma)$ when $\sigma\in G_H$ to
simplify the calculation.
Finally, the fact that $\varrho$ is ordinary at the rational prime $p$ (relative to a $p$-adic embedding of $H$ which sends $K$ to its completion at $\mathfrak{p}$)
implies that $\kappa$ is unramified at $\mathfrak{p}$. The relation  
\eqref{eqn:kappa-kappa} between $\kappa$ and $\kappa'$ implies that 
$\kappa'$ is unramified at $\mathfrak{p}'$, since any $\tau\in G_\Q\setminus G_K$ interchanges the
primes $\fp$ and $\fp'$ of $K$ above $p$.
\end{proof}
The space $H^1_{\fp'}(K, L_p(\psim))$ of global 
classes that are unramified at $\fp'$ is 
one-dimensional over $L_p$, and restriction to $G_H$ gives an isomorphism
\begin{equation}
\label{eqn:restrict}
 H^1_{\fp'}(K,L_p(\psim)) \lra \hom_{\fp'}(G_H,L_p(\psim))^{\Gal(H/K)}.
 \end{equation}
The target of this restriction map
can be described explicitly in terms of global class field theory, which identifies
$\hom_{\fp'}(G_H,L_p)$ with the continuous homomorphisms from the group $\A_H^\times$ 
 of 
id\`eles of $H$ which are trivial on principal elements and  on $\cO_{H_v}^\times$ for all
primes $v\nmid \fp$ of $H$. 
The space $\hom_{\fp'}(G_H,L_p)$ is of dimension $t:= \frac{[H:K]}{2}$. 
To describe it more concretely, let
 $\sigma_\infty$ denote  complex conjugation in $G=\Gal(H/K)$ (which
  is well-defined, independently of the choice of a complex embedding of $H$).
 Given a choice $\sigma_1, \ldots, \sigma_t$ of coset representatives for
 $\langle \sigma_\infty \rangle $ in $G$, 
 let $\kappa_j$ denote the id\`ele class character
 whose restriction to $\cO_{H_v}^\times$ is trivial for all $v\nmid \fp$ and 
 whose restriction to  $H_{\fp}^\times := (H\otimes_K K_{\fp})^\times$ 
is equal to 
  $$ \kappa_j(a) = \log_p(a^{\sigma_j}/ a^{\sigma_j\sigma_\infty}).$$
  The elements $\kappa_1,\ldots,\kappa_t$ form  an $L_p$  basis for
  $\hom_{\fp'}(G_H,L_p)$, and the 
one-dimensional target of  \eqref{eqn:restrict} is   spanned by
the function  $$ \kappa_{\psim} := \sum_{j=1}^t
\psim^{\!\!-1}(\sigma_j) \kappa_j,$$
whose restriction to $H_{\fp}^\times$ is given by
\begin{equation}
\label{eqn:cft}
 \kappa_{\psim}(a) = \sum_{\sigma\in G} \psim^{\!\!-1}(\sigma) \cdot \log_p(a^\sigma).
 \end{equation}
 
 \begin{lemma}
\label{lemma:cft}
For 
all primes $\ell$ of $\Q$ that are inert in $K$, and 
all $\lambda \in \Sigma_\ell$,
$$ \kappa_{\psim}(\frob_{\lambda}) = \log_p(u(\psim,\lambda)).$$
\end{lemma}
\begin{proof}
By global class field theory, the value of
 $\kappa_{\psim}(\frob_{\lambda})$ is equal to the image of
$\kappa_{\psim}$ on the id\`ele which is equal to the inverse of a local  uniformiser 
of $H_{\lambda}$ at $\lambda$,  and to $1$ everywhere else. 
This id\`ele class agrees, modulo the kernel of $\kappa_{\psim}$, with
the id\`ele which is trivial at all places of $H$ except $\fp$ and equal to 
$u(\lambda)$ in $H_{\fp}^\times$. The result now follows from 
\eqref{eqn:cft}.
\end{proof}


Assume from now on that $\kappa'$ has been scaled so that it is equal to the class $\kappa_{\psim}$ of  \eqref{eqn:cft}. 
The Galois representation $\tilde\varrho$ comes from an overconvergent weight one
    eigenform
 $$\tilde g_\alpha = g_\alpha +  g_\alpha^\flat \cdot \varepsilon$$
 with coefficients in the ring $L_p[\varepsilon]$ of dual numbers.  The modular form
 $g_\alpha^\flat$ is a
 generalised eigenform in the sense of the introduction,
 and its Fourier coefficients  acquire a Galois-theoretic 
 interpretation via the identity
 \begin{equation}
 \label{eqn:galois-interpretation}
 {\rm trace}(\tilde\varrho(\sigma_\ell)) = a_\ell(g_\alpha) +   a_\ell(g_\alpha^\flat) \cdot \varepsilon,
 \end{equation}
  where $\ell\nmid N$ is any rational prime and $\sigma_\ell\in G_\Q$ is a Frobenius element 
  at $\ell$ (attached to an arbitrary embedding of $\bar\Q$ into $\bar\Q_\ell$). Note that although $\tilde\varrho$ is
  always ramified at $p$, 
  \eqref{eqn:rho-tilde} ensures that the trace of $\tilde\varrho(\sigma_p)$  
  does not depend on the choice of Frobenius element at $p$. 

 If $\ell\nmid N$ is a prime of $K$ for which $\chiK(\ell)=1$, and hence factors as
  a product of two primes $\lambda$ and $\lambda'$ of $K$, then
  the Frobenius element $\frob_\ell$ attached to $\ell$ belongs to $G_K$, and hence
  by \eqref{eqn:rho-tilde},
  $$ {\rm trace}({\tilde \varrho}(\frob_\ell)) = \psi(\lambda) + \psi'(\lambda) = a_\ell(g_\alpha).$$
  The first assertion in Theorem 1 when $\ell$ is a split prime
  follows in light of 
  \eqref{eqn:galois-interpretation}.
  
Let now $\ell\nmid N$ be a prime of  $K$ for which $\chiK(\ell)=-1$. 
Let $H_\varrho$ be the abelian extension of $K$ which is fixed by the kernel of $\varrho$. 
The function $\eta$ gives rise to a function on the primes of 
$H_\varrho$ above $\ell$
by setting $\eta(\lambda):= \eta(\sigma_\lambda)$, where $\sigma_\lambda\in \Gal(H_\varrho/\Q)$
is the $\ell$-power Frobenius automorphism attached to the prime $\lambda$.
Observe that, if $\lambda_1= \sigma\lambda_2$ for some
$\sigma\in \Gal(H_\varrho/K)$, then
\begin{equation}
\label{eqn:trans-eta}
\eta(\lambda_1) = \eta(\sigma_{\lambda_1}) = \eta(\sigma \sigma_{\lambda_2}\sigma^{-1}) 
= \psim^{\!\!\!-1}(\sigma)\eta(\lambda_2).
\end{equation}
In particular, the value  $\eta(\lambda)$ depends only on the
 restriction of the ideal $\lambda$ to the ring class
field $H\subset H_\varrho$, and therefore $\eta$ can also be viewed as a function on the sets 
$\Sigma_\ell$ of primes of $H$ that were described in the introduction. 




Fix now an embedding $\iota_\ell: \bar\Q \hookrightarrow \bar\Q_\ell$. It determines a Frobenius
element $\sigma_\ell \in G_\Q$, whose square is the Frobenius element in $G_K$ attached to the prime  $\ell$ of 
 $K$. Let $\lambda\in \Sigma_\ell$ denote the prime of $H$ above $\ell$ determined by $\iota_\ell$.
 By  \eqref{eqn:galois-interpretation} and \eqref{eqn:rho-tilde},
 \begin{equation}
 \label{eqn:trace-al}
  {\rm trace}(\tilde\varrho(\sigma_\ell)) = (d_1(\sigma_\ell) + d_2(\sigma_\ell))  \cdot \varepsilon =  a_\ell(g_\alpha^\flat) \cdot \varepsilon,
  \end{equation}
 while a direct calculation of $\tilde\varrho(\sigma_\ell^2)$ using
 \eqref{eqn:rho-tilde} shows that
 $$ \mat{\psi(\sigma_\ell^2)}{ \psi'(\sigma_\ell^2) \kappa'(\sigma_\ell^2) \cdot \varepsilon}
 {\psi(\sigma_\ell^2)\kappa(\sigma_\ell^2) \cdot \varepsilon}{\psi'(\sigma_\ell^2) }
 = \mat{\eta(\sigma_\ell)\eta'(\sigma_\ell)}  { \eta'(\sigma_\ell) (d_1(\sigma_\ell) + d_2(\sigma_\ell)) \cdot \varepsilon} 
 {  \eta'(\sigma_\ell) (d_1(\sigma_\ell)+d_2(\sigma_\ell))\cdot \varepsilon} {\eta(\sigma_\ell)\eta'(\sigma_\ell)}.$$
 Comparing the upper right hand corners in this equality of matrices
 yields
 $$ \psi'(\sigma_\ell^2) \kappa'(\sigma_\ell^2) = \eta'(\sigma_\ell) (d_1(\sigma_\ell) + d_2(\sigma_\ell)),$$
 and hence by  \eqref{eqn:trace-al},
 $$ a_\ell(g_\alpha^\flat) = \eta(\sigma_\ell) \kappa'(\sigma_\ell^2) = \eta(\lambda) \kappa_{\psim}(\frob_\lambda) = \eta(\lambda)\log_p u(\psim,\lambda) = \log_p u(\psim, \ell),$$
where the penultimate equality follows from Lemma \ref{lemma:cft},
and the last from the definition of $u(\psim,\ell)$ given in 
\eqref{eqn:def-u-l}. 
The first assertion in Theorem 1  follows.
The second assertion is a direct consequence of the multiplicativity properties of the 
Fourier coefficients of $\tilde g_\alpha$, now that the values of $a_\ell(\tilde g_\alpha)$ 
for $\ell$ prime
are known. 
 
 
 
 
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\end{thebibliography}

\end{document}



 