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\title[Tate's last decade at Harvard]{Tate's last decade at Harvard:   \\
 \vspace{0.02in}
 Stark's  conjecture, elliptic curves,  \\ and class field theory }

\author{Henri Darmon }
%\date{Harvard University, Cambridge, MA, March 2025}
\dedicatory{
To Dick Gross and John Tate, with gratitude and admiration }
\begin{document}
 
\maketitle
\tableofcontents

\section*{Introduction}
John Tate  literally ``wrote the book"  on  the  three topics evoked in the title (\cite{TateStark}, \cite{Tate-EC}, 
\cite{ArtinTate}), but this   essay 
will only  touch on a   small part of his contributions: those that were  made   in the decade   from roughly 1977 to 1987 and  concern   special values of $L$-functions and their arithmetic interpretations.

Tate's popularization of the  Stark conjectures
in the late 70's   soon led to the $p$-adic variants proposed by Dick Gross \cite{gross-stark}  and Jean-Pierre Serre 
\cite{serre-zeta}, which in turn  likely  motivated Tate's 
subsequent collaboration 
with Barry Mazur and Jeremy Teitelbaum on 
$p$-adic analogues of the Birch and Swinnerton-Dyer  conjecture \cite{MTT}. The striking richness of the $p$-adic setting   provided fertile ground for the  ``refined conjectures"  of \cite{MT-bsd},  the  theme of the  
 influential
 Mazur--Tate seminar held during the winter semester of 1986,
where  tame refinements of  the      Stark conjectures  were also formulated \cite{gross-tower}. A posteriori, these refinements   provided a suggestive  conceptual framework for the theory of Euler Systems initiated by Thaine, Rubin and Kolyvagin  in the late 1980's.

Tate's work made it apparent that the conjectures of Stark and of Birch and Swinnerton-Dyer lend themselves to  numerous refinements and variants, with far-reach\-ing implications    for the  explicit  analytic construction of class fields, global units,  and  rational points on elliptic curves.


The directions
 Tate pursued during his final decade at Harvard     were   a central part of the intellectual landscape
  when  the author arrived   as a graduate student in 1987, a year before Tate left for the University of Texas. 
  What follows is a brief  survey of Tate's  scientific influence on the author, with occasional digressions into  topics that have emerged more recently, such as the surprising connection between Stark units and the design of optimal quantum measurements in Section \ref{sec:sics}, and the recent breakthroughs of Dasgupta and Kakde on the Brumer--Stark conjectures and Gross's ``tower of fields" conjecture. These developments illustrate the continuing vitality of the
   directions that Tate pioneered. We have not strived  for a comprehensive account  of Tate's impact on number theory beyond our narrow focus, and ask for the indulgence of the reader  for 
the important developments
our
subjective  treatment   has    inevitably  downplayed or omitted.
  
  \section*{Dedication}
   I asked John Tate to be my thesis advisor right after passing the Harvard preliminary exams. He stared   at me with a puzzled look before asking ``What is an advisor?" A philosophical question that struck me at the time as  disconcertingly modest, coming  from   one of the most acclaimed  mentors  on the planet! 
  He then proceeded to  offer the best career advice I have ever received: ``Why don't you try working with Dick Gross?"  This is how I joined the privileged ranks of Tate's mathematical descendants.
 Sadly,  Dick passed away a few months after the Tate Centennial meeting. 
  His deep insights and groundbreaking results have been a constant guide and inspiration throughout my scientific life, and it is   an honor to dedicate this essay to him as well as to Tate. May their memories be a blessing.
 

 
\section{Stark's conjecture}

Tate's  interest in 
  Stark's conjecture \cite{Stark1971} dates back to around 1977, possibly  motivated   by his prior interest in modular forms of weight one whose $L$-series provided an important 
 source of concrete examples and experimental evidence for the Stark conjecture at its inception.
Tate's   influential  graduate course 
 at Orsay in 1980
\cite{TateStark} shaped  the way in which   the subject is understood today. 


The Stark conjecture can be envisaged as a far-reaching generalisation of
   the Kronecker limit formula, which gives a closed form expression for  the value at $s=1$ --- or, equivalently by the functional equation,  the first derivative at $s=0$ ---   of the partial zeta function
attached to a (non-trivial)
  ideal class character of 
  an imaginary quadratic field $K$ in terms of {\em elliptic units} arising from the theory of complex multiplication.  
  
  Given any ideal class $\fA$ of $K$, let 
\begin{equation}
\label{eqn:zeta-a}
 L(\fA,s) =  \sum_{\fa\in \fA}  N(\fa)^{-s}
\end{equation}
be the partial zeta function of $\fA$,
the sum in \eqref{eqn:zeta-a}
being taken over the integral ideals in the class $\fA$, with $N(\fa)\in {\Z}^{>0}$ denoting the norm of  such an ideal.
The first Kronecker limit formula relates the Taylor expansion of this function at $s=1$ to the  values at quadratic imaginary arguments of the Dedekind eta function 
\begin{equation}
\label{eqn:eta}
\eta(z) = q^{1/24} \prod(1-q^n), \qquad q := e^{2\pi i z},\end{equation}
 a holomorphic modular form of weight $1/2$ whose $24$-th power is the Ramanujan Delta function $\Delta(z)$ of weight $12$ and level $1$.
More precisely, it  asserts that 
\begin{equation}
\label{eqn:klf1}
 L(\fA ,s) = \frac{\pi}{s-1} +  C - \log(\sqrt{y}|\eta(\tau_{\fA})|^2) + O(s-1),
 \end{equation}
where $C $ is a constant that does not depend on $\fA$, and $\tau_{\fA}=x+iy$ is a complex number attached to $\fA$ by the requirement that  $y>0$ and that $\Z+\Z\tau_{\fA}$ is a projective  $\cO_K$-module isomorphic to a fractional ideal in the class $\fA$.   Thanks to the theory of complex multiplication,   the ratios $\Delta(\tau_{\fA})/\Delta(\tau_{\fA'})$ attached to any pair $(\fA,\fA')$ of $\cO_K$-ideals 
 are closely related to  certain distinguished algebraic units in the Hilbert class field $H$ of $K$.
 



Replacing  the functions $L(\fA,s)$    by  a  linear combination 
\begin{equation}
L(\chi,s) := \sum_{\fA\in {\rm Cl}(K)} \chi^{-1}(\fA) L(\fA,s) 
\end{equation} 
weighted by a non-trivial   character $\chi$ of the class group of $K$ with values in a field $E_\chi\subset \C$, 
and exploiting the functional equation relating the behaviour of   $L(\chi,s)$ at $s=1$ and $s=0$,
one eventually obtains
 \begin{equation}
 \label{eqn:stark1} L'(\chi,0)  =
 \log|U_\chi|, 
 \end{equation}
 where 
 \begin{equation}
 U_\chi  \in E_\chi \otimes {\cO}_H^\times
 \end{equation} 
 is an {\em elliptic unit} constructed from suitable ratios of    values of the Dedekind $\eta$-function at CM points.
 
 
 Stark's bold insight is that an identity like \eqref{eqn:stark1} should continue to hold even when a suitable counterpart of the theory of complex multiplication is lacking and
 one disposes of no {\em independent expression} for the right hand side in terms of CM values of modular functions.
 
For example, the $L$-function $L(\chi,s)$
appearing in \eqref{eqn:stark1} is also the Hecke $L$-function attached to the theta series 
\begin{equation}\theta_\chi(z) := \sum_{\fa} \chi(\fa) q^{N(\fa)}, \qquad q = e^{2\pi i z},\end{equation}
a holomorphic eigenform of weight one  on a Hecke congruence subgroup of
$\SL_2(\Z)$. 
Replacing $\theta_\chi$ by a more general eigenform $g$ of weight one in \eqref{eqn:stark1},
Stark's conjecture suggests that
\begin{equation}
\label{eqn:stark2}
 L'(g,0) \stackrel{?}{=}
\log|U_g|, \qquad \mbox{ with } U_g \in E_g \otimes \cO_g^\times, 
\end{equation}
where  $E_g$ is the field generated by the Fourier coefficients of $g$ and $\cO_g$ is the ring of integers of the field  cut out  by the odd two-dimensional Artin representation attached to $g$ by the construction of Deligne-Serre \cite{deligne-serre}.
 
  

More generally, let $V$ be any finite-dimensional complex vector space (of dimension $n \ge 1$, say) endowed with an irreducible, non-trivial continuous action of the absolute Galois group
$G_{\Q}$ of $\Q$, and let \begin{equation}\varrho: G_\Q \to  {\rm Aut}(V) \simeq \mathrm{GL}_n(\mathbb C)\end{equation} be the associated Artin representation. Since $\varrho$  is continuous, it has finite image and factors through the Galois group $\mathrm{Gal}(H/\mathbb{Q})$ of a finite extension  $H$ of $\Q$, 
and the image under $\varrho$ of the Frobenius element   attached to a choice of prime $\lambda$ of $H$ above $\ell$ is a well defined linear endomorphism, denoted  $\sigma_\ell$,  on the space $V^{I_\lambda}$ of inertia invariants of $V$.
The characteristic polynomial  of $\sigma_\ell$ has coefficients in a finite (cyclotomic) extension of $\Q$, denoted $E
\subset \C$, and
the 
 Artin $L$-function associated to $\varrho$ is defined by the 
 Euler product 
\[
L(V,s) \;=\; \prod_{\ell}  \det(1- \sigma_\ell  \cdot \ell^{-s})^{-1},
\]
which  converges for $\Re(s)>1$ and extends to a meromorphic function
on $\mathbb{C}$.  

Fix an embedding  $H\subset\C$  and let $c$ denote the associated complex conjugation in 
${\rm Gal}(H/\Q)$, for which $H^{c=1}\subset \R$.
An extension of Dirichlet's unit theorem shows that
\[
r  := \dim_E \operatorname{Hom}_{G_\mathbb{Q}}\!\bigl(V, \cO_H^\times \otimes_\mathbb{Z} E \bigr) = \dim V^{c=1}.
\]
Stark conjectured \cite{Stark1971} that
    $L(V,s)$  vanishes to order $r$ at $s=0$, and that its leading term satisfies
\begin{equation}
\label{eqn:stark3}
L^{(r)}(V,0) =R(V) \pmod{E^\times},
\end{equation}
where $R(V)$ is the {\em Stark unit regulator} whose definition depends on the choice of   bases $\{ \phi_1,\dots,\phi_r \}$ and $\{w_1,\ldots,w_r\}$ 
 of the $E$-vector spaces
\[
\operatorname{Hom}_{G_\Q}\!\bigl(V, \cO_H^\times \otimes_\Z E \bigr) \quad \mbox{ and } \quad V^{c=1}
\]
respectively. Note that 
$\phi_i(w_j)$ can be viewed as an element of $\R^\times$ thanks to the chosen complex embedding of $H$.
The regulator $R(V)$ can then be  defined as
\begin{equation}
\label{eqn:stark-reg}
R(V) := \det\left(\bigl( \log \phi_i(w_j)\bigr) _{1 \leq i,j \leq r}\right).
\end{equation}

 



\section{Explicit class field theory}

When $r=1$, the Stark regulator reduces to the logarithm of a {\em single unit} in $H$, 
which can be recovered (up to scaling) from $L(V,1)$ or from $L'(V,0)$. 
 Stark's conjecture therefore suggests analytic constructions of units in the fields  cut out by such $V$, essentially by exponentiating $L'(V,0)$. In this way, equation \eqref{eqn:stark2}
provides the  basis for an explicit  albeit conjectural construction  of the  Artin representations attached to newforms of weight one including the exceptional ones  with non-dihedral projective image, leading to an effective procedure
for calculating the  Galois representations of  Deligne and Serre \cite{deligne-serre}. This application  may have piqued Tate's interest at the outset, in light of his prior interest in weight one cusp forms and their associated  Artin representations.


As Stark himself pointed out in \cite{stark-3}, the   scenario 
where $r=1$ is directly germane to explicit class field theory as well. Namely, consider the case where
  $V$ is induced from a one-dimensional  character $\chi:G_K\lra E^\times$  of 
a fixed ground field $K$.
Then $r = 1$ precisely when 
   $\chi$  is {\em odd} at all but one distinguished archimedean place of $K$,  denoted $\infty$.  
  The field $K$ is  either totally real when $\infty$ is itself real, or $\infty$ is its unique  complex place. In the latter scenario $K$ is referred to as an
  {\em almost totally real} (ATR) extension of $\Q$.
  
  
  
    Let $\infty=\infty_1$ and $\infty_2, \ldots, \infty_t$ be
   the archimedean places of $K$ and write 
   \begin{equation}
   v_1: K \hookrightarrow \C, \qquad
   v_2,\ldots, v_t: K \hookrightarrow \R \end{equation} for the associated   embeddings of $K$.  Class field theory identifies $\chi$ with a character of the generalised narrow ideal class group associated to a modulus $\fm$ of $K$ which is odd at the places $v_2, \ldots, v_t$, i.e.,  satisfies
\begin{equation}
\label{eqn:sign-chi}
 \chi((x)) = {\rm sgn}(x):= {\rm sign}(v_2(x)\cdots v_t(x)), \qquad \mbox{ for all } 
  x\in 1+\fm\cO_K.
  \end{equation}
Assume for simplicity of the exposition that $\fm=1$ so that 
  the field which is cut out by $\chi$ is contained in 
  the Hilbert class field of $K$ (in the narrow sense). The existence of a character $\chi$ 
  satisfying \eqref{eqn:sign-chi}   implies that $H$ is complex at all the archimedean places lying above $\infty_2, \ldots, \infty_t$.
 In particular, $H$ is  totally complex (but typically not   CM)  when $K$ is ATR.
  
  The $L$-functions $L(\chi,s)$ can be expressed as linear combinations of the odd
  partial $L$-functions attached to narrow ideal classes $\fA$  of $K$, defined as
  \begin{equation} L(\fA,s) = \sum_{I = x\fA } {\rm sgn}(x) {\rm Norm}(I)^{-s},\end{equation}
  where the sum is taken over the integral ideals $I$  in the class of $\fA$  in the wide sense. 
  
 Stark's conjecture    predicts the existence of a system of units $u_\fA\in \cO_H^\times\otimes \Q$ indexed by the narrow ideal classes $\fA$ which satisfy
\begin{equation}
\label{conj:stark4}
 L'(\fA,0) \stackrel{?}{=} \log|{\tilde v}_1(u_\fA)|, \qquad \mbox{ for all } \fA \in {\rm Cl}(K), \end{equation}
 where ${\tilde v}_1:H\lra \C$ is a suitable archimedean embedding of $H$ lying above  the distinguished place $v_1$.
 These units are further expected to be conjugate to each other under the action of ${\rm Gal}(H/K)$. More precisely, letting \begin{equation} {\rm rec}: {\rm Cl}^+(K) \lra {\rm Gal}(H/K)\end{equation}
 be the reciprocity law of global class field theory, it is expected that 
 \begin{equation} {\rm rec}(\fB) (u_{\fA}) \stackrel{?}{=}
 u_{\fA\fB^{-1}}, \quad
 \mbox{ for all } \fA, \fB \in {\rm Cl}^+(K). \end{equation}
 The  complex conjugations (Frobenius elements) 
 \begin{equation}c_1, \ldots, c_t \in {\rm Gal}(H/K)\end{equation}
 attached to the archimedean places  $\infty_1,\ldots, \infty_t$ of $K$   act on the units $u_{\fA}$ according to the rules
\begin{equation}
\label{eqn:signc}
 c_j(u_\fA) = \left\{
 \begin{array}{cl}
 u_\fA & \mbox{ if } j=1;\\
 u_{\fA}^{-1} & \mbox{ if } j=  2, \ldots, t.
 \end{array} \right.
\end{equation}
 When $K$ is totally real, it follows that the $u_{\fA}$ belong to the subfield $H^{c_1=1}$ which is real at all the places   above $\infty_1$, and therefore the quantities $\tilde v_1(u_\fA)$ can in principle be recovered (up to a mere sign ambiguity) from 
  \eqref{conj:stark4}.
This equation therefore provides a compelling approach to explicit class field theory and  Hilbert's twelfth problem for totally real fields.

When $K$ is an ATR extension (which encompasses the motivating scenario 
of \eqref{eqn:klf1}
and
 \eqref{eqn:stark1} where $t=1$ and $K$ is a quadratic imaginary field)
the Frobenius  $c_1$
attached to $\infty_1$ is the identity and 
\eqref{eqn:signc} places no restrictions on the complex numbers  
$\tilde v_1(u_{\fA})$.
This leads to the following natural question:
\begin{question}
\label{question:bcg}
When $K$ is an ATR extension, is there an analytic formula for the quantities $\tilde v_1(u_\fA)$ occurring in 
\eqref{conj:stark4} and not just their absolute values?
\end{question}
A positive answer  would constitute a non-trivial refinement of the Stark conjecture, and (as will be further explained in Section \ref{sec:mt-ht}
 below)
is somewhat in the spirit of the ``refined conjectures" that later emerged from the work of Tate in collaboration  with Barry Mazur.

 Question \ref{question:bcg} appears to have been raised explicitly for the first time in
Samit Dasgupta's Harvard undergraduate senior thesis \cite{dasgupta-thesis}, 
which examines its first non-trivial instance, where $K$ is a complex cubic field. (Cf.~the final paragraph in \S 5.5 of loc.cit.)
Tian Ren and Robert Sczech later proposed a conjectural analytic expression   for the units $u_{\fA}$ in this case
\cite{ren-sczech}  (up to a small root of unity ambiguity), through 
 a careful refinement of Shintani's cone decompositions.
 
 
The article \cite{charollois-darmon} considers the case where $K$ is a quadratic ATR extension of a totally real field $F$ of degree $n$ over $\Q$, expressing the Stark units ${\tilde v}_1(u_{\fA})\in \C^\times$ in terms of special values of 
certain {\em analytic cocycles} attached to Hilbert modular Eisenstein series of parallel weight two over $F$.
More precisely,   a ``transgression" of the $n$-cocycle encoding the periods of such  Eisenstein series  is used to construct certain $(n-1)$-cocycles
\begin{equation} 
\label{eqn:kappaeis}
\kappa_{eis} \in Z^{n-1}(\Gamma, \cO_{\cH}^\times), \qquad \Gamma=\SL_2(\cO_F),
\end{equation}
where $\cO_{\cH}^\times$ is the multiplicative group of analytic functions on the Poincar\'e upper half-plane $\cH$.
The Hilbert modular group $\Gamma$ attached to $F$  is viewed as a subgroup of $\SL_2(\R)$ via the real embedding $v_1:F\lra \R$ attached to the distinguished complex place $\infty$ of $K$, and
 thereby acts on $\cH$ by M\"obius transformations.
 
  Assume for simplicity that $F$ has narrow class number one. The choice of a basis of an  ideal class $\fA$ of $K$ gives rise to a torus $T\simeq K_1^\times$   in $\SL_{2/F}$  for which
 \begin{equation} T(F)\cap \Gamma = T(\cO_F) = ({\cO_K})^\times_1  \simeq \Z^{n-1} \times (\Z/2\Z),\end{equation}
 and $T$ has a unique fixed point $\tau_{\fA}$ in $\cH$. Restriction to $T(\cO_F)$ composed with   evaluation at
 $\tau_{\fA}$ gives rise to an {\em evaluation map}
 \begin{equation} {\rm ev}_{\fA}: H^{n-1}(\Gamma, \cO_H^\times) \lra H^{n-1}(T(\cO_F), \C^\times) = H^{n-1}(\Z^{n-1},\C^\times) \simeq \C^\times.\end{equation}
The main  conjecture which is formulated and tested experimentally in
 \cite{charollois-darmon} is that, up to a possible torsion ambiguity,
\begin{equation}{\rm ev}_{\fA}(\kappa_{\rm eis}) \stackrel{?}{=} {\tilde v}_1(u_{\fA}).
\end{equation}
 

A  more recent 
work of Bergeron, Charollois and Garcia \cite{BCG}  revisits the Ren-Sczech setting where $K$ is a complex cubic field and 
offers an explicit conjectural formula for the quantities ${\tilde v}_1(u_{\fA})$ which is   closer in spirit to the cocycle-based approach of
\cite{charollois-darmon}. It involves the special values of a remarkable 
cocycle \begin{equation} \kappa_{BCG} \in Z^{1}(\Gamma, \cA^\times), \qquad \Gamma = \SL_3(\Z),\end{equation}
where $\cA^\times$ is the multiplicative group of complex-valued functions on an archimedean symmetric space attached to $\SL_3(\R)$ which are meromorphic on a suitable region.
This cocycle is  constructed from the ``elliptic Gamma function" of Felder and Varchenko
\cite{felder-varchenko} 
and is described by explicit  infinite products
reminiscent of the product expansions of the Dedekind  eta  function of
\eqref{eqn:eta}.  It can be evaluated at tori in $\SL_3$ corresponding to ideal classes in {\em arbitrary} cubic  ATR extensions of $\Q$, leading  to a tantalising analogue of the Kronecker limit formula of 
\eqref{eqn:klf1} for complex cubic fields.


Pierre Morain has   extended the construction  of Bergeron, Charollois and Garcia    by constructing  a striking hierarchy of
cocycles \cite{morain}
\begin{equation} \kappa_{n} \in
 Z^{n-2}(\Gamma, \cA^\times), \qquad \Gamma = \SL_n(\Z),\end{equation}
which can be evaluated  at  tori attached to (narrow) ideal classes in  ATR fields  of  any degree $n\ge 3$. The approach of Ren and Sczech  has also been extended  to this setting in the recent PhD thesis of Kairi Giovanna Black \cite{black-thesis}.


The breakthroughs in  \cite{BCG} and 
\cite{morain} suggest  an affirmative answer to Question \ref{question:bcg},
supplying a  framework that is  more refined than leading terms of $L$-functions and is able to detect    the arguments and not just the absolute values of the quantities  ${\tilde v}_1(u_{\fA})$ occurring in \eqref{conj:stark4}.

Even when $K$ is totally real, there are good theoretical and practical reasons, some of which will be touched on in the next section,
 to want to refine Stark's conjecture 
by  removing the sign ambiguity in the formulae for Stark units.
 Gene Kopp's   ``Shintani-Faddeev cocycle"  in $H^1(\SL_2(\Z), \cA^\times)$, where $\cA$ is the group of analytic functions on $\cH$,
 leads to an intriguing counterpart  of the Kronecker limit formula for real quadratic fields \cite{kopp-shin}.  It would be interesting to present a unified perspective encompassing the Shintani-Faddeev  cocycle along with those
 of
   \cite{charollois-darmon},
   \cite{BCG}, and 
   \cite{morain}.
  

    
  \section{Equiangular configurations of complex lines}
  \label{sec:sics}
 
  The complex embeddings
  ${\tilde v}_j$ of $H$ above the real places $v_j$ for $j=2,\ldots, t$ do not play a role in the Stark conjecture
  \eqref{conj:stark4}
 or in  Question  \ref{question:bcg}.
  Because of \eqref{eqn:signc}, the images  ${\tilde v}_j(u_{\fA})$ of the Stark units $u_{\fA}$   are complex numbers of absolute value $1$, and hence they satisfy
  $\log|{\tilde v}_j(u_{\fA})|=0$.
  The following   is superficially similar to 
 Question  \ref{question:bcg}:
\begin{question}
\label{question:sczech}
When $K$ is totally real or ATR, is there an analytic formula for the quantities $\tilde v_j(u_\fA)$ for $2 \le j \le t$ arising from the Stark units $u_{\fA}$ ?\end{question}
In contrast to 
Question \ref{question:bcg} towards which decisive progress has been made  thanks to \cite{BCG} and \cite{morain},  Question
\ref{question:sczech} remains essentially untouched. 

Its importance   is underscored by a surprising recently discovered geometric application of Stark's conjecture  to  a question motivated by the theory of quantum designs, which provides a modern illustration of the continuing impact of Tate's ideas, and which
we cannot resist describing briefly.


Let $V$ be a complex hermitian space of dimension $d<\infty$.
The {\em complex angle} between a pair $(L_1,L_2)$ of complex lines  in $V$  is the one whose cosine is 
$|\langle e_1,e_2\rangle|$,
where $e_1$ and $e_2$ are unit vectors spanning 
$L_1$ and $L_2$ respectively.
 A collection $(L_1,\ldots, L_m)$ of lines in $V$ is said to be  {\em equiangular} if the angle between $L_i$ and $L_j$ is {\em independent} of the pair $(i,j)$ with $i\ne j$. Letting $e_j$ be a unit vector in $L_j$, the orthogonal projections $\pi_i$ onto $L_i$ satisfy
 \begin{equation} 
 \pi_j \pi_i \pi_j = \alpha^2  \pi_j, \quad \mbox{ for all } i\ne j, \qquad \mbox{ where } \alpha := |\langle e_i,e_j\rangle|. \end{equation}
  These projections are  readily seen to be linearly independent 
 in ${\rm End}(V)$, and it follows that $m\le d^2$.
 
 The following terminology dates back to \cite{RBSC},  where the connection with the theory of quantum measurements was first observed:
 \begin{definition}
  A maximal equiangular configuration $(L_1,\ldots, L_{d^2})$ is called a 
 SIC-POVM ({\em Symmetric Informationally Complete Positive Operator-Valued Measure}), or simply a   SIC  for short.
 \end{definition}
  It can be shown that the common angle  in a SIC is necessarily given by
 \begin{equation} \alpha =  \frac{1}{\sqrt{d+1}},  \end{equation}
 but the following existence question remains open:
 \begin{question}
 \label{question:zauner}
 Is there a SIC in every dimension $d$?
 \end{question}
It is  not  even known 
at present whether  SICs  exist for infinitely many $d$.
In his 1999 PhD thesis \cite{zauner-thesis,zauner}, Gerhard Zauner conjectured that 
Question \ref{question:zauner} has an affirmative answer,
and that some notable
  SICs  arise as the orbit  of a single line under the action of a finite {\em Heisenberg group} acting irreducibly on $V$.
  
  Assume henceforth, for ease of exposition, that $d$ is {\em an odd prime}.
The Heisenberg group  $G_d$ of cardinality $d^3$  is a  non-trivial central extension of $\Z/d\Z\times\Z/d\Z$ by $\Z/d\Z$.  It fits into a non-split exact sequence
 \begin{equation} 1 \lra Z \lra G_d \lra \Z/d\Z \times \Z/d\Z\lra 1, \qquad
 Z= \Z/d\Z,
 \end{equation}
 where $Z$ is both the center and the commutator subgroup of $G_d$.
 One model for $G_d$ realizes it as the group of
 $3\times 3$ upper-triangular unipotent matrices with entries in  the field $\Z/d\Z$.
 
  The  (elementary) finite counterpart of the  Stone-Von Neumann theorem asserts that an irreducible representation of $G_d$ is completely determined by its central character when the latter is non-trivial. Furthermore, this distinguished 
 representation is faithful, and 
 of dimension $d$. After 
 fixing a non-trivial central character $\psi:Z \lra \C^\times$,  assume that $V$ realises the irreducible representation of $G_d$ attached to $\psi$, and let 
 \begin{equation}
 \varrho:G_d\lra {\rm Aut}(V)
 \end{equation}
  be the resulting group homomorphism. The center $Z$ of $G_d$ acts trivially on the set of  lines in $V$,
 and   a line $L$ whose stabiliser is  equal to $Z$  has an  orbit under $G_d$  of size $d^2$.
A SIC of the form \begin{equation}S:= \{ g\cdot L \}_{g\in G_d}\end{equation} is called a {\em Heisenberg SIC},   the line $L$ is called a {\em fiducial line} attached to $S$, and a unit vector   $e\in L$ is called a {\em fiducial vector}. 
 
 The search for Heisenberg SICs is vastly facilitated by the fact that these special classes of SICs are entirely determined by  a single fiducial vector $e\in V$ rather than by a configuration of $d^2$ lines. The complex numbers
 \begin{equation} \alpha_g = \langle g 
 \cdot e,  e\rangle, \qquad g\in G_d,  \end{equation}
 whose absolute values are all equal to $\alpha$ for all $g\notin Z$, can furthermore be understood as a specific {\em matrix coefficient} in the representation $(V,\varrho)$. Finally, the fiducial line $L$ is readily recovered from the data of the $\alpha_g$, as the image of the rank one projector
 \begin{equation} \Pi:= \sum_{g\in G_d} \alpha_g \cdot \varrho(g).\end{equation}
 To find a Heisenberg SIC, it therefore suffices to find a matrix coefficient $\{\alpha_g\}_{g\in G_d}$
 for $(V,\varrho)$
 satisfying
 \begin{equation} |\alpha_g|^2 = \frac{1}{d+1}, 
 \qquad \mbox{ for all } g\in G_d-Z.\end{equation}
 A remarkable insight which was first proposed in \cite{AYZ} and 
 was   gradually sharpened  in a series of experimental works by many physicists   predicts that matrix coefficients with this property arise from suitable Stark units in a ray class field of the real quadratic field \begin{equation}
  K=\Q(\sqrt{(d+1)(d-3)}).
  \end{equation} 
  (Cf.~\cite{AFK} 
 for a definitive and up-to-date account, as well as the bibliography therein.) 
 The  quadratic order
 $\cO_D = \Z[\frac{D+\sqrt{D}}{2}]$  of discriminant $D:= (d+1)(d-3)$   is distinguished by the fact that it   has an  explicit unit 
 \begin{equation}\varepsilon_d = \frac{(d-1)+\sqrt{(d+1)(d-3)}}{2}
 \end{equation} 
 of relatively small height relative to $D$.   
 
 Let $H$ be the narrow Hilbert class field of $K$ and let $H_d$ be the narrow  ray class field of $K$ of conductor $(d)$.
 Global class field theory  identifies the
  Galois group of $H_d$ over $H$  with
  \begin{equation} {\rm Gal}(H_d/H) = (\cO_K/d)^\times/\langle \varepsilon_d\rangle. \end{equation}
 The fact that the   image of $\varepsilon_d$ 
 in $(\cO_K/d)^\times$ has order $3$ shows that 
  \begin{equation} 
  [H_d:H] = \left\{ \begin{array}{cl}
   \frac{d^2-1}{3}& \mbox{ if } d\equiv 1 \pmod{3},\\   \frac{d^2+1}{3}& \mbox{ if } d\equiv 2 \pmod{3}.
   \end{array}\right.
   \end{equation}
   (It also accounts for an empirically observed order $3$ symmetry in Heisenberg SIC-POVM's, called the {\em Zauner symmetry.})
 Assume from now on that the prime $d$ is congruent to $2$ modulo $3$.
  
  Let   $u_d \in \cO_{H_d}^\times$ be a   unit of $H_d$ and let 
 \begin{equation} 
 \label{eqn:alphag}
 \alpha_g = \psi(i) \cdot {\rm rec}(j+k\varepsilon_d) \left(\frac{u_d}{\sqrt{d+1}}\right),
 \qquad \mbox{ for } g = \left(\begin{array}{ccc}
 1 & j & i \\
 0 & 1 & k \\
 0 & 0 & 1 
 \end{array}\right).
 \end{equation} 
 The following conjecture, which is known in the field as the {\em twisted convolution conjecture}, asserts something remarkable about   Stark units attached to the ray class field $H_d$. (We state it here in a somewhat vague form, and more precise statements can be found   in \cite{AFK}.)
 \begin{conjecture}
 There is an explicit unit $u_d$, whose square is a Stark unit in $H_d$, and for which 
 the   function $\alpha_g$ of \eqref{eqn:alphag}
 is  a matrix coefficient for $(V,\varrho)$. 
 \end{conjecture}
 This   intriguing property of   Stark units  seems to have entirely escaped the attention of number theorists   and was discovered experimentally by physicists engaged  in the numerical search for Heisenberg SICs. It is now the basis for the most  efficient 
 numerical calculation of Heisenberg SICs, notably the recent {\tt Zauner.jl} package written in the computer language Julia that is documented in 
 \cite{AFK}.
 
 
The numbers $\alpha_g$
arising from a Stark unit are obtained from an analytic calculation of their images under a real 
embedding $\tilde v_1$ of $H_d$, as the RM values of Kopp's 
Shintani-Faddeev modular cocycle. These real
numbers are of varying magnitudes  and the kernel of the rank one projector
 \begin{equation}  \Pi_1:= \sum_{g\in G} \tilde{v}_1(\alpha_g) \cdot \varrho(g)\end{equation}
 is far from being a fiducial line for a Heisenberg SIC.
The Stark conjecture predicts, on the other hand, that the complex numbers $\tilde v_2(\alpha_g)$
all lie on the circle of radius $1/\sqrt{d+1}$,
for any complex embedding 
 ${\tilde v}_2$ of $H_d$ that lies above $\infty_2$.
Since the  rank one property is also
satisfied by the projector
\begin{equation} \Pi_2:= \sum_{g\in G} \tilde{v}_2(\alpha_g) \cdot \varrho(g),\end{equation}
the image of the projector $\Pi_2$
 is a fiducial line for a Heisenberg SIC.

 Note that
the quantities $\alpha_g$ are calculated in practice 
by first coaxing the 
real numbers ${\tilde v}_1(\alpha_g)$ out of
the RM values of a modular cocycle encoding 
 the leading terms of suitable abelian   $L$-functions,  recognizing these real numbers as algebraic numbers in $H_d$, and finally applying to them the   complex embedding 
${\tilde v}_2$.  A direct analytic expression for the quantities ${\tilde v}_2(\alpha_g)$, in the spirit of a positive answer to Question 
\ref{question:sczech}, would lead to a more efficient, robust, and purely analytic  procedure for calculating the still mysterious Heisenberg SICs that ostensibly arise from 
Stark units.


 
 
  
 
 
 
 
 
 
 



\section{The $p$-adic Stark conjecture}
\label{sec:padicstark}
Tate's popularisation  of the Stark conjecture    spurred the study of its $p$-adic variants. 
The time for this was ripe  in the early 80's, since
  $p$-adic $L$-functions  
 attached to    finite order characters   of   totally real fields  had recently been constructed   by Pierrette Cassou-Nogu\`es \cite{cassou} by interpolating Shintani's explicit formulae, and by 
 Serre \cite{serre-lnm} and Deligne-Ribet 
 \cite{deligne-ribet}
following an approach of Siegel  exploiting the occurrence of  the relevant classical   $L$-values  as constant terms of Hilbert   Eisenstein series. It was thus understood how to associate    a   $p$-adic 
 $L$-function  $L_p(\chi,s)$ to any totally real field $K$ and finite order character $\chi$ of  $K$ which is either totally even or totally odd. These $p$-adic $L$-functions (following a somewhat non-standard normalisation)  interpolate the special values $L(\chi,j)$ at    negative  integers
 satisfying
 \begin{equation}
  j\equiv \left\{
 \begin{array}{ll}
 1 \pmod{p-1} & \mbox{ if } \chi \mbox{ is even},\\
 0 \pmod{p-1} & \mbox{ if $\chi$ is odd}. 
 \end{array}\right.
 \end{equation}
      The parity restriction on $\chi$ is required to ensure that the classical $L$-function $L(\chi,s)$ admits sufficiently many non-zero critical values at negative integers   to allow for their $p$-adic interpolation.
 
   
 Two entirely different $p$-adic analogues of the Stark conjectures were proposed almost simultaneously:  by  Jean-Pierre Serre,
  with whom Tate  maintained a regular correspondence   over many decades (cf.~\cite[Letter of Sept.12 1978]{serre-tate} and  \cite{serre-zeta}),  and by Dick Gross \cite{gross-stark}, who was Tate's PhD student from 1974 to 1978.
 These conjectures concern the behaviour of abelian $L$-series at the points $s=1$ and $s=0$ respectively, which, in contrast to the archimedean setting, are not related to each other in a simple way under the  functional equation. 

 Serre's $p$-adic Stark conjecture considers the behaviour at $s=1$ of 
 $p$-adic Artin $L$-series. In the special case  of a  non-trivial    even  character $\chi$, which cuts out a totally real abelian extension $H$  of $K$, the induced representation $V={\rm Ind}_H^\Q(\chi)$ occurs with multiplicity 
 $n:=[K:\Q]$ in the units of the Galois closure $H'$ of $H$.
Serre conjectures that $L_p(\chi,1)$   is equal to a non-zero algebraic multiple  of a $p$-adic regulator $R_p(V)$,   defined
by the same formula as in 
  \eqref{eqn:stark-reg} for $R(V)$, but replacing complex by $p$-adic logarithms (after fixing a  $p$-adic embedding of $H'$). 
  Serre's regulator  involves  the $p$-adic logarithms 
  of {\em genuine units}.  In part because of this, the $p$-adic Stark conjecture at $s=1$    remains just as wide open as its   archimedean counterpart. (Cf.~also \cite{solomon} for a more in-depth discussion and a more precise statement in the special case of abelian characters of a totally real field.)

Gross's $p$-adic Stark conjecture considers the behaviour at $s=0$ of the  $p$-adic $L$-function of a {\em totally odd}   character $\chi$, which cuts out a CM abelian extension $H$  of the totally real field $K$. 
 In this setting, the  induced representation $V={\rm Ind}_H^\Q(\chi)$ now occurs with {\em multiplicity zero}
    in the units of  $H'$, reflecting the fact that the classical $L$-function $L(\chi,s)$ is non-vanishing at $s=0$.
    
  Choose a prime $p$ at which $V$ is unramified, and let $\sigma_p$ be an associated Frobenius element in ${\rm Aut}(V)$ attached to a prime of $H'$ above $p$. The $p$-adic $L$-function
    $L_p(\chi,s)$ interpolates the critical  values 
     of $L(\chi,s)$ at suitable negative integers, with the Euler factors at the primes above $p$ removed. 
  This Euler factor  vanishes at $s=0$ precisely when 
  there is a prime $\fp$ of $K$ above $p$ for which $\chi(\fp)=1$. In this case, 
 $V$ occurs with non-zero multiplicity in the $p$-unit group of $H'$. Let
  \begin{eqnarray*}
   r_p &=&    \dim_E \hom_{G_\Q}(V, (\cO_{H'}[1/p])^\times\otimes E) \\
   &=& \#\{\fp|p \ \mbox{ s.t.}  \ \chi(\fp)=1\} = \dim_E V^{\sigma_p=1} 
   \end{eqnarray*}
   denote this multiplicity.
Gross   conjectures that $L_p(\chi,s)$ vanishes to order exactly $r_p$ at $s=0$ and that its leading term    is equal to a non-zero algebraic multiple  of a $p$-adic regulator $R_p^{\rm Gr}(V)$. This regulator is  
built from $p$-units rather than actual units, and  
depends on a choice of   bases $\{ \phi_1^{(p)},\dots,\phi_{r_p}^{(p)}\}$ and $\{w_1,\ldots,w_{r_p}\}$ 
 of the $E$-vector spaces
\begin{equation}
\operatorname{Hom}_{G_\Q}\!\bigl(V, (\cO_{H'}[1/p])^\times \otimes E \bigr) \quad \mbox{ and } \quad V^{\sigma_p=1}
\end{equation}
respectively. Note that 
$\phi_i^{(p)}(w_j)$ can be viewed as an element of $\Q_p^\times$ thanks to the chosen $p$-adic embedding of $H$.
The Gross regulator $R_p^{\rm Gr}(V)$ can then be  defined just as in 
\eqref{eqn:stark-reg}:\begin{equation}
\label{eqn:gross-reg}
R_p^{\rm Gr}(V) := \det\left(\bigl( \log_p \phi_i^{(p)}(w_j)\bigr) _{1 \leq i,j \leq r_p}\right).
\end{equation}
A rough version of Gross's conjecture asserts that
\begin{equation}
\lim_{s\lra 0} s^{-r_p}L_p(V,s)   = R_p^{\rm Gr}(V) \pmod{\bar\Q^\times}.
\end{equation}

The Gross regulator is
 of a fundamentally different nature than the Serre regulator $R_p(V)$, since it  involves the $p$-adic logarithms of $p$-units  rather than of genuine units.
 Unlike Serre's $p$-adic Stark conjecture at $s=1$, Gross's conjecture turns out to be more tractable than its archimedean counterpart, essentially because  Gross-Stark $p$-units can often be ``read off" from the $p$-adic deformation theory of the Artin representation of $G_K$ attached to $\chi$ from global class field theory. This key fact is exploited in 
 \cite{DDP}, as well as in 
 \cite{DKV}, 
 \cite{DLR-advances}, and
 \cite{DLR-heidelberg}.
     
     In the case where $r_p=1$ which is the most germane to explicit class field theory for $K$, Gross's conjecture is proved in \cite{DDP} (under certain technical assumptions on $\chi$ which were later removed by Kevin Ventullo). 
     The case of general $r_p$ was  subsequently disposed of 
      in a series of breakthroughs  \cite{DKV}, \cite{DK-bs}.
  All of these works owe much to the deformation theoretic methods used in the proof by Greenberg and Stevens of the ``exceptional zero conjecture" of Mazur, Tate and Teitelbaum,  which will be touched on in the next section.
     


  

 
  

 
 

\section{The Mazur-Tate-Teitelbaum conjecture}
\label{sec:mtt}
The existence of two fundamentally different $p$-adic  analogues of the Stark conjecture underscores the richness of  
 the non-archimedean  setting and its propensity to reveal new phenomena not  readily apparent in the complex picture.
This may have 
led   Tate to reflect on  $p$-adic analogues of the Birch and Swinnerton--Dyer conjecture in the early 1980's, leading to a  highly influential   collaboration with Barry  Mazur and Jeremy Teitelbaum  which uncovered a plethora of  
unexpected 
features.
This sentiment is expressed in the following passage from the introduction of \cite{MTT}:
\begin{quote}
It seemed to us, then, to be an appropriate time to embark on the project of 
formulating a $p$-adic 
analogue of the conjecture of Birch and Swinnerton-Dyer, and gathering numerical data in its support. It also seemed, at the outset, that this would be a relatively routine project. The project has proved to be anything but routine, and this article is an attempt to report on our findings so far."
\end{quote}

The $p$-adic $L$-function attached to an elliptic curve $E$ -- or rather, to the associated weight two cusp form $f_E$, whose existence at the time was only conjectural --  had already been constructed in \cite{MSwD} as a $p$-adic 
Mellin transform of a measure on $\Z_p^\times$ arising from the {\em modular symbol} attached to $f_E$.
This modular symbol is the  $\Gamma_0(N)$-invariant 
additive function
\begin{equation}
m_E\{ \ ,\ \}:   \PP_1(\Q) \times \PP_1(\Q) \lra \C
 \end{equation}
 defined by 
 \begin{equation}
 m_E\{ x,y \} :=  \int_{x}^y 2\pi i f_E(z)dz.
 \end{equation}
 It takes values in a   lattice 
  $\Lambda_E\subset\C$  commensurable with the 
  N\'eron lattice  of $E$.
Let $\Omega_E^+$ (resp.~$\Omega_E^-$) be the positive generator  of  the projection of $\Lambda_E$ to the real  (resp.~imaginary)  axis
in $\C$,  so that
\begin{equation}
\label{eqn:def-Omega}
\Lambda_E \subset \Z\cdot \Omega_E^+ \oplus \Z 
\cdot \Omega_E^- \ \ \ 
\mbox{ 
with index } \ \le 2. 
\end{equation}
The periods  $\Omega_E^+$ and $\Omega_E^-$ 
can be used to extract
 a pair $m_E^\pm\{\ , \ \}$ of $\Z$-valued modular symbols from  
$m_E\{\ , \ \}$ 
by setting
\begin{equation}
\label{eqndef-periods}
 m_E\{x,y\}_E = m_E^+\{x,y\} \cdot\Omega_E^+ + m_E^-\{x,y\}  \cdot \Omega_E^- \qquad \mbox{ for all } x,y
 \in \PP_1(\Q).
\end{equation}
The $\Z$-valued symbol $m_E^+\{\ \ \}$ gives an integral measure $\mu_{f_E}$  on $\Z_p^\times$ by setting
\begin{equation}
 \int_{a+p^n \Z_p} d\mu_{f_E} := \{a/p^n,\infty\}_E^+,
 \end{equation}
and $L_p(E,s)$ is the 
Mazur-Mellin transform of this measure:
\begin{equation}
 L_p(E,s) = 
\int_{\Z_p^\times} \langle x \rangle^{s-1} d\mu_{f_E}(x), \quad \mbox{ with }  \langle x \rangle := x \lim_{n\rightarrow \infty}x^{-p^n}.
\end{equation}
 

With the $p$-adic $L$-function    in hand, the naive
$p$-adic variant of the Birch Swinnerton--Dyer conjecture would predict that $L_p(E,s)$ has a zero of order $r:={\rm rank}(E(\Q))$ and that the leading term $L_p^{(r)}(E,1)$  can be expressed in terms of the usual quantities arising in the Birch and Swinnerton Dyer conjecture -- the cardinality of the conjecturally finite Shafarevich-Tate group of $E$,  and a {\em regulator} whose definition is the same as the classical regulator, after replacing the N\'eron-Tate canonical height by a $p$-adic height pairing 
\begin{equation}
\label{eqn:padic-height}
\langle \ , \ \rangle_p: E(\Q) \times E(\Q) \lra \Q_p.
\end{equation}
This is indeed what the  $p$-adic conjecture predicts in general,    with one seemingly minor but crucial exception:
 when $E$ has 
 {\em split multiplicative reduction at $p$} --- i.e., when the prime $p$ divides 
$N$ exactly  and the $p$-th Fourier coefficient of $f_E$ is equal to  $1$ --- the order of vanishing of $L_p(E,s)$ seems to be thrown off, and to {\em increase by one}.
Thus $L_p(E,s)$ acquires what the authors of \cite{MTT} refer to as an  {\em exceptional zero}.
On the ``arithmetic side", the elliptic curve $E$ is endowed 
with Tate's celebrated 
$p$-adic uniformisation
\begin{equation}
\label{eqn:tate-par} \Phi_{\rm Tate}: \Q_p^\times/q^{\Z} \lra E(\Q_p),
\end{equation}
where $q\in p\Z_p$ is the 
{\em Tate period} of $E$ which is one of the key {\em dramatis personae} in the contributions by Brian Conrad and Dustin Clausen in this volume.


To account for the leading term of $L_p(E,s)$ 
  in the presence of an exceptional zero, 
Mazur, Tate, and Teitelbaum 
 introduce the {\em extended Mordell-Weil group} 
 \begin{equation}E^\dag(\Q) := 
 \Q_p^\times \times_
 {\Phi_{{\rm Tate}}} E(\Q) =\{ (z,P)\in \Q_p^\times \times E(\Q)  \mbox{ such that  } \Phi_{{\rm Tate}}(z) =P \},
 \end{equation}
 which fits into an exact sequence
 \begin{equation} 1 \lra q^\Z \lra E^{\dag}(\Q) \lra E(\Q) \lra 1.\end{equation}
 It is an abelian group of rank $r+1$ when $E$ has split multiplicative reduction at $p$, and of rank $r$ otherwise.
 The main prediction of
 \cite{MTT} is
 \begin{conjecture}[Mazur-Tate-Teitelbaum]
 \label{conj:MTT}
 The $p$-adic $L$-function satisfies
 $${\rm ord}_{s=1} L_p(E,s) = 
 {\rm rank}(E^\dag(\Q)).$$
 \end{conjecture}
 To account for the leading term of $L_p(E,s)$ in the presence of an exceptional zero,  the $p$-adic height pairing of \eqref{eqn:padic-height}
 is extended to (a finite index subgroup of) $E^{\dag}(\Q)$ by the rules
 \begin{equation} \langle (z_1, P_1), (z_2, P_2)\rangle_p = \langle P_1,P_2\rangle_p, 
 \qquad \mbox{ whenever } z_1, z_2\in \Z_p^\times,\end{equation}
 together with the complementary rules for the extra element $q = (q,0)\in E^\dag(\Q)$ corresponding to the Tate period:
 \begin{equation}
 \langle q, (P,z)\rangle_p = \log_p(z), \qquad
 \langle q,q \rangle_p = \log_p(q)/{\rm ord}_p(q), \end{equation}
 where $\log_p:\Q_p^\times/p^\Z \lra \Q_p$
 is Iwasawa's branch of the $p$-adic logarithm satisfying $\log_p(p)=0$.
 The {\em extended regulator} is the determinant
 \begin{equation} R_p^\dag(E) = \det\left(\langle x_i,x_j\rangle_p)_{1\le i,j \le r+1}\right),\end{equation}
 where $x_1,\ldots,x_{r+1}$ is a $\Z$-basis of $E^\dag(\Q)$.  A more precise version  of 
 Conjecture \ref{conj:MTT}
 asserts that
 the leading term of $L_p(E,s)$ is a product of this regulator times the other terms that appear in the classical Birch and Swinnerton-Dyer conjecture like the cardinality of the conjecturally finite Shafarevich-Tate group of $E$.
  
 In the special case where $r=0$ and $p$ is a prime of split multiplicative reduction for $E$, the Mazur-Tate-Teitelbaum conjecture asserts that $L_p(E,s)$ has a simple zero at $s=1$ and that
\begin{equation}
\label{eqn:exceptional-zero}
L_p'(E,1)  = \frac{\log_p(q)}{\operatorname{ord}_p(q)} \cdot \frac{L(E,1)}{2\cdot \Omega_E^+}.
\end{equation}
This strikingly concrete statement asserts that the 
modular symbol attached to $E$, obtained from the modular parametrisation of $E$ over $\C$, ``knows about" the period $q$ that Tate had discovered more than a decade earlier in his  $p$-adic uniformisation theory of elliptic curves!
As the authors comment in the  introduction to \cite{MTT},  
\begin{quote}
It is quite remarkable to see the $p$-adic digits of this [Tate] $p$-adic period be ``reproduced" in the print-out of a computing machine programmed to compute [$\ldots$] certain expressions involving modular symbols."
\end{quote}
The extensive    computer calculations carried out 
 by
the authors of \cite{MTT}  to test their predictions
is one of the high points of what can now be seen as the twilight of the 
 ``heroic era" in computational number theory. 
 
 Formula \eqref{eqn:exceptional-zero} 
 came to be known as the ``exceptional zero conjecture". It has turned out to be enormously 
 impactful,  and has been generalised to a wide variety of settings.  The original  conjecture was  proved in an influential work of  Greenberg and Stevens \cite{greenberg-stevens} which exploits several important new ingredients: most significantly,   the theory of $p$-adic deformations of modular forms and their associated Galois representations and $p$-adic $L$-functions.



  
 For general $r$, the Mazur--Tate--Teitelbaum conjecture seems   as difficult as the original Birch and Swinnerton-Dyer conjecture, even if one inequality -- 
 \begin{equation}
 \label{eqn:kato-ineq}
  {\rm ord}_{s=1} L_p(E,s)
 \ge {\rm rank}(E^\dag(\Q)),
 \end{equation}
 has been proved by Kazuya 
 Kato \cite{kato} using ideas from the theory of ``Euler systems". 
 Kato's proof rests on the Iwasawa-theoretic interpretation of the $p$-adic $L$-function $L_p(E,s)$, as encoding the behaviour of $E$ over
 the cyclotomic $\Z_p$-extension of $\Q$, which is contained in the field 
 \begin{equation}
 \label{eqn:cyclotomic-Zp}
  \Q_\infty := \bigcup_{n=1}^\infty \Q_n, \qquad \Q_n\subset \Q(e^{2\pi i /p^n}),
  \end{equation}
  where $\Q_n$ is the maximal $p$-power degree subfield  of the field of $p^n$-th roots of unity.  
 The Iwasawa main conjecture for $L_p(E,s)$ relates this analytic invariant to the {\em characteristic power series} of the $p$-Selmer group of $E$ over $\Q_\infty$, viewed as a module over the Iwasawa algebra $\Z_p[\![{\rm Gal}(\Q_\infty/\Q)]\!]$.  Kato does not prove the full main conjecture but shows that $L_p(E,s)$ is divisible by the characteristic power series of the pro-$p$ Selmer group, which
 directly implies 
\eqref{eqn:kato-ineq} but is somewhat stronger and more precise.
 
 
 The mechanism whereby the rank of $E$  forces vanishing of the associated $L$-function at the central point
 is thus better understood than in the complex setting, where no elliptic curve $E$ with 
 Hasse-Weil $L$-function
  satisfying   ${\rm ord}_{s=1}L(E,s)>3$ has so far been
shown to exist. The reverse  inequality in 
\eqref{eqn:kato-ineq}   also lies much
 deeper, and does not follow  from the corresponding inequality in the Main Conjecture.

 
 
 
 

\section{The Mazur-Tate conjecture}
\label{sec:MT}
In \cite{MT-bsd}, Mazur and Tate
propose to replace the $\Z_p$-extension  
of \eqref{eqn:cyclotomic-Zp}  
by a {\em finite  abelian extension} $H$ of 
$\Q$  with Galois group $G$,
the Iwasawa algebra by the integral group ring 
$\Z[G]$, 
 and the 
$p$-adic $L$-function
by a ``theta element" in   $\Z[G]$, which plays the role of the Stickelberger element in   the Brumer-Stark conjecture.





The setting considered primarily in \cite{MT-bsd} is
the one where $H=H_M:=\Q(\zeta_M)^+$ is the real subfield of the $M$-th cyclotomic field, so that 
\begin{equation}
G:=G_M = (\Z/M\Z)^\times/\langle \pm 1\rangle, 
\end{equation}
and where $E$ is a (modular) elliptic curve over $\Q$.
Like the Mazur-Swinnerton-Dyer $p$-adic $L$-function of 
\cite{MTT}, the  resulting theta element 
\begin{equation}
\label{eqn:thetaEM}
\theta_{E,M} := \sum_{a\in (\Z/M\Z)^\times}  m_E^+\{a/M, \infty\} \cdot [a] 
\ \ \in \ \ \Z[G_M]
\end{equation}
  is constructed from  the modular symbol attached to $E$, and satisfies the interpolation property
\begin{equation}
\label{eqn:interpolation-mt}
\chi(\theta_{E,M}) =  \frac{\tau(\chi) L(E,\bar\chi,1)}{2 \cdot\Omega_E^+}  \qquad \mbox{ for all even primitive } \chi:G_M \lra \C^\times,
\end{equation}
where $\tau(\chi)$ is a Gauss sum attached to $\chi$.
In particular,  $\theta_{E,M}$  belongs to the augmentation ideal $I$ of $\Z[G_M]$ whenever $L(E,1)=0$. The ``refined Birch and Swinnerton-Dyer conjecture" of \cite{MT-bsd}
asserts that $\theta_{E,M}$ vanishes to order at least $r := {\rm rank}(E(\Q))$ at the trivial character, namely, that 
\begin{equation}
\label{eqn:mt-original}\theta_{E,M} \stackrel{?}{\in} I^r.\end{equation}
It also proposes a formula for the leading term of $\theta_{E,M}$, i.e., its natural image in $I^r/I^{r+1}$,   involving the same quantities as in the Birch and Swinnerton--Dyer conjecture, but with the N\'eron-Tate regulator replaced by a 
finite ``Mazur-Tate regulator". 
This regulator is defined in terms of the {\em Mazur-Tate height pairing}
that will be discussed in slightly more detail in Section \ref{sec:mt-ht} below:
\begin{equation}
\label{eqn:mtht}
\langle\ , \ \rangle_{E,M}: E_M(\Q) \times E(\Q) \lra G_M = I/I^2,
\end{equation}
where $E_M(\Q)$ is the group of points in the connected component of the N\'eron model of $E$ over $\Z$ that reduce to the identity modulo $M$. 
The Mazur-Tate regulator is the determinant of the associated $r\times r$ pairing matrix,   with entries  in $I/I^2$, and therefore  naturally lies  
in $I^r/I^{r+1}$. 

The point of view introduced in \cite{MT-bsd}   is broadly applicable and reveals   a   variety of new and intriguing arithmetic phenomena. 
It was extended almost immediately to the setting of abelian $L$-series of totally real fields by
Dick Gross, leading to ``tame refinements" \cite{gross-tower} of the $p$-adic Gross-Stark conjecture evoked in 
Section \ref{sec:padicstark}.


It was also Dick Gross who suggested  to the author that  transposing the Mazur-Tate conjecture  to a setting where $\Q$ is replaced by an imaginary quadratic field $K$ would    provide  a congenial framework for understanding the ``Euler system" arguments of Kolyvagin,  which 
burst on the scene shortly after
the Mazur-Tate seminar of 1986.
 In Kolyvagin's approach, certain global objects (the ``Kolyvagin cohomology classes") indexed by the collection of  ring class fields of $K$ and constructed from Heegner points are used to control the  Selmer group of $E$ over $K$. When $E$ has odd (analytic) rank over $K$, the author's PhD thesis  \cite{darmon-thesis} associates a theta element $\Theta_{E,K,c}'$ to the ring class field $H_c$ of $K$  of any conductor $c$  that is relatively prime to the conductor of $E$. Writing 
    $G_c={\rm Gal}(H_c/K)$, this theta element 
is obtained
by fixing  a Heegner point $P_c\in E(H_c)$ of conductor $c$, letting
\begin{equation} \theta_{E,c}' = \sum_{\sigma\in G_c} P_c^\sigma \otimes \sigma, \qquad
\bar\theta_{E,c}' = \sum_{\sigma\in G_c} P_c^\sigma \otimes \sigma^{-1} 
\end{equation}
be the formal resolvent elements in $E(H_c)\otimes \Z[G_c]$  
arising from this 
Heegner point, 
and defining
\begin{equation} \Theta_{E,K,c}' := \theta'_{E,c} \otimes \bar\theta'_{E,c}  \in E(H_c)^{\otimes 2}\otimes \Z[G_c]. 
\end{equation}
Let 
$$h^{\otimes 2}: E(H_c)^{\otimes 2} \lra \R$$ 
be the canonical N\'eron-Tate height, extended to an 
$\R$-valued linear functional on $E(H_c)^{\otimes 2}\otimes \R$.  
The Gross-Zagier formula
\cite{gross-zagier} for
Heegner points over $H_c$ suggests that 
$\Theta_{E,K,c}'$ satisfies the interpolation property
analogous to \eqref{eqn:interpolation-mt}
\begin{equation}
 (h^{\otimes2} \otimes\chi)(\Theta_{E,K,c}') =
 \frac{L'(E/K,\chi,1)}{\Omega_E^+\Omega_E^-}  \pmod{\bar\Q^\times},\end{equation}
 where $\Omega_E^+$ and $\Omega_E^-$ are as in 
 \eqref{eqn:def-Omega}.

For this reason, $\Theta'_{E,K,c}$ 
can be envisaged as a Mazur-Tate style avatar of the first derivative $L'(E/K,s)$. 
The main result of the author's thesis
\cite{darmon-thesis} asserts that 
\begin{equation}
\label{eqn:darmon-thesis}
\Theta_{E,K,c}' \mbox{ belongs to }E(H_c)^{\otimes 2} \otimes I^{r-1}, \quad \mbox{ where }
r := {\rm rank}(E(K)),\end{equation}
under suitable technical conditions (and after inverting some problematic small primes in the integral group ring).

The Euler system techniques of Kato were later applied in 
\cite{ota} to 
extend the formal argument  in the proof of \eqref{eqn:darmon-thesis} to the original setting of the
Mazur-Tate conjecture, thereby establishing
\eqref{eqn:mt-original}.


  
Building on the foundational work of Gross \cite{gross-tower},
 Mazur-Tate style variants in the  setting of unit groups of number fields and the  Stark conjectures have been further explored in the cyclotomic setting 
 (\cite{darmon-cjm},  \cite{mazur-rubin-Gm}, \cite{sano}, $\ldots$) 
 Recent breakthroughs of Dasgupta and Kakde 
  have established the Brumer-Stark conjecture 
  \cite{DK-bs-ref},
\cite{DKSW}
   and the Mazur-Tate 
 refinement of the Gross-Stark conjecture of \cite{gross-tower} for totally odd abelian extensions of totally real fields \cite{DK-bs}.
 
 
  A recent  conjecture of Harris and Venkatesh \cite{HV} motivated by Venkatesh's theory of ``derived Hecke operators" suggests a tantalising   Mazur-Tate analogue of Serre's $p$-adic  Stark conjecture at $s=1$ 
  for the Artin representation attached to the adjoint of a weight one  newform. This might    lead to a more systematic framework for the Mazur-Tate conjecture for units in number fields, refining  the two types of $p$-adic Stark conjectures evoked in Section \ref{sec:padicstark}:
Gross's variant   at $s=0$ and Serre's at $s=1$. 
(See  \cite{zhang} for some first steps towards bringing this  analogy into  sharper focus.)




   

\section{A Stark conjecture for elliptic curves}
Just as with the Mazur-Tate conjectures, it
 is natural to explore the Mazur-Tate-Teitelbaum conjecture 
 beyond the  cyclotomic setting. A particularly appealing set-up
 involves replacing the 
 cyclotomic $\Z_p$-extension of $\Q$ 
 in 
 \eqref{eqn:cyclotomic-Zp}
  with  the  {\em anticyclotomic} $\Z_p$-extension of a quadratic imaginary field $K$ (of discriminant $D$, say).
This $\Z_p$-extension is given by
\begin{equation}
\label{eqn:anticyclotomic-Zp}
 K_\infty := \bigcup_{n=1}^\infty K_n,
\qquad K_n \subset K\left( j(\Z + p^n \cO_K)\right), 
\end{equation}
where $K_n$ is the largest $p$-power degree  extension of $K$ contained in the ring class field of $K$ of conductor $p^n$. The latter extension is generated over $K$ by the $j$-invariants of elliptic curves with endomorphism ring equal to the  order of conductor $p^n$ in $\cO_K$. 
The availability of a systematic supply of {\em Heegner points} in $ E(K_n)$ under suitable hypotheses suggest that the anticyclotomic setting might involve features not apparent in the original cyclotomic case, 
and singles it out for special attention \cite{BD-mumford-tate}.

 
  The relevant $p$-adic $L$-function attached to $(E,K_\infty)$ is denoted 
  $\Lp(E/K_\infty,s)$,
and can also be viewed as belonging to the Iwasawa algebra of $G_\infty := {\rm Gal}(K_\infty/K)$,
\begin{equation}\Z_p[\![G_\infty]\!] :=\lim_{\leftarrow} \Z_p[{\rm Gal}(K_n/K)] \simeq \Z_p[\![T]\!]. \end{equation}
It
   interpolates
 the central critical values $L(E/K,\chi,1)$ as $\chi$ ranges over the continuous 
 finite order characters of $G_\infty$. 
 More accurately, it interpolates their {\em square roots}: this is why it is denoted $\Lp(E/K_\infty,s)$ rather than
 \begin{equation}L_p(E/K_\infty,s):= 
 \Lp(E/K_\infty,s) \Lp(E/K_\infty,s)^*,\end{equation}
 where $\lambda\mapsto \lambda^*$ denotes the involution on $\Z_p[\![G_\infty]\!]$ sending each group-like element $g\in G_\infty$ to its inverse.
 The availability  of such 
 ``square root $p$-adic $L$-functions" is a   feature  of the anticyclotomic setting that gives rise to some of its more compelling  arithmetic applications.
 
 When $E$ has good reduction at $p$, it can be observed that
  $\Lp(E,K_\infty)$
  vanishes identically
 unless the sign in the functional equation for $L(E/K,s)$ is
 equal to $1$. 
 But when $E$ has multiplicative reduction at $p$ and  $K_p:= K\otimes \Q_p$ is the quadratic unramified extension of $\Q_p$, an interesting {\em sign change phenomenon}  ensures that the power series
 $\Lp(E,K_\infty)$ can only be  non-zero  when   the classical $L$-series  $L(E/K,s)$  
 has sign $-1$ in its functional equation, and hence a {\em zero of odd order} at $s=1$. In that case, $L(E/K, 1)=0$ and  $\Lp(E,K_\infty)$ likewise vanishes at  the trivial character: in other words, it belongs to the augmentation ideal $I$
 of the completed group ring $\Z_p[[G_\infty]]$. It    then becomes natural to consider its {\em first derivative} with respect to $s$. In terms of the Iwasawa algebra description of $\Lp(E/K_\infty)$,  this derivative
  corresponds to  the canonical image of
 $\Lp(E,K_\infty)$ in   
 \begin{equation}I/I^2 = G_\infty = K_{p,1}^\times,\end{equation} where  the first identification is a general algebraic fact and the second follows from   global class field theory.
   
 Because $E$ has split multiplicative reduction over $K_p$, it is equipped with the $p$-adic analytic
 {\em Tate uniformisation} 
$ \Phi_{\rm Tate}$
of the previous section.

Suppose that $E(K)$ is of rank one. The extended Mordell-Weil group 
$E^\dag(K)$ is spanned (up to torsion) by $q\in \Q_p^\times$ and by a Mordell-Weil generator $P\in E(K)$. 
The role of the cyclotomic $p$-adic height pairing that enters into the Mazur-Tate conjecture 
is now played by an 
{\em anticyclotomic} $p$-adic height
\begin{equation} \langle\ , \ \rangle_K: E^\dag(K) \times E^\dag(K) \lra G_\infty,\end{equation}
whose definition owes everything to the general framework for $p$-adic height pairings initiated by Mazur and Tate.

Remarkably, in this specific instance it turns out that both the point $P$ and the period $q$ are
{\em isotropic} relative to the (extended) anticyclotomic $p$-adic height:
\begin{equation} \langle q,q\rangle_K = \langle P,P\rangle_K = 
0, \end{equation}
while $\langle q,P\rangle_K$
is essentially given by
\begin{equation} \langle q,P\rangle_K := 
\log_p({\tilde P}), \qquad 
\mbox{ where } \Phi_{\rm Tate}({\tilde P}) = P.\end{equation}
The Mazur-Tate regulator in this special case acquires a remarkably simple form, namely, it is given by
\begin{equation} R^\dag(E/K) = \det\left(
\begin{array}{cc}
0 & \log_p(\tilde P) \\
\log_p(\tilde P) & 0 
\end{array} \right) = -\log_p(\tilde P)^2.\end{equation}
The resulting exceptional 
zero conjecture leads to the following theorem:
\begin{theorem}[\cite{bd-gz}]
\label{thm:bd-gzp}
The leading term 
 \begin{equation}\Lp'(E,K_\infty)\in  G_\infty = K_{p,1}^\times\end{equation}
   maps to a global point in $E(K)$ under the Tate uniformisation.
  \end{theorem}
  This result can be viewed   as a $p$-adic 
  analogue of the rank one Stark conjecture, since it recovers a global point on $E$ by applying the Tate ``exponential" to the first derivative of a suitable (anticyclotomic) $p$-adic $L$-function.
  Similar phenomena had  been uncovered earlier  by 
  Karl Rubin \cite{rubin}
  (in the setting of elliptic curves with complex multiplication)
  and  Perrin-Riou 
  \cite{perrin-riou} 
  for modular elliptic curves.
  
  The proof of Theorem 
  \ref{thm:bd-gzp}  relies on an explicit $p$-adic analytic description of {\em Heegner points} arising from the Cerednik-Drinfeld theory of $p$-adic uniformisation of Shimura curves, and it can thus also be viewed as a $p$-adic variant of the Gross-Zagier formula
  \cite{gross-zagier}. 
  That 
the global  point that accounts for the leading term of $\Lp(E,K_\infty)$  
     should be obtained via  a   Heegner point construction is hardly surprising -- to this day, Heegner points provide the only 
 systematic way to construct global points on elliptic curves with direct ties to $L$-function behaviour. But Theorem 
 \ref{thm:bd-gzp} suggests the possibility of transposing the approach of Stark's conjecture to elliptic curves, and of extracting (conjectural) global points on elliptic curves from leading terms of $p$-adic $L$-series in settings where Heegner 
 points arising from the  theory of complex multiplication are entirely unavailable. 
 It is in this spirit that ``elliptic Stark conjectures"   involving Mordell-Weil groups over the fields cut out by  Artin representations attached to modular forms of weight one   were later
  explored in 
  \cite{DLR-pi}.
The notion of ``Stark-Heegner points" over ring class fields of real quadratic fields that will be touched on  in Sections \ref{sec:mod-par} and \ref{sec:stark-heegner}  emerges from a similar perspective.
  
 
 


\section{Mazur-Tate heights}
\label{sec:mt-ht}
In the process of formulating the $p$-adic Birch and Swinnerton-Dyer conjecture of
\cite{MTT}  and its tame variants \cite{MT-bsd}, 
Mazur and Tate were led to a foundational study of N\'eron-Tate style height pairings on the Mordell-Weil group of an elliptic curve over a number field $K$ \cite{MT-heights}. In that level of generality, they  associate a canonical bilinear pairing to any modulus $\fm\subset \cO_K$: 

\begin{equation}
\label{eqn:mtht-bis}
 \langle \ ,\ \rangle_{E,\fm}: E_{\fm}(K) \times E(K) \lra C_{\fm} :=   U_\fm\backslash {\mathbb A}_K^\times/K^\times,
 \end{equation}
where $E_\fm(K)$ is the subgroup of $E(K)$ in the connected component of the N\'eron model of $E$  consisting of points that reduce to $1$ modulo the ideal $\fm$, and $U(\fm)$ is the standard compact open subgroup of the id\'eles of $K$ attached to  $\fm$. The classical (additive) N\'eron-Tate height is recovered by composing
$\langle \ , \rangle_{E,\fm}$ with the logarithm of the id\`elic norm to 
$\R^\times$, and the pairings of \eqref{eqn:mtht} with values in finite Galois groups 
that arise  in 
the statement of the Mazur-Tate conjecture are likewise obtained by composing with the reciprocity map of global class field theory.

 In the special case where $K$ is a real quadratic field (assumed to be of class number one,    to simplify the statements), the pairing \eqref{eqn:mtht-bis} can be parlayed into a pairing with values in a circle group,
 \begin{equation} \langle \ , \ \rangle_{\rm circ}: 
 E(K)\times E(K) \lra \R/\Z\log(\varepsilon_K),\end{equation}
 where $\varepsilon_K$ is the fundamental unit of norm $1$ in $K$, by projecting onto the connected component of the norm $1$  elements of the id\'ele class group.
 
  Assume that all the primes dividing the conductor $N$ of $E$ split in $K/\Q$. This implies that the ring $M_0(N)\subset M_2(\Z)$ of matrices that are upper-triangular modulo $N$ contains a subring isomorphic to $\cO_K$, which is well-defined up to conjugation. The image
  of $\varepsilon_K\in (\cO_K^\times)_1 $ in 
  $\Gamma_0(N)= M_0(N)_1^\times$ gives an element 
  $\gamma_K$ which has two fixed points $\tau_K$ and $\tau_K'$  on the 
  boundary  of the complex upper half-plane.
  
  Let $f$ be the normalised weight two newform on $\Gamma_0(N)$ that is associated to $E$ by  modularity.  The period integral 
  \begin{equation} I_{E,K} =\int_x^{\gamma_K x} f(z) dz\end{equation}
  is an explicit non-zero multiple of 
  $L(E/K,1)$.
 If $I_{E,K}=0$,
   \cite[Lemma 2.6]{BD-circle} shows 
   that the image of the ``derived period integral"
  \begin{equation}
  \label{eqn:IEKp}
   I_{E,K}' = \int_x^{\gamma_K x} \log
  \left(\frac{z-\tau_K}{z-\tau_K'}\right) f(z) dz  \quad 
  \in \quad \R/ \Omega_E^+ \log(\varepsilon_K)     
  \end{equation}
  does not depend on the choice of base point $x\in \PP_1(\Q)$ that was made to define it.
 It can be viewed as a Mazur-Tate style avatar of the ``square root" of the {\em second derivative} of $L(E/K,s)$ 
 at $s=1$. This interpretation motivates the following conjecture
 \begin{conjecture}[\cite{BD-circle}, Conj 3.1]
  \label{conj:BD-circle}
  The period integral of
  \eqref{eqn:IEKp} vanishes when ${\rm rank}(E(K))>2$ and satisfies
 \begin{equation}
 \label{eqn:mt-circle}
  I'_{E,K} \stackrel{?}{=} 
  \langle P^+,P^-\rangle_{\rm circ} \cdot \Omega_E^+ \pmod{\Q^\times\Omega_E^+ \log(\varepsilon_K)}
      \end{equation}
      otherwise,
 where $P^+$ and $P^-$ 
 are Mordell-Weil generators of the eigenspaces $E(K)^+ = E(\Q)$ and $E(K)^-$ for the action of ${\rm Gal}(K/\Q)$. 
 \end{conjecture}
 Conjecture \ref{conj:BD-circle}
  can be envisaged as   an archimedean fragment of a refined Birch and Swinnerton-Dyer  conjecture involving the Mazur-Tate circle pairing, but it is unclear how to make sense of  higher regulators for this  pairing to obtain non-trivial invariants when ${\rm rank}(E(K))>2$.
 
   
 
 
 Like the arguments of Stark units that arise in Questions \ref{question:bcg} and
 \ref{question:sczech},
 the Mazur-Tate circle pairing takes its values in a circle group
 and is not readily extracted from the leading terms of $L$-functions.
Conjecture  \ref{conj:BD-circle}
 is therefore   in somewhat the same spirit as these two questions about Stark units,  providing  a loose analogue of them in the setting of elliptic curves.

 
 
    
\section{The exceptional zero conjecture, revisited}
\label{sec:mod-par}
\medskip
The exceptional zero conjecture \eqref{eqn:exceptional-zero}  of Mazur, Tate and Teitelbaum  is profoundly evocative, and has been framed in several fruitful ways (cf.~\cite{orton} and \cite{breuil}  for example). 
As the authors of 
\cite{MTT} point out,  
it reveals a surprising interplay between the modular symbol of an elliptic curve $E$ arising from its    modular parametrisation over $\C$  and Tate's $p$-adic uniformisation of $E$, suggesting that the modular symbol might be parlayed 
into a $p$-adic variant of the  parametrisation of Shimura-Taniyama-Weil. It turns out that \eqref{eqn:exceptional-zero} can indeed be recast in this light, as we now proceed to explain.
 

Assume for simplicity that $E$ has prime conductor $p$.
The modularity  of $E$  yields
 a cusp form $f_E$ of weight two on $\Gamma_0(p)$, 
 and two 
  period homomorphisms
   \begin{equation} \varphi_E^+, \quad \varphi_E^-  \ \ \in \ \ H^1(\Gamma_0(p),\Z),\end{equation}
   satisfying
 \begin{equation} \int_{z_0}^{\gamma z_0}   f_E(z) dz  \ =  \ 
 \varphi_E^+(\gamma)\cdot  \Omega_E^+ + \varphi_E^-(\gamma) \cdot \Omega_E^-, \quad \mbox{ for all } \gamma\in \Gamma_0(p),\end{equation}
 where $\Omega_E^+$ and
 $\Omega_E^-$ are related to the N\'eron period lattice $\Lambda_E$ of $E$ as in 
 \eqref{eqn:def-Omega}.
 In other words, the class of 
 \begin{equation} \Omega_E^+ \cdot \varphi_E^+ + \Omega_E^- \cdot \varphi_E^- \   \in \  H^1(\Gamma_0(p),\C)\end{equation}
 becomes trivial in   $H^1(\Gamma_0(p),\cA_\infty)$, 
 where $\cA_\infty$ is the space of holomorphic functions on the Poincar\'e upper half-plane equipped with the left translation action $\gamma \cdot F(\tau) := F(\gamma^{-1}\tau)$:
 \begin{equation}
 \label{eqn:perC}
  \Omega_E^+ \cdot \varphi_E^+ + \Omega_E^- \cdot \varphi_E^-  = d F_E, \qquad F_E(\tau) := \int_{i\infty}^\tau  f_E(z) dz.
  \end{equation}
     This property characterises the  lattice 
  $\Lambda_{E}$
    up to scaling and commensurability, and 
    $F_E$ is just a complex realisation of the 
    Weil uniformisation of $E$:
   \begin{equation}
   \label{eqn:weilc}
    F_E: \Gamma_0(p)\backslash \cH \lra \C/\Lambda_E = E(\C).
    \end{equation}       
   \medskip
 It is natural to ask if  the Tate  period 
$q$ of \eqref{eqn:tate-par} admits a similar characterisation: for instance, if 
 the  period homomorphisms
\begin{equation}
\label{eqn:failed}
 q^{\varphi_E^+}, \ q^{\varphi_E^-} \ \in \ H^1(\Gamma_0(p), \C_p^\times) 
 \end{equation} 
can be  trivialised   after replacing $\cA_\infty$ by the multiplicative group $\cA^\times$ of {\em rigid analytic} functions on the Drinfeld 
   $p$-adic upper-half plane 
 \begin{equation}\cH_p:= \PP_1(\C_p) - \PP_1(\Q_p).\end{equation} 
 The fact that $\Gamma_0(p)$ does not act discretely on $\cH_p$   precludes the existence of functions in $\cA^\times$ 
 satisfying the $p$-adic analogue of \eqref{eqn:perC}.
A $p$-adic avatar of $F_E$ is obtained instead
 from the cohomology of
 the {\em Ihara group}
\begin{equation}\Gamma \ =\  \SL_2(\Z[1/p]) \ = \
 \SL_2(\Z) *_{\Gamma_0(p)} \SL_2(\Z).\end{equation}
 The Mayer-Vietoris sequence in group cohomology associated to its
  description as
  an amalgamated product of two copies of $\SL_2(\Z)$ along a common 
copy of $\Gamma_0(p)$  leads to 
 a canonical inclusion with finite cokernel
\begin{equation} H^1(\Gamma_0(p),\Z) \hookrightarrow H^2(\Gamma,\Z). \end{equation}
The
 homomorphisms $\varphi_E^+$ and $\varphi_E^-$  can thus be transferred to classes
$\alpha_E^+$ and $\alpha_E^- \in H^2(\Gamma,\Z)$, for which
the following $p$-adic analogue of 
\eqref{eqn:perC}  becomes available:
\begin{theorem}[\cite{darmon-hpxh}]
 \label{thm:uniformisation}
The classes $q^{\alpha_E^\pm}\in H^2(\Gamma,\C_p^\times)$ become  trivial in $H^2(\Gamma,\cA^\times)$. 
In particular, there
are one-cochains $J_E^\pm:\Gamma\lra \cA^\times$  satisfying  
\begin{equation} q^{\alpha_E^+} = d J_E^+, 
\qquad
q^{\alpha_E^-} = d J_E^-.\end{equation}
\end{theorem}
Although the statement of this theorem resonates  with \eqref{eqn:perC}, its proof is in a very different spirit. The complex periods of $E$
are related to those of a weight two cusp form
 by realising $E$ as a quotient of a modular jacobian,  so the proof of 
  \eqref{eqn:perC} relies on the full force of Falting's proof of the Tate conjecture for curves. 
 Theorem \ref{thm:uniformisation}
 is shown in 
 \cite{darmon-hpxh} to be a direct formal consequence of
 the exceptional zero conjecture \eqref{eqn:exceptional-zero} of \cite{MTT}, whose   proof by Ralph Greenberg and Glenn Stevens 
 \cite{greenberg-stevens} uses     techniques of Galois deformation theory,
 notably, the two-variable Mazur-Kitagawa $p$-adic $L$-function attached to a Hida family interpolating the newform $f_E$ in weight two. Although in a sense  Theorem 
 \ref{thm:uniformisation}  
was conjectured by Mazur, Tate and Teitelbaum, and proved  in 
\cite{greenberg-stevens}
(more than a decade before it was stated!),   
the geometry 
 underlying Theorem \ref{thm:uniformisation}   remains unclear, lending it
  an air of mystery which has yet to be dispelled.

Just as the function $F_E$ becomes $\Gamma_0(p)$-invariant modulo periods and therefore gives rise to a class
\begin{equation}
\label{eqn:FE}
 F_E \ \in \ H^0(\Gamma_0(p), \cA_\infty/\Lambda_{E}),\end{equation}
 the one-cochains $J_E^+$ and $J_E^-$ of
Theorem \ref{thm:uniformisation}
 satisfy the one-cocycle relation modulo $q^\Z$,  and therefore give rise to classes
\begin{equation} J_E^+, \ J_E^- \ \in \  H^1(\Gamma, \cA^\times/q^\Z)\end{equation}
  analogous to the modular parametrisation
 $F_E$ of
 \eqref{eqn:perC} and 
 \eqref{eqn:FE}.
  They are called the (even and odd) {\em elliptic 
  cocycles} attached to $E$.
  In passing from  $F_E$ to its $p$-adic counterparts $J_E^+$ and $J_E^-$, 
  the
  shift in cohomological degrees
is even more significant than the transition from complex to $p$-adic analysis.



 \section{Stark-Heegner points}
 \label{sec:stark-heegner}
 \medskip   
  The values of the complex uniformisation $F_E$  of 
  \eqref{eqn:weilc} at quadratic imaginary arguments  of the Poincar\'e upper half-plane  give rise to a plentiful collection of
   global points on $E$ defined over ring class fields of imaginary quadratic fields. The elliptic cocycles  $J_E^\pm$ can likewise be evaluated at {\em real quadratic} elements of $\cH_p$, which are characterised by the fact that their stabiliser in $\Gamma$ is an abelian group of rank one, with a generator $\gamma_\tau$ (the so-called {\em automorph} of $\tau$). The value of $J_E= J_E^+$ or $J_E^-$  at $\tau$ is defined to be
\begin{equation} J_E[\tau] := J_E(\gamma_\tau)(\tau) \in \C_p^\times/q^\Z = E(\C_p).\end{equation}
\begin{conjecture}[\cite{darmon-hpxh}] 
\label{conj:stark-heegner}
 The value  $J_E[\tau]$ at a real quadratic 
  $\tau\in \cH_p$ is a  global point  on $E$  defined over an explicit  ring class field of   $K:=\Q(\tau)$. 
 \end{conjecture}
 These  ``real multiplication values" are called {\em Stark-Heegner points}. They  are expected to 
  behave ``just like" the Heegner points $F_E(\tau)$
  when $\tau$ is imaginary quadratic, but lie beyond the scope of the theory of complex multiplication.
  
While a proof of Conjecture \ref{conj:stark-heegner}
would seem to require    substantial new ideas,   extensive numerical evidence has been gathered in its support, thanks to efficient algorithms \cite{darmon-pollack}
 for calculating elliptic cocycles based on the theory of ``overconvergent modular symbols" elaborated by Robert Pollack and Glenn Stevens.
 For 
the elliptic curve
\begin{equation} E_{11} = 11A: y^2+y = x^3 -x^2 -10 x -20\end{equation}
of conductor $11$ 
and the field $\Q(\sqrt{101})$ of class number one, a numerical evaluation reveals that
\begin{equation} J_{E_{11}}^+\left[\frac{1+\sqrt{101}}{2}\right] = (x/t^2, y/t^3)  \pmod{11^{200}},\end{equation} 
 where 
\begin{eqnarray*}
x &=& 1081624136644692539667084685116849 \\
y &=& -1939146297774921836916098998070620047276215775500 \\ && - 450348132717625197271325875616860240657045635493 \sqrt{101} \\
t &=& 15711350731963510, 
\end{eqnarray*}
  reconstructing a global point of fairly large height in $E_{11}(\Q(\sqrt{101}))$ without any a priori knowledge of it.
  
The real quadratic  field $K=\Q(\sqrt{1297})$ has class number $11$ and its Hilbert class field $H$ is therefore a degree $11$ extension of $K$.
Let $E_{37}$ be the elliptic curve
 \begin{equation}
 E_{37} = 37A: y^2+y = x^3 -x \end{equation}
 of conductor $37$.
 An analysis of signs in functional equations shows
 that  
 \begin{equation}
  {\rm ord}_{s=1} L(E_{37}/H, s) \ge 11,
  \end{equation}
 suggesting that $E_{37}(H)$ has rank $\ge 11$.
  A numerical calculation reveals that 
\begin{equation} J_{E_{37}}^+\left[\frac{1+\sqrt{1297}}{2}\right] =
 (x,y)  \pmod{37^{50}},\end{equation}
where $x$ satisfies the polynomial
\begin{eqnarray*} && \!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\! 
961 x^{11} -4035 x^{10} -3868 x^9 +19376 x^8 + 13229 x^7 
-27966 x^6   - 21675 x^5  \\
&& \qquad +11403 x^4 + 11859 x^3 +1391 x^2 -369 x - 37, 
  \end{eqnarray*}
whose roots generate  $H$ and are the $x$ coordinates of global points on $E_{37}(H)$.

\medskip
Samit Dasgupta's PhD thesis \cite{dasgupta-phd}
\cite{darmon-dasgupta}
puts the   analogy between Stark-Heegner points and Gross-Stark units into sharper focus by replacing the modular symbols attached to the cusp form $f_E$ in the  construction of  $J_E^\pm$ by the Dedekind-Rademacher 
homomorphism encoding the periods of weight two Eisenstein series. The   values of the resulting ``Dedekind-Rademacher cocycle" at elements of  $K$   turn out to be Gross-Stark $p$-units in ring class fields of $K$, whose local norms to $\Q_p$ encode the derivatives at $s=0$ of $p$-adic $L$-series attached to (totally odd) ring class characters of $K$. Since this formalism leads to a conjectural recipe for the Gross-Stark units themselves and not just their norms to $\Q_p$, it provides a genuine strengthening of the Gross-Stark conjecture and points to a positive answer to a $p$-adic analogue of 
Question \ref{question:bcg}.
The construction
of \cite{dasgupta-phd} is now unconditional thanks to the proof by Dasgupta and Kakde of Gross's tame refinement of the Gross-Stark conjecture alluded to at the end of 
Sec.~\ref{sec:MT}.
(See also \cite{DPV}.)

\medskip
 Dasgupta's thesis is a prototypical instance of the fruitful dictionary between  unit groups and elliptic curves, exemplified  by the passage from 
 the Mazur-Tate element of 
 \eqref{eqn:thetaEM}
 to Tate's Stickelberger element in the Brumer-Stark conjecture.
The Eisenstein cocycle
 \eqref{eqn:kappaeis}
 of \cite{charollois-darmon} was guided by a similar analogy 
 with the elliptic cocycle
 of \cite{darmon-logan} attached to a Hilbert cuspidal eigenform (attached, say, to an elliptic curve $E$ over the totally real field $F$),
 whose values at  points 
 in a  quadratic ATR extension $K$  of $F$ lead to a conjectural   archimedean recipe for points on $E$ over abelian extensions of $K$. 


 
 
 
\medskip
The rich store of parallel constructions of units in number fields and  global
  points on elliptic curves  in settings  that  lie   outside the   purview of   the theory of complex multiplication hints at  a    synthesis of the 
 themes   Tate explored   in his last decade at Harvard: the Stark conjectures and $p$-adic variants of the Birch and Swinnerton-Dyer conjecture.
The  influence of Tate's work during this period manifests itself  not just through  the depth and importance of his theorems, 
   but  in how his perspectives have  set the stage  for later developments and continue to frame our understanding of   the subject.
   




 
  
 
\begin{thebibliography}{DLR15a}

\bibitem[AFK26]{AFK}
M.~Appleby, S.T.~Flammia, and G.S.~Kopp. 
{\em A constructive approach to Zauner's conjecture via the Stark conjectures}.
Forum Math Pi, to appear.


\bibitem[AT67]{ArtinTate} E.~Artin, J.~Tate, 
\emph{Class Field Theory}, 
AMS Chelsea Publishing, 2nd ed., 2009.

\bibitem[AYZ13]{AYZ}
D.M.~Appleby, H.~Yadsan-Appleby, and
G.~Zauner.
{\em Galois automorphisms of a symmetric measurement}.
Quant.~Inf.~Comput. {\bf 13} 672--720 (2013)

\bibitem[BCG23]{BCG}
N.~Bergeron, P.~Charollois, and L.~Garcia. 
{\em Elliptic units for complex cubic fields},
preprint.

\bibitem[BD96]{BD-mumford-tate}
M.~Bertolini and H.~Darmon.
{\em Heegner points on Mumford-Tate curves}. 
Invent. Math. {\bf 126} (1996), no. 3, 413--456.

\bibitem[BD98]{bd-gz}
M.~Bertolini and H.~Darmon.
{\em Heegner points, $p$-adic $L$-functions, and the Cerednik-Drinfeld uniformization}.
Invent. Math. {\bf 131} (1998), no. 3, 453--491.

\bibitem[BD04]{BD-circle}
M.~Bertolini and H.~Darmon, {\em A Birch and Swinnerton-Dyer conjecture for the Mazur-Tate circle pairing}. Duke Math. J. {\bf 122} (2004), no. 1, 181--204.

\bibitem[Bl25]{black-thesis}
K.~G.~Black, 
{\em A refinement of the Stark conjecture over ATR fields}.
PhD thesis, Duke University, 2025.


\bibitem[Br10]{breuil}
C.~Breuil. 
{\em S\'erie sp\'eciale $p$-adique et cohomologie \'etale compl\'et\'ee}.
Ast\'erisque {\bf 331} (2010) 65--115.


\bibitem[CD08]{charollois-darmon}
P.~Charollois and H.~Darmon.
{\em Arguments des unit\'es de Stark et p\'eriodes de s\'eries d'Eisenstein}.
Algebra and Number Theory {\bf 2} (2008) no.6, 655-688.

 \bibitem[CN79]{cassou}
 P.~Cassou-Nogu\`es.
 {\em Valeurs aux entiers n\'egatifs des fonctions z\^etaet fonctions z\^eta
 $p$-adiques},
 Inventiones Math.~{\bf 51} (1979) 29--59.
 
 \bibitem[Dar92]{darmon-thesis}
 H.~Darmon.
 {\em  A refined conjecture of Mazur-Tate type for Heegner points}.
Invent. Math. {\bf 110} (1992), no. 1, 123--146.
 
 \bibitem[Dar95]{darmon-cjm}
 H.~Darmon.
 {\em Thaine's method for circular units and a conjecture of Gross}. Canad.~J.~Math {\bf 47}
 (1995) 302--317.
 

\bibitem[Dar01]{darmon-hpxh} 
H.~Darmon, 
\emph{Integration on $\mathcal{H}_p \times \mathcal{H}$ and arithmetic applications}, 
Ann. of Math. (2) \textbf{154} (2001), 589--639.

\bibitem[Das99]{dasgupta-thesis}
S.~Dasgupta.
{\em Stark's conjectures}.
Undergraduate senior thesis, Harvard University, 1999.

\bibitem[Das04]{dasgupta-phd}
S.~Dasgupta.
{\em Gross-Stark Units, Stark-Heegner Points, and Class Fields of Real Quadratic Fields},
PhD. Thesis, University of California-Berkeley, May 2004. 

\bibitem[DD06]{darmon-dasgupta}
 H.~Darmon and S.~Dasgupta. {\em
Elliptic Units for Real Quadratic Fields}.
Annals of Mathematics, {\bf 163} (2006), no. 1, 301--345. 

\bibitem[DDP11]{DDP}
S.~Dasgupta, H.~Darmon, and R.~Pollack. 
{\em Hilbert modular forms and the Gross-Stark conjecture}.
Ann. of Math.~{\bf 174} (2011) 439--484.

\bibitem[DKV18]{DKV}
S.~Dasgupta, M.~Kakde, and K.~Ventullo. 
{\em On the Gross-Stark conjecture}.
Ann. of Math.(2) {\bf 188} (2018) 833--870.


\bibitem[DK23]{DK-bs-ref}
S.~Dasgupta and M.~Kakde. 
{\em On the Brumer--Stark Conjecture and Refinements}. 
Proceedings of the ICM.   (2022) 1768--1799.  




\bibitem[DK24]{DK-bs}
S.~Dasgupta and M.~Kakde, 
{\em 
Brumer-Stark units and explicit class field theory.}  Duke Math. J. {\bf 173} (2024), no. 8, 1477--1555.

 
 
 \bibitem[DKSW]{DKSW}
S.~Dasgupta, M.~Kakde, J.~Silliman, and J.~Wang. {\em The Brumer--Stark Conjecture over $\Z$}. To appear.


\bibitem[DL03]{darmon-logan}
H.~Darmon and A.~Logan.
{\em Periods of Hilbert modular forms and rational points on elliptic curves}.
Int.~Math.~Res.~Not. (2003)   {\bf 40}, 2153--2180.
 






\bibitem[DLR15a]{DLR-advances}
H.~Darmon, A.~Lauder, and V.~Rotger. 
{\em Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields}. 
Adv.~Math.~{\bf 283} (2015) 130--142.

 \bibitem[DLR15b]{DLR-pi}
 H.~Darmon, A.~Lauder and V.~Rotger.
 {\em 
Stark points and $p$-adic iterated integrals attached to modular forms of weight one}.
Forum of Mathematics, Pi (2015), Vol. 3, e8, 95 pages.



\bibitem[DLR17]{DLR-heidelberg}
H.~Darmon, A.~Lauder, and V.~Rotger.
{\em First order $p$-adic deformations of weight one newforms}.
Contrib.~Math.~Comput.~Sci {\bf 10} Springer (2017) 39--80.



\bibitem[DP06]{darmon-pollack}
H.~Darmon and R.~Pollack, 
 {\em Efficient calculation of Stark-Heegner points via overconvergent modular symbols}
Israel J. Math. {\bf 153} (2006), 319--354.

\bibitem[DPV24]{DPV}
H.~Darmon, A.~Pozzi, and J.~Vonk. {\em The values of the Dedekind-Rademacher cocycle at real multiplication points.}
J. Eur. Math. Soc. (JEMS) {\bf 26} (2024), no. 10, 3987--4032.
 


\bibitem[DR80]{deligne-ribet}
P.~Deligne and K.~Ribet.
{\em Values of abelian $L$-functions at negative integers over totally real
fields}. 
Invent.~Math.~{\bf 59} (1980) 227--286.

\bibitem[DS74]{deligne-serre}
P.~Deligne and J.-P.~Serre.
{\em Formes modulaires de poids 1}. 
Ann.~Sci.~Ecole Norm.~Sup. (4) {\bf 7} (1974) 507--530.



\bibitem[FV00]{felder-varchenko}
G.~Felder and A.~Varchenko.
{\em The elliptic Gamma function $\SL(3,\Z)\ltimes \Z^3$}. 
Advances in Math {\bf 156}(1):44--76 (2000)




\bibitem[Gr81]{gross-stark}
B.H. Gross. {\em $p$-adic $L$-series at $s=0$}. 
J. Fac. Sci. Univ. Tokyo Sect. IA Math.  {\bf 28} (1981), no. 3, 979--994.

\bibitem[Gr88]{gross-tower}
B.H.Gross. {\em On the values of abelian $L$-functions at $s=0$}. 
J. Fac. Sci. Univ. Tokyo Sect. IA Math. {\bf 35} (1988), no. 1, 177--197.
 
\bibitem[GS93]{greenberg-stevens} R.~Greenberg and G.~Stevens, 
\emph{$p$-adic $L$-functions and $p$-adic periods of modular forms}, 
Invent. Math. \textbf{111} (1993), 407--447.

 

\bibitem[GZ86]{gross-zagier}
B.H.~Gross and D.B.~Zagier, 
\emph{Heegner points and derivatives of $L$-series}, 
Invent. Math. \textbf{84} (1986), 225--320.

\bibitem[HV19]{HV}M.~Harris and  A.~Venkatesh, 
\emph{Derived Hecke algebra for weight one forms}, 
Exp. Math. \textbf{28} (2019), 342--361. 

\bibitem[Ka04]{kato}
K.~Kato, 
{\em $p$-adic Hodge theory and values of zeta functions of modular forms}. In
Cohomologies p-adiques et applications arithm\'tiques. III
Ast\'erisque {\bf 295} (2004), ix, 117--290.

\bibitem[Kol90]{Kolyvagin1990} V.~Kolyvagin, 
\emph{Euler systems}, 
The Grothendieck Festschrift, Vol. II, 
Progr. Math. {\bf 87}, Birkh\"auser (1990) 435--483.

\bibitem[Kop21]{kopp}
G.S.~Kopp. 
{\em SIC-POVMs and the Stark conjectures}.
Int.~Math.~Res.~Not.~IMRN
{\bf 18} (2021) 13812--13838.

\bibitem[Kop25]{kopp-shin}
G.S.~Kopp.
 {\em
The Shintani--Faddeev modular cocycle: Stark units from  $q$-Pochhammer ratios}.
Submitted.


\bibitem[Ma72]{mazur}
B.~Mazur. 
{\em Rational points on abelian varieties with values in towers of number fields}.
Invent.~Math. {\bf 18} (1972) 183--266.


\bibitem[Mo24]{morain}
P.~Morain.
{\em Elliptic units above fields with exactly one complex place}.
Preprint, to appear.

\bibitem[MR11]{mazur-rubin-Gm}
B.~Mazur and K.~Rubin.
{\em Refined class number formulas and Kolyvagin systems}.
Compos.~Math.{\bf 147} (2011) 56--74.

\bibitem[MS74]{MSwD}
B.~Mazur and P.~Swinnerton-Dyer.
{\em Arithmetic of Weil curves}.
Invent.~Math.~{\bf 25} 
(1974) 1--61.

\bibitem[MT83]{MT-heights}
B.~Mazur and J.~Tate.
{\em Canonical height pairings via biextensions}.
Progr.~Math.~{\bf 35} Birkh\"auserBoston (1983) 195--237.

\bibitem[MT87]{MT-bsd} B.~Mazur and J.~Tate, 
\emph{Refined conjectures of the ``Birch and Swinnerton--Dyer type''}, 
Duke Math. J. \textbf{54} (1987), 711--750.

\bibitem[MTT86]{MTT} B.~Mazur, J.~Tate, and J.~Teitelbaum, 
\emph{On $p$-adic analogues of the conjectures of Birch and Swinnerton--Dyer}, 
Invent. Math. \textbf{84} (1986), 1--48.

\bibitem[Or04]{orton}
L.~Orton.
{\em An elementary proof 
of a weak exceptional zero conjecture}. 
Canad.~J.~Math {\bf 56} (2004) 373--405.

\bibitem[Ot18]{ota}
K.~Ota. 
{\em Kato's Euler system and the Mazur-Tate refined conjecture of BSD type.}
Amer.~J.~Math. {\bf 140} 
(2018) 495--542.

\bibitem[PR93]{perrin-riou}
B.~Perrin-Riou.
{\em Fonctions $L$ $p$-adiques d'une courbe elliptique et points rationnels.}
Ann.~Inst.~Fourier (Grenoble) {\bf 43} (1993) no. 4, 945--995.


\bibitem[RBSC04]{RBSC}
J.M.~Renes, R.~Blume-Kohout, A.J.~Scott,and C.M.~Caves.
{\em Symmetric informationally complete 
quantum measurements}.
J.~Math.~Phys. {\bf 45} (6): 2171--2180. (2004)
 




\bibitem[RS09]{ren-sczech}
T.~Ren and R.~Sczech. 
{\em A refinement of Stark's conjecture over complex cubic number fields.}
J. Number Theory {\bf 129}
(2009) no.~4, 831--857.

\bibitem[Ru92]{rubin}
K.~Rubin.
{\em  $p$-adic $L$-functions and rational points on elliptic curves with complex multiplication}.
Invent. Math. {\bf 107} (1992), no. 2, 323--350.
 
 
\bibitem[Sa14]{sano}
T.~Sano. 
{\em A generalization of Darmon's conjecture or Euler systems for general
$p$-adic representations}.
J.~Number Theory {\bf 144} (2014) 281--324.


\bibitem[Se73]{serre-lnm}
J.-P.~Serre.
{\em Formes modulaires et fonctions z\^eta 
$p$-adiques}. Lecture Notes in Math {\bf 350} (1973) 191--268.

\bibitem[Se78]{serre-zeta}
 J.-P. Serre, {\em Sur le r\'esidu de la fonction z\^eta $p$-adique d'un corps de nombres}, C.R. Acad.~Sc.~Paris s\'erie A, {\bf 287} (1978)  83--126.

\bibitem[So02]{solomon}
D.~Solomon. {\em
$p$-adic Abelian Stark conjectures at $s = 1$}.
Annales Inst.~Fourier {\bf 52}, no 2 (2002), p. 379--417.
 
\bibitem[ST15]{serre-tate}
J.-P.~Serre and J.~Tate.
Correspondance Serre-Tate, Vol II., P.~Colmez and J.P.~Serre eds., in 
Documents Math\'ematiques, Soci\'et\'e Math\'ematique de France (2015).

 

\bibitem[St71]{Stark1971} H.M.~Stark, 
\emph{$L$-functions at $s=1$. I. $L$-functions for quadratic characters}, 
Adv. Math. \textbf{7} (1971), 301--343.


\bibitem[St76]{stark-3}
H.M.~Stark. 
{\em $L$-functions at $s=1$. III. Totally real fields and Hilbert's twelfth 
problem.}
Advances in Math {\bf 22}
(1976) no. 1, 64--84.




\bibitem[Ta74]{Tate-EC}
J.~Tate, {\em 
The arithmetic of elliptic curves}.
Invent. Math. {\bf 23} (1974), 179--206.
 
\bibitem[Ta84]{TateStark} J.~Tate, 
\emph{Les conjectures de Stark sur les fonctions $L$ d'Artin en $s=0$}, 
Progress in Math. 47, Birkh\"auser, 1984. 

\bibitem[Za99]{zauner-thesis}
G.~Zauner. 
{\em Quantendesigns --- Grundz\"uge einer nichtcommutativen Designtheorie.}
PhD thesis, University of
Vienna, 1999.

\bibitem[Za11]{zauner}
G.~Zauner.
{\em Quantum designs: foundations of a noncommutative design theory}.
Int. J. Quantum Inf. {\bf 9} (2011), no. 1, 445--507.

\bibitem[Zh26]{zhang}
R.~Zhang.
{\em The Harris--Venkatesh conjecture for derived Hecke operators II: a unified Stark conjecture}. 
Preprint, submitted.

\end{thebibliography}

\end{document}
 