1.1 Setting the stage

Let $(p,k)$ be an irregular pair – that is, $p$ is an irregular prime such that $p$ divides the numerator of $B_{k}$ , the $k$ th Bernoulli number. In [Reference RibetRib76], Ribet proved the converse of Herbrand’s theorem which predicts the non-triviality of a particular eigenspace of the $p$ -part of the class group of $\mathbb{Q}(\unicode[STIX]{x1D707}_{p})$ under the action of $\operatorname{Gal}(\mathbb{Q}(\unicode[STIX]{x1D707}_{p})/\mathbb{Q})$ . His method exploited a congruence between an Eisenstein series and a cuspform. Ribet worked in weight 2 at level $p$ , but we describe the idea here in weight $k$ and level $1$ ; namely, let

$$\begin{eqnarray}E_{k}=-\frac{B_{k}}{2k}+\mathop{\sum }_{n\geqslant 1}\unicode[STIX]{x1D70E}_{k-1}(n)q^{n}\end{eqnarray}$$

denote the Eisenstein series of weight $k>2$ on $\operatorname{SL}_{2}(\mathbb{Z})$ . Under the assumption that $(p,k)$ is an irregular pair, the constant term of $E_{k}$ vanishes mod $p$ , and thus $E_{k}$ ‘looks like’ a cuspform modulo  $p$ . One can make this idea precise and prove the existence of a cuspidal eigenform $g_{k}$ of weight $k$ and level 1 which is congruent to the Eisenstein series $E_{k}$ . The mod $p$ Galois representation of $g_{k}$ is then reducible, and from this Galois representation one can extract the desired unramified extension of $\mathbb{Q}(\unicode[STIX]{x1D707}_{p})$ .

Wiles in [Reference WilesWil90] pushed this argument further by looking at the whole Hida family of the $g_{k}$ as $k$ varies $p$ -adically. By analyzing the intersection of the Eisenstein and cuspidal branches of this Hida family (in the Hilbert modular case), Wiles proved the main conjecture over totally real fields.

In this paper, rather than looking at the Iwasawa theory of class groups, we will instead focus on the Iwasawa theory of these famous cuspforms $g_{k}$ in their own right. Namely, we will examine the $p$ -adic $L$ -functions and Selmer groups of the $g_{k}$ as $k$ varies $p$ -adically. In fact, within the paper, we will study a larger collection of cuspforms with reducible residual representations by allowing for congruences with Eisenstein series with characters. However, to keep things simple, for the remainder of the introduction we will continue to work with tame level $N=1$ .

To further set the stage, let us recall the Hida theoretic picture of congruences between Eisenstein series and cuspforms. Namely, to put the $E_{k}$ in a $p$ -adic family, we must consider $E_{k}^{\operatorname{ord}}:=E_{k}(z)-p^{k-1}E_{k}(pz)$ , the ordinary $p$ -stabilization to $\unicode[STIX]{x1D6E4}_{0}(p)$ . Explicitly, we have

$$\begin{eqnarray}E_{k}^{\operatorname{ord}}=-(1-p^{k-1})\frac{B_{k}}{2k}+\mathop{\sum }_{n\geqslant 1}\unicode[STIX]{x1D70E}_{k-1}^{(p)}(n)q^{n}\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{k-1}^{(p)}(n)$ is the sum of the $(k-1)$ th powers of the prime-to- $p$ divisors of $n$ . Since $d^{k}$ is a $p$ -adically continuous function in $k$ as long as $(d,p)=1$ , we see that the functions $\unicode[STIX]{x1D70E}_{k-1}^{(p)}(n)$ satisfy nice $p$ -adic properties.

Specifically, let ${\mathcal{W}}=\operatorname{Hom}_{\operatorname{cont}}(\mathbb{Z}_{p}^{\times },\mathbb{C}_{p}^{\times })$ denote weight space which is isomorphic to $p-1$ copies of the open unit disc of $\mathbb{C}_{p}$ . Let ${\mathcal{W}}_{j}$ denote the disc of weights with tame character $\unicode[STIX]{x1D714}^{j}$ where $\unicode[STIX]{x1D714}$ is the mod $p$ cyclotomic character. We view $\mathbb{Z}{\hookrightarrow}{\mathcal{W}}$ by identifying $k$ with the character $z\mapsto z^{k}$ . Then there exist rigid analytic functions $A_{n}$ on ${\mathcal{W}}$ such that $A_{n}(k)=\unicode[STIX]{x1D70E}_{k-1}^{(p)}(n)$ for each $n\geqslant 1$ .

The constant term of $E_{k}^{\operatorname{ord}}$ also enjoys nice $p$ -adic properties. For $k>2$ even, we have $-(1-p^{k-1})(B_{k}/2k)=\frac{1}{2}\unicode[STIX]{x1D701}_{p}(k)$ where $\unicode[STIX]{x1D701}_{p}(\unicode[STIX]{x1D705})$ is the $p$ -adic $\unicode[STIX]{x1D701}$ -function as in [Reference ColmezCol00, Reference Bellaïche and DasguptaBD15]. In particular, the formal $q$ -expansion

$$\begin{eqnarray}{\mathcal{E}}_{\unicode[STIX]{x1D705}}=\frac{1}{2}\unicode[STIX]{x1D701}_{p}(\unicode[STIX]{x1D705})+\mathop{\sum }_{n\geqslant 1}A_{n}(\unicode[STIX]{x1D705})q^{n}\end{eqnarray}$$

defines the $p$ -adic Eisenstein family in the weight variable $\unicode[STIX]{x1D705}$ over all even components of weight space.

Note further that if $\unicode[STIX]{x1D705}_{0}$ is a zero of $\unicode[STIX]{x1D701}_{p}(\unicode[STIX]{x1D705})$ (which necessarily cannot be a classical weight), then the weight $\unicode[STIX]{x1D705}_{0}$ Eisenstein series ${\mathcal{E}}_{\unicode[STIX]{x1D705}_{0}}$ ‘looks cuspidal’. In the spirit of Ribet’s proof of the converse to Herbrand’s theorem, one can show that there is a cuspidal Hida family which specializes at weight $\unicode[STIX]{x1D705}_{0}$ to an Eisenstein series. That is, the weights of the crossing points of the cuspidal and Eisenstein branches of the Hida family occur precisely at the zeros of $\unicode[STIX]{x1D701}_{p}(\unicode[STIX]{x1D705})$ .

We set some notation. Let $\mathbb{T}$ denote the universal ordinary Hecke algebra of tame level $N=1$ acting on ordinary modular forms of all weights (as in [Reference HidaHid86]) which is a $\unicode[STIX]{x1D6EC}$ -module where $\unicode[STIX]{x1D6EC}=\mathbb{Z}_{p}[[1+p\mathbb{Z}_{p}]]$ . Let $\mathbb{T}^{c}$ denote the quotient of $\mathbb{T}$ corresponding to the cuspidal Hecke algebra, and let $\mathfrak{m}^{c}\subseteq \mathbb{T}^{c}$ denote some maximal ideal containing the cuspidal Eisenstein ideal. Note that the choice of such an ideal implicitly chooses some disc ${\mathcal{W}}_{j}$ of weight space over which our Hida family sits and for which $(p,j)$ is an irregular pair.

For simplicity and concreteness, we begin by imposing the hypothesis

(Cuspidal rank one) $$\begin{eqnarray}\dim _{\unicode[STIX]{x1D6EC}}\mathbb{T}_{\mathfrak{m}^{c}}^{c}=1\end{eqnarray}$$

so that the geometry of the Hida family is as simple as possible. We note that this condition is equivalent to there being a unique cuspidal eigenform $f_{k}$ in weight $k$ congruent to $E_{k}^{\operatorname{ord}}$ for one (equivalently any) $k\equiv j\;(\text{mod}\,p-1)$ . The condition (Cuspidal rank one) certainly does not hold for all irregular pairs $(p,k)$ . But we checked that it does hold for all irregular pairs $(p,k)$ with $p<10^{5}$ with the exception of $p=547$ and $k=486$ . In this case, $\dim _{\mathbb{Z}_{p}}\mathbb{T}_{\mathfrak{m}^{c}}^{c}=2$ .

1.2 $p$ -adic $L$ -functions and mod $p$ multiplicity one

Each of the cuspforms $f_{k}$ have an associated $p$ -adic $L$ -function $L_{p}^{+}(f_{k})$ , which can equivalently be viewed as a $\mathbb{Z}_{p}$ -valued measure on $\mathbb{Z}_{p}^{\times }$ or as an element in the completed group algebra $\mathbb{Z}_{p}[[\mathbb{Z}_{p}^{\times }]]$ . The $+$ superscript here indicates that this is the $p$ -adic $L$ -function supported only on even characters. Moreover, these $p$ -adic $L$ -functions vary $p$ -adic analytically in $k$ , yielding a two-variable $p$ -adic $L$ -function as in [Reference KitagawaKit94, Reference Greenberg and StevensGS93].

Under (Cuspidal rank one), we can view the two-variable $p$ -adic $L$ -function $L_{p}^{+}(\unicode[STIX]{x1D705})$ as an element in $\mathbb{Z}_{p}[[\unicode[STIX]{x1D6E4}_{w,j}\times \mathbb{Z}_{p}^{\times }]]$ where $\unicode[STIX]{x1D6E4}_{w,j}=1+p\mathbb{Z}_{p}$ . Here, we think of $\unicode[STIX]{x1D705}$ as ranging over weights in ${\mathcal{W}}_{j}$ . That is, under this assumption, we can conflate a parameter of the Hida family with a parameter of ${\mathcal{W}}_{j}$ . At each classical weight in ${\mathcal{W}}_{j}$ , the two-variable $p$ -adic $L$ -function specializes to a one-variable $p$ -adic $L$ -function. That is, for $k\in \mathbb{Z}^{{\geqslant}2}\cap {\mathcal{W}}_{j}$ we have

$$\begin{eqnarray}L_{p}^{+}(\unicode[STIX]{x1D705})|_{\unicode[STIX]{x1D705}=k}=L_{p}^{+}(f_{k}).\end{eqnarray}$$

One can also naturally attach $p$ -adic $L$ -functions to Eisenstein series (e.g. via overconvergent modular symbols as in [Reference Greenberg and StevensGS93]), and somewhat surprisingly the result is simply the zero distribution (see Theorem 2.3).Footnote ¹

The key idea of this paper is to use the vanishing of the $p$ -adic $L$ -functions on the Eisenstein branch and the intersection points of the Eisenstein and cuspidal branches to make deductions about the cuspidal $p$ -adic $L$ -functions. Indeed, if there were a two-variable $p$ -adic $L$ -function defined simultaneously over both the Eisenstein and cuspidal branches, the vanishing of the $p$ -adic $L$ -function on the Eisenstein branch would imply the vanishing of the $p$ -adic $L$ -functions at the crossing points of the branches. This in turn would mean that $L_{p}^{+}(f_{k})$ becomes more and more divisible by $p$ as $k$ moves close to one of these crossing points – i.e. to one of the zeros of $\unicode[STIX]{x1D701}_{p}(\unicode[STIX]{x1D705})$ . In the language of Iwasawa theory, this means that the $\unicode[STIX]{x1D707}$ -invariants in the Hida family blow up as you approach these zeros!

We now set some further notation and introduce a mod $p$ multiplicity-one condition which helps explain this phenomenon. Let $\mathbb{T}_{k}$ (respectively, $\mathbb{T}_{k}^{c}$ ) denote the full Hecke algebra over $\mathbb{Z}_{p}$ which acts faithfully on $M_{k}(\unicode[STIX]{x1D6E4}_{0}(p))^{\operatorname{ord}}$ (respectively, $S_{k}(\unicode[STIX]{x1D6E4}_{0}(p))^{\operatorname{ord}}$ ). Let $\mathfrak{m}_{k}$ denote the maximal ideal of $\mathbb{T}_{k}$ corresponding to $E_{k}^{\operatorname{ord}}$ , and let $\mathfrak{m}_{k}^{c}$ denote the image of $\mathfrak{m}_{k}$ in $\mathbb{T}_{k}^{c}$ . Let $(p,k)$ be an irregular pair, in which case $\mathfrak{m}_{k}^{c}$ is a maximal ideal (i.e. it is a proper ideal). We consider the condition

(Mult One) $$\begin{eqnarray}\dim _{\mathbb{F}_{p}}(H_{c}^{1}(\operatorname{SL}_{2}(\mathbb{Z}),{\mathcal{P}}_{k-2}^{\vee }\,\otimes \,\mathbb{F}_{p})^{+}[\mathfrak{m}_{k}])=1\end{eqnarray}$$

where ${\mathcal{P}}_{g}$ is the collection of $\mathbb{Q}_{p}$ -polynomials of degree less than or equal to $g$ which preserve $\mathbb{Z}_{p}$ . We will see that (Mult One) holds for one irregular pair $(p,k)$ if and only if it holds for all irregular pairs $(p,k)$ as $k$ runs over a fixed residue class mod $p-1$ (Theorem 3.9).

To see the relevance of condition (Mult One), a weaker version of the argument on $\unicode[STIX]{x1D707}$ -invariants given above would be to simply say that since $f_{k}$ and $E_{k}^{\operatorname{ord}}$ satisfy a congruence modulo $p$ , one would hope for their $p$ -adic $L$ -functions to also satisfy a congruence. Since the $p$ -adic $L$ -function of $E_{k}^{\operatorname{ord}}$ is 0, we would then deduce that the $\unicode[STIX]{x1D707}(f_{k})>0$ . However, the implication that a congruence of forms leads to a congruence of $p$ -adic $L$ -functions is a subtle one which is typically proven via a congruence between their associated modular symbols – that is, via (Mult One). Mod $p$ multiplicity one is now well established in the residually irreducible case (e.g. [Reference WilesWil95]), but is more subtle in the residually reducible case, and not always true for tame level greater than 1. But we do note that (Cuspidal rank one) implies (Mult One) (see Lemma 3.24 and Theorem 3.11).

We now state a precise theorem relating the $\unicode[STIX]{x1D707}$ -invariants of the $f_{k}$ with the $p$ -adic $\unicode[STIX]{x1D701}$ -function. For $a$ even, we write $\unicode[STIX]{x1D707}(f,\unicode[STIX]{x1D714}^{a})$ for the $\unicode[STIX]{x1D707}$ -invariant of the projection of $L_{p}^{+}(f)$ to the $\unicode[STIX]{x1D714}^{a}$ -component of the semi-local ring $\mathbb{Z}_{p}[[\mathbb{Z}_{p}^{\times }]]$ .

Theorem 1.1. Fix an irregular pair $(p,j)$ and assume that (Cuspidal rank one) holds for this pair. Then

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{p}(\unicode[STIX]{x1D705})\text{ divides }L_{p}^{+}(\unicode[STIX]{x1D705})\end{eqnarray}$$

in $\mathbb{Z}_{p}[[\unicode[STIX]{x1D6E4}_{w,j}\times \mathbb{Z}_{p}^{\times }]]$ . In particular,

$$\begin{eqnarray}\unicode[STIX]{x1D707}(f_{k},\unicode[STIX]{x1D714}^{a})\geqslant \operatorname{ord}_{p}(\unicode[STIX]{x1D701}_{p}(k))=\operatorname{ord}_{p}(B_{k}/k)\end{eqnarray}$$

for each even $a$ with $0\leqslant a\leqslant p-1$ and for classical $k\equiv j\;(\text{mod}\,p-1)$ .

Note that this theorem makes precise the previous claim that the $\unicode[STIX]{x1D707}$ -invariants blow up as one approaches the zeros of the $p$ -adic $\unicode[STIX]{x1D701}$ -function as the above lower bound blows up as one approaches these zeros.

Our method of proof of this theorem is to build a two-variable $p$ -adic $L$ -function on the full Hida family (i.e. on both the cuspidal and Eisenstein branches). We follow the construction of Greenberg and Stevens, but keep careful track over the integrality of all of the objects. Then (Mult One) implies that some large space of overconvergent modular symbols is free as a Hecke module. This freeness allows us to build a two-variable $p$ -adic $L$ -function over the full Hida algebra.

With this two-variable $p$ -adic $L$ -function in hand, the vanishing of the $p$ -adic $L$ -functions of the Eisenstein series implies the divisibility of Theorem 1.1. We note that the last claim of the theorem follows immediately from this divisibility. Indeed, when one specializes $\unicode[STIX]{x1D705}$ to some classical weight $k$ , one gets that the number $\unicode[STIX]{x1D701}_{p}(k)$ divides $L_{p}^{+}(f_{k})$ in $\mathbb{Z}_{p}[[\mathbb{Z}_{p}^{\times }]]$ – which exactly gives the above lower bound on the $\unicode[STIX]{x1D707}$ -invariant.

We now give an upper bound on $\unicode[STIX]{x1D707}$ -invariants which holds even without our (Cuspidal rank one) assumption. In what follows, we write $\unicode[STIX]{x1D707}(f)$ for $\unicode[STIX]{x1D707}(f,\unicode[STIX]{x1D714}^{0})$ .

Theorem 1.2. Fix an irregular pair $(p,j)$ and assume that (Mult One) holds for this pair. Then

$$\begin{eqnarray}\unicode[STIX]{x1D707}(f_{k})\leqslant \operatorname{ord}_{p}(a_{p}(f_{k})-1)\end{eqnarray}$$

for all classical $k\equiv j\;(\text{mod}\,p-1)$ .

We sketch the simple proof of this theorem. Assuming (Mult One), one shows that $L(f_{k},1)/\unicode[STIX]{x1D6FA}_{f}^{+}$ is a $p$ -adic unit (as one knows that the plus modular symbol attached to $f$ is congruent to a boundary symbol mod $p$ ). The interpolation formula for the $p$ -adic $L$ -function thus gives that $L_{p}(f_{k},\mathbf{1})$ has valuation $\operatorname{ord}_{p}(a_{p}(k)-1)$ which in turns gives our upper bound on $\unicode[STIX]{x1D707}$ .

Thus under (Cuspidal rank one) (which implies (Mult One)), we get the string of inequalities

(1) $$\begin{eqnarray}\operatorname{ord}_{p}(\unicode[STIX]{x1D701}_{p}(k))\leqslant \unicode[STIX]{x1D707}(f_{k})\leqslant \operatorname{ord}_{p}(a_{p}(f_{k})-1).\end{eqnarray}$$

We note that since $a_{p}(E_{\unicode[STIX]{x1D705}}^{\operatorname{ord}})=1$ for all $\unicode[STIX]{x1D705}\in {\mathcal{W}}$ , we have that $\unicode[STIX]{x1D701}_{p}(\unicode[STIX]{x1D705})$ divides $a_{p}(f_{\unicode[STIX]{x1D705}})-1$ in $\mathbb{Z}_{p}[[\unicode[STIX]{x1D6E4}_{w}]]$ which is consistent with the above inequalities.

Given the philosophy that $\unicode[STIX]{x1D707}$ -invariants are ‘as small as they can be’, it is tempting to make the following conjecture. We will also further justify this conjecture after analyzing the algebraic side of the situation.

Conjecture 1.3. For an irregular pair $(p,j)$ satisfying (Cuspidal rank one) we have

$$\begin{eqnarray}\unicode[STIX]{x1D707}(f_{k},\unicode[STIX]{x1D714}^{a})=\operatorname{ord}_{p}(\unicode[STIX]{x1D701}_{p}(k))\end{eqnarray}$$

for $a$ even and for any classical $k\equiv j\;(\text{mod}\,p-1)$ .

Besides giving an elegant formula for $\unicode[STIX]{x1D707}$ -invariants in a Hida family, the above conjecture has an appealing philosophical feel. Namely, the non-trivial $\unicode[STIX]{x1D707}$ -invariant is explained away through $p$ -adic variation. For an isolated form, a non-trivial $\unicode[STIX]{x1D707}$ -invariant almost feels like an error in the choice of the complex period. However, in the family, one sees these $\unicode[STIX]{x1D707}$ -invariants explicitly arising from the special values of some interesting analytic function. Further, by Ferrero and Washington [Reference Ferrero and WashingtonFW79], the $\unicode[STIX]{x1D707}$ -invariant of $\unicode[STIX]{x1D701}_{p}(\unicode[STIX]{x1D705})$ is 0, and so all divisibility by a power of $p$ vanishes in the family. We further note that Conjecture 1.3 implies the weaker statement that the two-variable $\unicode[STIX]{x1D707}$ -invariant of $L_{p}^{+}(\unicode[STIX]{x1D705})$ vanishes.

We note that this conjecture holds for $a=0$ for all irregular pairs $(p,k)$ with $p<2000$ . Unfortunately, this provides scant evidence for the conjecture as in this range, $\operatorname{ord}_{p}(a_{p}(f_{k})-1)=1$ , and thus the lower bound equals the upper bound in the inequalities in (1), forcing Conjecture 1.3 to hold.

However, when $a>0$ , we have no a priori upper bound on the $\unicode[STIX]{x1D707}$ -invariant (i.e. Theorem 1.2 does not apply) and thus we must instead compute $\unicode[STIX]{x1D707}$ -invariants of $p$ -adic $L$ -functions to verify that our conjecture holds. In this case, we verified that our conjecture holds for $p<750$ (for all even  $a$ ). For details, see § 3.9 where we describe the extensive computation we did using the algorithms in [Reference Pollack and StevensPS11] on overconvergent modular symbols.

Returning to the case of trivial tame character (i.e.  $a=0$ ), when the lower and upper bounds of (1) meet, the $p$ -adic $L$ -functions of the $f_{k}$ turn out to be very simple. In what follows, $L_{p}^{+}(\unicode[STIX]{x1D705},\unicode[STIX]{x1D714}^{a})$ denotes the projection of $L_{p}^{+}(\unicode[STIX]{x1D705})$ to the $\unicode[STIX]{x1D714}^{a}$ -component of $\mathbb{Z}_{p}[[\unicode[STIX]{x1D6E4}_{w,j}\times \mathbb{Z}_{p}^{\times }]]$ .

Theorem 1.4. Fix an irregular pair $(p,j)$ satisfying (Cuspidal rank one) and for which

$$\begin{eqnarray}\operatorname{ord}_{p}(\unicode[STIX]{x1D701}_{p}(j))=\operatorname{ord}_{p}(a_{p}(f_{j})-1).\end{eqnarray}$$

Then

$$\begin{eqnarray}L_{p}^{+}(\unicode[STIX]{x1D705},\unicode[STIX]{x1D714}^{0})=\unicode[STIX]{x1D701}_{p}(\unicode[STIX]{x1D705})\cdot U\end{eqnarray}$$

where $U$ is a unit in $\mathbb{Z}_{p}[[\unicode[STIX]{x1D6E4}_{w,j}\times (1+p\mathbb{Z}_{p})]]$ . In particular, for every classical $k\equiv j\;(\text{mod}\,p-1)$ , we have that

$$\begin{eqnarray}\unicode[STIX]{x1D706}(f_{k})=0\quad \text{and}\quad \unicode[STIX]{x1D707}(f_{k})=\operatorname{ord}_{p}(\unicode[STIX]{x1D701}_{p}(k))=\operatorname{ord}_{p}(B_{k}/k).\end{eqnarray}$$

That is, up to a unit, $L_{p}^{+}(f_{k},\unicode[STIX]{x1D714}^{0})$ is simply a power of $p$ .

We sketch the proof here. For our fixed $j$ , our assumption combined with (1) tells us the value of the $\unicode[STIX]{x1D707}$ -invariant exactly:

$$\begin{eqnarray}\operatorname{ord}_{p}(\unicode[STIX]{x1D701}_{p}(j))=\unicode[STIX]{x1D707}(f_{j})=\operatorname{ord}_{p}(a_{p}(j)-1).\end{eqnarray}$$

But, as argued above, we know that the value of $L_{p}^{+}(f_{j})$ at the trivial character has $p$ -adic valuation equal to $\operatorname{ord}_{p}(a_{p}(j)-1)$ . Since this valuation equals the $\unicode[STIX]{x1D707}$ -invariant, we get that $\unicode[STIX]{x1D706}(f_{j})=0$ . Then the divisibility of Theorem 3.26 tells us that the projection of

$$\begin{eqnarray}\frac{L_{p}^{+}(\unicode[STIX]{x1D705})}{\unicode[STIX]{x1D701}_{p}(\unicode[STIX]{x1D705})}\biggr|_{\unicode[STIX]{x1D705}=j}=\frac{L_{p}^{+}(f_{j})}{\unicode[STIX]{x1D701}_{p}(j)}\end{eqnarray}$$

to $\mathbb{Z}_{p}[[1+p\mathbb{Z}_{p}]]$ has vanishing $\unicode[STIX]{x1D707}$ - and $\unicode[STIX]{x1D706}$ -invariants since $\unicode[STIX]{x1D707}(f_{j})=\operatorname{ord}_{p}(\unicode[STIX]{x1D701}_{p}(j))$ , and is thus a unit. Since $L_{p}^{+}(\unicode[STIX]{x1D705})/\unicode[STIX]{x1D701}_{p}(\unicode[STIX]{x1D705})$ specializes to a unit at one weight, it must itself be a unit, as desired.

We note that it appears reasonable to believe that the case where the upper and lower bounds of (1) meet is the ‘generic’ case. Indeed, if $e=\operatorname{ord}_{p}(\unicode[STIX]{x1D701}_{p}(k))$ , then $p^{e}$ necessarily divides $a_{p}(f_{k})-1$ , and one might imagine that $(a_{p}(f_{k})-1)/p^{e}$ is a random number modulo $p$ . If this is the case, then for any given $p$ , the probability is $1/p$ for these bounds not to meet. One then expects there to be infinitely many $p$ such that the bounds in (1) fail to meet, but, given how slowly $\sum _{p}(1/p)$ diverges, such examples will be extremely rare.

1.3 A more general picture

We now discuss the more general case where (Cuspidal rank one) is no longer assumed to hold, but for this introduction we retain the tame level $N=1$ . In trying to repeat the arguments from earlier in the introduction without assuming (Cuspidal rank one), several new difficulties emerge. First, we no longer have that (Mult One) is automatically satisfied, which is key to our construction of a two-variable $p$ -adic $L$ -function over both the cuspidal and Eisenstein families. Second, the intersections of the cuspidal branches and the Eisenstein branch are no longer controlled solely by the $p$ -adic $\unicode[STIX]{x1D701}$ -function. Indeed, if there are multiple branches of the cuspidal family, some of them may not even intersect the Eisenstein family.

To remedy the first of these problems, we introduce a Gorenstein hypothesis,

(Goren) $$\begin{eqnarray}\mathbb{T}_{k,\mathfrak{m}_{k}}\text{is Gorenstein},\end{eqnarray}$$

where $(p,k)$ is some irregular pair. We note that (Goren) holds for one irregular pair $(p,k)$ if and only if it holds for all $(p,k)$ as $k$ varies over a fixed residue class mod $p-1$ if and only if $\mathbb{T}_{\mathfrak{m}}$ is Gorenstein where $\mathfrak{m}\subseteq \mathbb{T}$ is the maximal ideal attached to $E_{k}$ .

The connection between Gorenstein hypotheses and mod $p$ multiplicity one in the residually irreducible case has been understood for a while now as in [Reference MazurMaz78, Reference TilouineTil97, Reference WilesWil95]. In our residually reducible case, we have that (Goren) implies (Mult One) which follows from work by Ohta [Reference OhtaOht99, Reference OhtaOht00] and Sharifi [Reference SharifiSha11] (see Theorem 3.11).

We also note that in this more general setting we cannot consider our cuspidal two-variable $p$ -adic $L$ -function as an elements in $\mathbb{Z}_{p}[[\unicode[STIX]{x1D6E4}_{w,j}\times \mathbb{Z}_{p}^{\times }]]$ since we are not able to conflate the Hida family with weight space. Instead, we will consider the two-variable $p$ -adic $L$ -function as an element $L_{p}^{+}(\mathfrak{m}^{c})$ in $\mathbb{T}_{\mathfrak{m}^{c}}^{c}[[\mathbb{Z}_{p}^{\times }]]$ . The weight variable of $L_{p}^{+}(\mathfrak{m})$ then varies over the spectrum of $\mathbb{T}_{\mathfrak{m}^{c}}^{c}$ ; that is, for $\mathfrak{p}$ a height-one prime of $\mathbb{T}_{\mathfrak{m}^{c}}^{c}$ of residual characteristic 0, we can ‘evaluate’ $L_{p}^{+}(\mathfrak{m}^{c})$ at $\mathfrak{p}$ by looking at its image in $(\mathbb{T}_{\mathfrak{m}^{c}}^{c}/\mathfrak{p})[[\mathbb{Z}_{p}^{\times }]]$ . Here $\mathbb{T}_{\mathfrak{m}^{c}}^{c}/\mathfrak{p}$ is a finite extension of $\mathbb{Z}_{p}$ . Moreover, if $\mathfrak{p}_{f}\subseteq \mathbb{T}_{\mathfrak{m}^{c}}^{c}$ is the prime ideal associated with a classical ordinary eigenform $f$ , then the image of $L_{p}^{+}(\mathfrak{m}^{c})$ in $\mathbb{T}_{\mathfrak{m}^{c}}^{c}/\mathfrak{p}_{f}[[\mathbb{Z}_{p}^{\times }]]$ recovers the single variable $p$ -adic $L$ -function $L_{p}^{+}(f)$ .

For the second problem, to find a replacement for the $p$ -adic $\unicode[STIX]{x1D701}$ -function to control the intersection of the Eisenstein and cuspidal branches, we seek a single element in the cuspidal Hecke algebra $\mathbb{T}_{\mathfrak{m}^{c}}^{c}$ which measures congruences between Eisenstein series and cuspforms. Essentially, by definition, we can simply use a generator, if one exists, of the cuspidal Eisenstein ideal in $\mathbb{T}_{\mathfrak{m}^{c}}^{c}$ . That is, assume the Eisenstein ideal of $\mathbb{T}_{\mathfrak{m}^{c}}^{c}$ is principal, generated by, say, $L_{\operatorname{eis}}$ . Fix a weight- $k$ cuspidal eigenform $f$ congruent to $E_{k}^{\operatorname{ord}}$ and let $\mathfrak{p}_{f}\subseteq \mathbb{T}^{c}$ denote the corresponding prime ideal. Then $\mathbb{T}_{\mathfrak{m}^{c}}^{c}/\mathfrak{p}_{f}$ is some finite extension of $\mathbb{Z}_{p}$ . Write ${\mathcal{O}}_{f}$ for the ring of integers in the field of fractions of $\mathbb{T}_{\mathfrak{m}^{c}}^{c}/\mathfrak{p}_{f}$ , and write $L_{\operatorname{eis}}(\mathfrak{p}_{f})$ to denote the image of $L_{\operatorname{eis}}$ in ${\mathcal{O}}_{f}$ . We then have that $E_{k}^{\operatorname{ord}}$ is congruent to $f$ modulo $L_{\operatorname{eis}}(\mathfrak{p}_{f}){\mathcal{O}}_{f}$ .

Unfortunately, there is no a priori reason why the Eisenstein ideal is principal. To ensure the principality of this ideal we introduce another Gorenstein condition,

(Cusp Goren) $$\begin{eqnarray}\mathbb{T}_{k,\mathfrak{m}_{k}^{c}}^{c}\text{is Gorenstein},\end{eqnarray}$$

where $(p,k)$ is some irregular pair. Again, by Hida theory, (Cusp Goren) holds for one irregular pair $(p,k)$ if and only if it holds for all $(p,k)$ as $k$ varies over a fixed residue class mod $p-1$ if and only if $\mathbb{T}_{\mathfrak{m}^{c}}^{c}$ is Gorenstein.

Now, by work by Ohta [Reference OhtaOht05], we have that (Goren) and (Cusp Goren) hold if and only if the Eisenstein ideal is principal (see Theorem 3.5). We also note that Vandiver’s conjecture implies that both (Goren) and (Cusp Goren) hold (see Proposition 3.7), and so we can offer no examples where these hypotheses fail when $N=1$ . However, when $N>1$ one does not expect these hypotheses to hold generally (see [Reference WakeWak15, Corollary 1.4]).

The following is our generalization of Theorem 1.1. Let $\unicode[STIX]{x1D71B}$ denote a uniformizer of ${\mathcal{O}}_{f}$ .

Theorem 1.5. Fix an irregular pair $(p,j)$ and assume that (Goren) and (Cusp Goren) hold for this pair, and let $\mathfrak{m}^{c}$ denote the corresponding maximal ideal of $\mathbb{T}^{c}$ . Let $L_{\operatorname{eis}}$ denote a generator of the cuspidal Eisenstein ideal of $\mathbb{T}_{\mathfrak{m}^{c}}^{c}$ . Then

$$\begin{eqnarray}L_{\operatorname{eis}}\text{ divides }L_{p}^{+}(\mathfrak{m}^{c})\end{eqnarray}$$

in $\mathbb{T}_{\mathfrak{m}^{c}}^{c}[[\mathbb{Z}_{p}^{\times }]]$ . In particular,

$$\begin{eqnarray}\unicode[STIX]{x1D707}(f,\unicode[STIX]{x1D714}^{a})\geqslant \operatorname{ord}_{\unicode[STIX]{x1D71B}}(L_{\operatorname{ eis}}(\mathfrak{p}_{f}))\end{eqnarray}$$

for all even $a$ with $0\leqslant a\leqslant p-2$ , and all $f$ in the Hida family corresponding to $\mathfrak{m}^{c}$ .

We again conjecture that the above inequality is in fact an equality (see Conjecture 3.16). Further, this lower bound meets the upper bound of Theorem 1.2 if and only if the cuspidal Eisenstein ideal is generated by $U_{p}-1$ . In this case, our Gorenstein hypotheses are automatically satisfied and we have the following generalization of Theorem 1.4 which asserts that the Iwasawa theory in this case is as simple as it can be.

Theorem 1.6. For an irregular pair $(p,j)$ for which $U_{p}-1$ generates the cuspidal Eisenstein ideal of $\mathbb{T}_{\mathfrak{m}^{c}}^{c}$ , we have

$$\begin{eqnarray}\unicode[STIX]{x1D706}(f)=0\quad \text{and}\quad \unicode[STIX]{x1D707}(f)=\operatorname{ord}_{p}(a_{p}(f)-1)\end{eqnarray}$$

for all $f$ in the corresponding Hida family.

1.4 Selmer groups

We return to the situation where $(p,j)$ is an irregular pair which satisfies (Goren) and (Cusp Goren) and let $L_{\operatorname{eis}}$ denote a generator of the associated cuspidal Eisenstein ideal. Let $f$ denote some classical form which belongs to the corresponding Hida family.

On the algebraic side, we note that there is not a well-defined Selmer group attached to $f$ . Indeed, within the Galois representation $\unicode[STIX]{x1D70C}_{f}:G_{\mathbb{Q}}\rightarrow \operatorname{Aut}(V_{f})$ , one must choose a Galois stable lattice $T\subseteq V_{f}$ . One then attaches a Selmer group to $V_{f}/T$ ; we denote this Selmer group by $\operatorname{Sel}(\mathbb{Q}_{\infty },V_{f}/T)\subseteq H^{1}(\mathbb{Q}_{\infty },V_{f}/T)$ , where $\mathbb{Q}_{\infty }$ is the cyclotomic $\mathbb{Z}_{p}$ -extension. Changing the lattice $T$ can change the $\unicode[STIX]{x1D707}$ -invariant of the corresponding Selmer group.

In this setting we have the following collection of lower bounds on the possible algebraic $\unicode[STIX]{x1D707}$ -invariants that can occur.

Theorem 1.7. Let $(p,j)$ denote an irregular pair satisfying (Goren) and (Cusp Goren) and let $f$ denote a classical eigenform in the corresponding Hida family. Then there exists a chain of lattice $T_{0}\subseteq T_{1}\subseteq \cdots \subseteq T_{m}$ in $V_{f}$ with $m=\operatorname{ord}_{\unicode[STIX]{x1D71B}}(L_{\operatorname{eis}}(\mathfrak{p}_{f}))$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D707}(\operatorname{Sel}(\mathbb{Q}_{\infty },V_{f}/T_{r})^{\vee })\geqslant r\end{eqnarray}$$

for $0\leqslant r\leqslant m$ .

This theorem is proven as follows. For each $r$ in the above range, there is a lattice $T_{r}$ such that $T_{r}/p^{r}T_{r}$ is a reducible representation with a cyclic submodule of size $p^{r}$ which is odd and ramified at $p$ . Using Greenberg’s methods as in [Reference GreenbergGre99, Proposition 5.7], the lower bound on $\unicode[STIX]{x1D707}$ follows.

Greenberg, in the case of elliptic curves, would then conjecture that this lower bound on $\unicode[STIX]{x1D707}$ is actually an equality (see [Reference GreenbergGre99, Conjecture 1.11 and p. 70]). Combining this conjecture with Theorem 1.1, we see that if there is any main conjecture defined over $\mathbb{Z}_{p}$ for $f$ , we must be using the lattice $T_{r}$ with maximal $\unicode[STIX]{x1D707}$ -invariant. Set $T$ equal to such a lattice and let $A_{f}=V_{f}/T$ .

We now state an upper bound for the algebraic $\unicode[STIX]{x1D707}$ -invariant. To do so, we need to invoke Vandiver’s conjecture. For $r$ even, we write the $\unicode[STIX]{x1D714}^{r}$ -part of Vandiver’s conjecture as follows:

Theorem 1.8. Let $(p,j)$ be an irregular pair for which $(\text{Vand}\;\unicode[STIX]{x1D714}^{2-j})$ holds. For $f$ a classical form in the corresponding Hida family, we have

$$\begin{eqnarray}\unicode[STIX]{x1D707}(\operatorname{Sel}(\mathbb{Q}_{\infty },A_{f})^{\vee })\leqslant \operatorname{ord}_{p}(a_{p}(f)-1).\end{eqnarray}$$

Our method of proof is analogous to the proof of Theorem 1.2. There we bounded the $\unicode[STIX]{x1D707}$ -invariant of the $p$ -adic $L$ -function by computing the $p$ -adic valuation of its special value at the trivial character. Here we instead look at $\operatorname{Sel}(\mathbb{Q}_{\infty },A_{f})^{\unicode[STIX]{x1D6E4}}$ and by bounding its size we get a bound on the $\unicode[STIX]{x1D707}$ -invariant (see Lemma 5.7). To bound the size of $\operatorname{Sel}(\mathbb{Q}_{\infty },A_{f})^{\unicode[STIX]{x1D6E4}}$ , we use a control theorem to compare this group with $\operatorname{Sel}(\mathbb{Q},A_{f})$ which we in turn control via Vandiver’s conjecture.

We now state the algebraic analogue of Conjecture 1.3 for our chosen lattice, and note that this is essentially Greenberg’s conjecture on $\unicode[STIX]{x1D707}$ -invariants in this special case.

Conjecture 1.9. For an irregular pair $(p,j)$ satisfying (Goren) and (Cusp Goren) and $f$ a classical form in the corresponding Hida family, we have

$$\begin{eqnarray}\unicode[STIX]{x1D707}(\operatorname{Sel}(\mathbb{Q}_{\infty },A_{f})^{\vee })=\operatorname{ord}_{p}(L_{\operatorname{eis}}(\mathfrak{p}_{f})).\end{eqnarray}$$

Finally, we state a theorem in the case where the lower and upper bounds meet.

Theorem 1.10. For an irregular pair $(p,j)$ such that $U_{p}-1$ generates the corresponding cuspidal Eisenstein ideal, we have

$$\begin{eqnarray}\operatorname{Sel}(\mathbb{Q}_{\infty },A_{f})^{\vee }\cong \unicode[STIX]{x1D6EC}/(a_{p}(f)-1)\unicode[STIX]{x1D6EC}\end{eqnarray}$$

for each classical $f$ in the corresponding Hida family. In particular,

$$\begin{eqnarray}\unicode[STIX]{x1D707}(\operatorname{Sel}(\mathbb{Q}_{\infty },A_{f})^{\vee })=\operatorname{ord}_{p}(a_{p}(f)-1)\quad \text{and}\quad \unicode[STIX]{x1D706}(\operatorname{Sel}(\mathbb{Q}_{\infty },A_{f})^{\vee })=0.\end{eqnarray}$$

We note that we do not need to assume Vandiver’s conjecture for the above theorem as our assumption that $U_{p}-1$ generates the cuspidal Eisenstein ideal implies that the relevant class group vanishes by the results in [Reference WakeWak15].

Combining Theorems 1.4 and 1.10 yields the following result.

Corollary 1.11 (The main conjecture).

For an irregular pair $(p,j)$ satisfying $(\text{Vand}\;\unicode[STIX]{x1D714}^{2-j})$ and such that $U_{p}-1$ generates the corresponding cuspidal Eisenstein ideal, we have

$$\begin{eqnarray}\operatorname{char}_{\unicode[STIX]{x1D6EC}}(\operatorname{Sel}(\mathbb{Q}_{\infty },A_{f})^{\vee })=L_{p}^{+}(f,\unicode[STIX]{x1D714}^{0})\unicode[STIX]{x1D6EC}\end{eqnarray}$$

for each classical $f$ in the corresponding Hida family.

We note that the above main conjecture over $\unicode[STIX]{x1D6EC}[1/p]$ follows from Kato’s work [Reference KatoKat04, Theorem 17.4.2] as the $p$ -adic $L$ -function is a unit up to a power of $p$ . However, neither [Reference KatoKat04] nor [Reference Skinner and UrbanSU14] control what happens with the algebraic and analytic $\unicode[STIX]{x1D707}$ -invariants in the residually reducible case.

This paper is organized as follows. In the following section we recall Stevens’ theory of overconvergent modular symbols and its connection to $p$ -adic $L$ -functions. The heavy lifting of constructing two-variable $p$ -adic $L$ -functions is relegated to Appendix A. In the third section we carry out our analysis of analytic $\unicode[STIX]{x1D707}$ -invariants sketched in the introduction. In the fourth section we recall the basic definitions and facts about Selmer groups. In the fifth section we carry out our analysis of algebraic $\unicode[STIX]{x1D707}$ -invariants.