Consequences of functional equations for pairs of p-adic L-functions

TL;DR

This paper explores the implications of functional equations of p-adic L-functions for elliptic curves at supersingular primes, revealing relationships between their leading terms, parity of vanishing orders, and invariance under twists.

Contribution

It establishes new theoretical consequences of functional equations for p-adic L-functions, including parity results and invariance properties, advancing understanding in Iwasawa theory.

Findings

Relationship between leading and sub-leading terms of p-adic L-functions

Parity result for orders of vanishing at supersingular primes

Invariance of Iwasawa invariants under conjugate twists

Abstract

We prove consequences of functional equations of p-adic L-functions for elliptic curves at supersingular primes p. The results include a relationship between the leading and sub-leading terms (for which we use ideas of Wuthrich and Bianchi), a parity result of orders of vanishing, and invariance of Iwasaswa invariants under conjugate twists of the p-adic L-functions.

Equations

$$L_{\ast }(E,\psi ,s)={\int }_{{\mathbb{Z}}_{p,M}^{\times }}\psi (x)⟨x{⟩}^{s-1}\mathrm{d}{\mu }_{E,\ast }.$$

$$L_{\ast }(E,\psi ,T):={\int }_{{\mathbb{Z}}_{p,M}^{\times }}\psi (x)(1+T{)}^{\frac{\log ⟨x⟩}{\log \kappa (\gamma )}}\mathrm{d}{\mu }_{E,\ast }.$$

$$L_{\ast }(E,\psi ,T)=-(1+T{)}^{-{\log }_{\gamma }(Q)}\overline{\psi }(-Q)c_Q{L}_{\ast }(E,\overline{\psi },\frac{1}{1+T}-1),$$

$$L_{\ast }(E,0)=\frac{1}{\Omega }{\left(1-\frac{1}{\ast }\right)}^{2}L(E,1).$$

$$\left(L_{\alpha }(E,\psi ,T),L_{\beta }(E,\psi ,T)\right)=\left(L_p^{♯}(E,\psi ,T),L_p^{♭}(E,\psi ,T)\right)\mathcal{L}o{g}_{\alpha ,\beta }(1+T)$$

$$L_p^{♯}(E,\psi ,T)$$

$$L_p^{♭}(E,\psi ,T)$$

$$L_p^{♯}(E,\chi ,T)=a_{♯}{T}^{m_{♯}}+b_{♯}{T}^{m_{♯}+1}+\cdots$$

$$L_p^{♭}(E,\chi ,T)=a_{♭}{T}^{m_{♭}}+b_{♭}{T}^{m_{♭}+1}+\cdots ,$$

$$b_{♯}=\frac{-a_{♯}}{2}\left({\log }_{\gamma }(Q)+m_{♯}\right)\text{ and }{b}_{♭}=\frac{-a_{♭}}{2}\left({\log }_{\gamma }(Q)+m_{♭}\right).$$

$$L_p^{+}(E,T)$$

$$L_p^{-}(E,T)$$

$$W^{+}(1+T)$$

$$W^{-}(1+T)$$

$$L_p^{+}(E,T)$$

$$L_p^{-}(E,T)$$

$$b_{m+}=-\frac{a_{m_{+}}}{2}\left({\log }_{\gamma }(N)-\frac{p}{1+p}+m_{+}\right)\text{and }{b}_{m_{-}}=-\frac{a_{m_{-}}}{2}\left({\log }_{\gamma }(N)-\frac{1}{1+p}+m_{-}\right).$$

$$b_{m+}=\frac{-a_{m_{+}}}{2}\left(-(F^{+}{)}^{\prime }(0)+m_{+}\right)\text{ and }{b}_{m_{-}}=\frac{-a_{m_{-}}}{2}\left(-(F^{-}{)}^{\prime }(0)+m_{-}\right).$$

$$(F^{+}{)}^{\prime }(0)=-{\log }_{\gamma }(N)+\frac{p}{p+1}\text{ and }(F^{-}{)}^{\prime }(0)=-{\log }_{\gamma }(N)+\frac{1}{p+1}\text{, while}$$

$$(F^{-}{)}^{\prime }(0)=-{\log }_{\gamma }(N)+\frac{p^2-p-1}{p+1}\text{ for p=2}.$$

$$g(T)=uf\left(\frac{1}{1+T}-1\right)$$

$$\left(\frac{1}{1+T}-1-\zeta \right)=U(T)T-\zeta ,$$

$$U(T)T-\zeta =\left(T+\frac{\zeta }{1+\zeta }\right)\times V_{\zeta }(T)$$

$$\{\text{zeros of f(T) of multiplicity n in U}\}\subset \{\text{zeros of g(T) of multiplicity ⩾n in U}\}.$$

$$g(T)=uf\left(\frac{1}{1+T}-1\right)$$

Full text

1. Introduction

In a beautifully short paper [9], Wuthrich related the leading term of the $L$-function of an elliptic curve $E$ with the second non-zero term in the Taylor expansion about $1$, which he called the sub-leading term. Bianchi adapted Wuthrich’s ideas to $p$-adic $L$-functions of $E$. In case $p$ is a prime of ordinary reduction, Bianchi was able to go further. As a consequence of the nice behavior (integrality) of the $p$-adic $L$-function, Bianchi found a fact endemic to the $p$-adic world: The twist of the $p$-adic $L$-function by a character $\psi$ has the same $\mu$-invariant as the twist by the conjugate character $\bar{\psi}$.

A natural question is what happens when $p$ is a prime of supersingular (i.e. non-ordinary) reduction. The $p$-adic $L$-function for the ordinary case has two direct supersingular analogues. They are not power series with integral coefficients, so we can’t speak of $\mu$-invariants. However, one can construct a pair $(L^{\sharp},L^{\flat})$ of integral $p$-adic $L$-functions [6]. The point of this article is thus to develop the ideas of Wuthrich and Bianchi in the setting of these $\sharp/\flat$ $L$-functions $L^{\sharp/\flat}$.

We relate the leading and sub-leading coefficients of $L^{\sharp/\flat}$ in the spirit of Bianchi and Wuthrich. Analogously to Bianchi, we deduce from this that the $\mu$-invariants of the twists of $L^{\sharp/\flat}$ by $\psi$ and $\bar{\psi}$ are the same. Do the $\lambda$-invariants also stay the same? We prove that the answer is yes. Further, we show that this $\lambda$-invariance holds also for the ordinary $p$-adic $L$-function. Another new consequence is that the two non-integral $p$-adic $L$-functions in the supersingular case vanish to the same order modulo $2$ at the critical point. We can say the same about the functions $L^{\sharp/\flat}$.

The most important tool in the proofs is the functional equation. In [6], two slightly different versions of pairs $(L^{\sharp},L^{\flat})$ were defined. We work with the first version, which is amenable to functional equations. The pair $(L^{\sharp},L^{\flat})$ we work with in this paper was denoted differently in [6] as $(\widehat{L}^{\sharp},\widehat{L}^{\flat})$, to record a completion that makes the functional equation work. But we drop the hats for notational convenience. The second non-completed version reduces to Pollack’s $p$-adic $L$-functions $L_{p}^{\pm}$ defined when $a_{p}:=p+1-\#E(\mathbf{F}_{p})=0$. The vanishing of $a_{p}$ guarantees that $L_{p}^{\pm}$ also fits a functional equation, allowing us to work out the results for $L_{p}^{\pm}$ as well.

Bianchi’s idea for proving the invariance of the $\mu$-invariant is to show that the responsible term is in the same position for both power series in question. An alternative idea due to Pollack – to simply observe that the two $p$-adic $L$-functions are Galois conjugates – may be employed in the ordinary case and the case $a_{p}=0$ assuming further that $p\equiv 3\mod 4$; his methods show further that the $\lambda$-invariants are the same in these cases as well.

By contrast, our methods count zeroes in the unit disk and work in all cases, for both the $\mu$ and the $\lambda$-invariants. The arguments may be of separate interest, and the reader may find them in the appendix.

Acknowledgment. We would like to thank Antonio Lei for putting us in touch.

2. Notation

Let $E$ be an elliptic curve over the field of rational numbers $\mathbb{Q}$ and let $p$ be a prime of good supersingular reduction for $E$. We denote by $\alpha$ and $\beta$ the two roots of the Hecke polynomial $X^{2}-a_{p}X+p$. Amice and Vélu, and Višik constructed two $p$-adic $L$-functions $L_{\alpha}(E,T)$ and $L_{\beta}(E,T)$ that each interpolate the special values of (cyclotomic twists of) the $L$-function of $E$. We denote their twists by Dirichlet characters $\psi$ by $L_{\alpha}(E,\psi,T)$ and $L_{\beta}(E,\psi,T)$, and make the convention of simply dropping the symbol $\psi$ when $\psi$ is the trivial character. See [3, I, paragraph 13]. We let $\overline{\psi}$ be the complex-conjugate of $\psi$. When the character of interrest is either trivial or quadratic, we denote it by $\chi$ to avoid confusion. $U$ denotes the open unit disk in $\mathbb{C}_{p}$, and we let $\Lambda=\mathbb{Z}_{p}[[T]]$. In the following section, we recall the main properties of the functions $\widehat{L}_{p}^{\sharp}(E,\psi,T)$ and $\widehat{L}_{p}^{\flat}(E,\psi,T)$ from [6], which we denote without the hats as $L_{p}^{\sharp}(E,\psi,T)$ and $L_{p}^{\flat}(E,\psi,T)$ for convenience.

3. Background

In this section, we briefly recall the $p$-adic $L$-functions of Amice–Vélu and Višik. See [3] or the original papers [8], [1]. We then recall the $\sharp/\flat$ $p$-adic $L$-functions of [6]. Their two essential properties are their relation with the Amice–Vélu–Višik $p$-adic $L$-functions and their functional equation. In fact, we go a tiny bit beyond simply recalling them. While [6] only worked with twists by powers of the Teichmuller character, we state the results for the twists by any Dirichlet character $\psi$.

Amice–Vélu and Višik’s constructed $p$-adic distributions $\mu_{E,*}$ on $\mathbb{Z}_{p}^{\times}$ attached to $E$ for $*\in\{\alpha,\beta\}$. For $x\in\mathbb{Z}_{p}^{\times}$, denote by $\langle x\rangle$ the projection of $x$ on $1+p\mathbb{Z}_{p}$ (or on $1+4\mathbb{Z}_{2}$ when $p=2$). If $\psi$ is a Dirichlet character of conductor $p^{k}M$ with $p\nmid M$, their $p$-adic $L$-function $L_{*}(E,\psi,s)$ is then given by

[TABLE]

where $\mathbb{Z}_{p,M}^{\times}=\mathbb{Z}_{p}^{\times}\times(\mathbb{Z}/M\mathbb{Z})^{\times}$. It is a $p$-adic locally analytic function in the variable $s\in\mathbb{Z}_{p}$. Let $\Gamma$ be the Galois group $\text{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})\cong\mathbb{Z}_{p}$ of the cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}$. Choose $\gamma$ a topological generator of $\Gamma$ and let $\kappa$ be the cyclotomic character. The change of variables $T=\kappa(\gamma)^{s-1}-1$ transforms $L_{*}(E,\psi,s)$ into the power series

[TABLE]

From now on, all functions are in the variable $T$.

Choose $c_{Q}\in\{\pm 1\}$ so that $f|_{\left(\begin{smallmatrix}0&-1\\ Q&0\end{smallmatrix}\right)}=c_{Q}f$. Here $f$ is the modular form attached to $E$, and $Q$ is the largest divisor of the level $N$ of $E$ that is coprime to both $p$ and the conductor of $\psi$. The function $L_{*}(E,\psi,T)$ fits the functional equation

[TABLE]

[3, Sections 5 and 17]. We put $\log_{\gamma}(x)=\frac{\log\langle x\rangle}{\log\kappa(\gamma)}$ to ease notation.

Once $p$-adic numbers are identified with complex numbers, the special values of $L_{*}(E,T)$ at $T=\zeta_{p^{n}}-1$ can be related to special values of complex $L$-series. (More precisely, they are multiples by algebraic numbers normalized by a transcendental period $\Omega$ of the special values of twists of the Hasse-Weil $L$-functions at the complex value $1$ – see [3, Section 14].) We record the property for $T=0$:

[TABLE]

Unlike in the ordinary case, the power series coefficients of $L_{*}(E,\psi,T)$ are unbounded so that $L_{*}(E,\psi,T)\notin\Lambda$ (and even $\notin\Lambda\otimes\mathbb{C}_{p}$). The main theorem of [6] remedies this:

Theorem 3.1**.**

We have the factorization

[TABLE]

for two power series $L_{p}^{\sharp}(E,\psi,T),L_{p}^{\flat}(E,\psi,T)\in\Lambda$, where $\mathcal{L}og_{\alpha,\beta}(1+T)$ is an explicit $2\times 2$ matrix of functions converging on $U$.

Proof.

This is [6, Theorem 2.14], where the statement is proved when $\psi$ is a power for the Teichmuller character. The same proof applies to arbitrary $\psi$. Note that $\mathcal{L}og_{\alpha,\beta}(1+T)$ was denoted $\widehat{\mathcal{L}og}_{\alpha,\beta}(1+T)$ in [6, Section 4]. We don’t recall its definition in this paper since it is not needed. ∎

The $\sharp/\flat$ $L$-functions satisfy a functional equation similar to $L_{\alpha/\beta}(f,\psi,T)$.

Theorem 3.2**.**

Let $E$ be an elliptic curve over $\mathbb{Q}$ of level $N$ and $p$ a good supersingular prime. Let $\psi$ be a Dirichlet character, and let $Q$ be the largest divisor of $N$ that is coprime to both $p$ and the conductor of $\psi$. Then

[TABLE]

Proof.

This follows from the same methods as in [6, Theorem 5.19]. ∎

Remark 3.3**.**

The original uncompleted $\sharp/\flat$ $p$-adic $L$-functions constructed in [7] do not satisfy a nice functional equation, but there is an explicit relationship between these uncompleted and our completed $p$-adic $L$-functions, cf. [6, Corollary 5.11].

4. The main results

In this section, we state the main results of the paper. The first is the parity of the orders of vanishing at the critical point of the $\sharp/\flat$ $p$-adic $L$-functions, the second is the relation between the leading and sub-leading terms in each of them, and the third is the invariance of the Iwasawa invariants under the substitution $\psi\mapsto\overline{\psi}$, where $\psi$ is any Dirichlet character

Let $\chi$ be either the trivial character or a quadratic character. We let $Q$ and $N$ be as in the previous section: $Q$ denotes the largest divisor of the conductor $N$ of $E$ that is coprime to both $p$ and the conductor of $\chi$. Consider the Taylor expansions

[TABLE]

and

[TABLE]

where $m_{\sharp/\flat}$ denote the orders of vanishing at $T=0$ of $L^{\sharp/\flat}(E,\chi,T)$.

Theorem 4.1**.**

The order of vanishing of $L_{p}^{\sharp}(E,\chi,T)$ at $T=0$ has the same parity as the order of vanishing of $L_{p}^{\flat}(E,\chi,T)$ at $T=0$.

Proof.

Differentiating the functional equations in Theorem 3.2 $m_{\sharp}$ resp. $m_{\flat}$ times and evaluating at $T=0$, we obtain $(-1)^{m_{\sharp}}=-c_{Q}$ and $(-1)^{m_{\flat}}=-c_{Q}$, cf. [2, Proof of Theorem 4.1]. Thus, $m_{\sharp}\equiv m_{\flat}\mod 2$. ∎

Corollary 4.2**.**

The parities of the orders of vanishing of $L_{\alpha}(E,\chi,T)$ and $L_{\beta}(E,\chi,T)$ are the same.

Proof.

We obtain this by using the same arguments and the functional equations for $L_{\alpha}(E,\chi,T)$ and $L_{\beta}(E,\chi,T)$, cf. [3, Section 17].∎

Corollary 4.3**.**

When $a_{p}=0$, the functions of Pollack $L_{p}^{\pm}(E,T)$ have the same parities of orders of vanishing.

Proof.

By [6, Corollary 5.1], $L_{p}^{\pm}(E,T)$ equal $L_{p}^{\sharp/\flat}(E,T)$ up to units. ∎

Theorem 4.4**.**

We have

[TABLE]

Proof.

We differentiate the functional equations in Theorem 3.2 $m_{\sharp}+1$ resp. $m_{\flat}+1$ times and then evaluate at $T=0$. This is analogous to [2, Proof of Theorem 4.1]. ∎

We now let $\psi$ denote any Dirichlet character.

Definition 4.5**.**

For the following theorem, we make the following conventions: We denote the $\lambda$-invariants of $L_{p}^{\sharp}(E,\psi,T)$ and $L_{p}^{\flat}(E,\psi,T)$ by $\lambda_{\sharp}$ and $\lambda_{\flat}$, and the $\lambda$-invariants of $L_{p}^{\sharp}(E,\overline{\psi},T)$ and $L_{p}^{\flat}(E,\overline{\psi},T)$ by $\overline{\lambda}_{\sharp}$ and $\overline{\lambda}_{\flat}$. We define the integers $\mu_{\sharp},\mu_{\flat},\overline{\mu}_{\sharp},$ and $\overline{\mu}_{\flat}$ similarly.

Theorem 4.6**.**

The Iwasawa invariants of $L_{p}^{\sharp}(E,\psi,T)$ and $L_{p}^{\flat}(E,\psi,T)$ stay invariant under the substitution $\psi\mapsto\overline{\psi}$. More precisely,

1. (1)

We have $\lambda_{\sharp}=\overline{\lambda}_{\sharp}$ and $\lambda_{\flat}=\overline{\lambda}_{\flat}$. 2. (2)

We have $\mu_{\sharp}=\overline{\mu}_{\sharp}$ and $\mu_{\flat}=\overline{\mu}_{\flat}$.

Proof.

We carry out the proof for the $\sharp$ $p$-adic $L$-functions. By Theorem 3.2, $L_{p}^{\sharp}(E,\psi,T)$ and $L_{p}^{\sharp}(E,\overline{\psi},\frac{1}{1+T}-1)$ differ by multiplication by an element of the form $-(1+T)^{-\log_{\gamma}(Q)}\overline{\psi}(-Q)c_{Q}$. By definition, $\overline{\psi}(-Q)c_{Q}$ is a unit. The term $-(1+T)^{-\log_{\gamma}(Q)}$ is a unit power series.

The $\lambda$-invariance and the $\mu$-invariance are treated as general propositions in the appendix, see Proposition A.1 and Proposition A.3. ∎

Corollary 4.7**.**

The above theorem holds if we replace the $\sharp/\flat$ $p$-adic $L$-functions by their non-completed counterparts constructed in [7].

Proof.

Recall from Remark 3.3 that the non-completed $\sharp/\flat$ $p$-adic $L$-functions do not satisfy a direct functional equation. However, the $\lambda$- and $\mu$-invariants of the completed and non-completed $\sharp/\flat$ $p$-adic $L$-functions are the same, cf. [5, Remark 2.11]. ∎

When $p$ is prime of good ordinary reduction for $E$, there is only one $p$-adic $L$-function, $L_{p}(E,\psi,T)$, coming from the unit root of $X^{2}-a_{p}X+p$. Since proposition A.1 and proposition A.3 are general statements about integral power series, they also apply to $L_{p}(E,\psi,T)$. Note that the invariance of $\mu$ was already known by Bianchi, albeit by using a different argument.

Corollary 4.8**.**

Let $p$ be a prime of ordinary reduction for $E$. Theorem 4.6 also holds if we replace the $\sharp/\flat$ $p$-adic $L$-functions by their analogue in the ordinary case $L_{p}(E,\psi,T)$.

Proof.

Since $L_{p}(E,\psi,T)$ is an integral power series and satisfies the same functional equation as $L_{p}^{\sharp/\flat}$, the proof of theorem 4.6 can be used in this case as well. ∎

Remark 4.9** (Pollack).**

One can obtain the invariance of Iwasawa invariants in the ordinary case (Corollary 4.8) and in the subcase $a_{p}=0$ and $p\equiv 3\mod 4$ of the supersingular case (Theorem 4.6) by simply observing that the $\psi$ and $\overline{\psi}$ are Galois-conjugate characters, cf. the arguments in [4, Lemma 6.7].

5. The case $a_{p}=0$

When $a_{p}=0$ and $\psi$ is the trivial character, the functions $L_{p}^{\sharp/\flat}$ are Pollack’s $L^{\pm}$ functions multiplied by a unit, cf. [6, Corollary 5.1]. See [4] for the original definition of $L^{\pm}$. The functional equations for the plus/minus $L$-functions are [6, Section 5.1]:

[TABLE]

where $W^{\pm}(1+T)$ are units given by

[TABLE]

Theorem 5.1**.**

Let $E$ be an elliptic curve over $\mathbb{Q}$ of conductor $N$ and $p$ a prime such that $(p,N)=1$ and $a_{p}=0$. Write

[TABLE]

where $m_{\pm}$ denotes the order of vanishing of $L_{p}^{\pm}(E,T)$ at $T=0$. We then have

[TABLE]

Proof.

We let $F^{\pm}(T)=(1+T)^{-\log_{\gamma}(N)}W^{\pm}(1+T)$. We follow the same arguments as in the proofs of Theorems 4.1 and 4.4. Differentiating $m_{+}$ times the functional equation (1) and $m_{-}$ times (2) on both sides and evaluating at $T=0$ gives $-c_{N}=(-1)^{m_{\pm}}$. Then, differentiating $m_{\pm}+1$ times on both sides of the respective functional equation and evaluating at $T=0$ yields

[TABLE]

But for odd $p$,

[TABLE]

[TABLE]

The coincidence $\frac{2^{2}-2-1}{2+1}=\frac{1}{2+1}$ shows that the result is the same for $p=2$! ∎

Appendix A Invariance of Iwasawa invariants under functional equations

This appendix contains results used in the proof of Theorem 4.6, which may be of general interest.

Proposition A.1** ($\lambda$-invariance).**

Let $f(T)$ and $g(T)\in\Lambda$ be related by

[TABLE]

for a unit $u\in\Lambda^{\times}$. Then we have that $\lambda(f(T))=\lambda(g(T))$.

Proof.

We must prove that $f(T)$ and $g(T)$ have the same number of zeroes in $U$ (counted with multiplicity). Let $\zeta$ be such a zero for $f(T)$, i.e. let $(T-\zeta)$ be a factor of $f(T)$ in $\overline{\mathbb{Z}}_{p}[T]$. Then $-\frac{\zeta}{1+\zeta}$ is a zero for $g(T)$ in $U$. We want to show that the factor $(T-\zeta)$ appears in $f(T)$ as often as $\left(T+\frac{\zeta}{1+\zeta}\right)$ does in $g(T)$.

For this, consider the term

[TABLE]

where $U(T)=\frac{(1+T)^{-1}-1}{T}\in\Lambda^{\times}$. Our term vanishes at $T=-\frac{\zeta}{1+\zeta}$, so

[TABLE]

for some $V_{\zeta}(T)\in\overline{\mathbb{Z}}_{p}[T]$. We introduced $U(T)$ in this proof because it helps us evaluate $V_{\zeta}(T)$ at $T=0$: $V_{\zeta}(0)=-(1+\zeta)\in\overline{\mathbb{Z}}_{p}^{\times}$, so $V_{\zeta}(T)$ is a unit.

Under $T\mapsto\frac{1}{1+T}-1$, we thus have $(T-\zeta)^{m}\mapsto\left(T+\frac{\zeta}{1+\zeta}\right)^{m}V_{\zeta}^{m}(T)$. Thus,

[TABLE]

Carrying out the argument with $f$ and $g$ reversed gives us a bijection between zeroes of $f$ and $g$ in $U$ that respects their multiplicities. ∎

Lemma A.2** (Unit invariance).**

Let $U(T)\in\Lambda^{\times}$. Then $U(\frac{1}{1+T}-1)\in\Lambda^{\times}$.

Proof.

Evaluation at $T=0$. ∎

Proposition A.3** ($\mu$-invariance).**

Let $f(T)$ and $g(T)\in\Lambda$ be related by

[TABLE]

for a unit $u\in\Lambda^{\times}$. Then we have that $\mu(f(T))=\mu(g(T))$.

Proof.

Write $f(T)=p^{\mu}D(T)U(T)$ for an integer $\mu$, $D(T)$ distinguished, and $U(T)\in\Lambda^{\times}$. In $\overline{\mathbb{Z}}_{p}[[T]]$, we factor $f(T)=p^{\mu}\left(\prod_{\zeta}(T-\zeta)\right)U(T)$. From the proof of the $\lambda$-invariance lemma, we have $g(T)=up^{\mu}\left(\prod_{\zeta}(T+\frac{\zeta}{1+\zeta})\prod_{\zeta}V_{\zeta}(T)\right)U(\frac{1}{1+T}-1)$. The term $\prod_{\zeta}\left(T+\frac{\zeta}{1+\zeta}\right)V_{\zeta}(T)$ is in $\Lambda$ and has $\mu$-invariant zero, while $U(\frac{1}{1+T}-1)$ is a unit by the unit-invariance lemma above. The result follows. ∎