Congruences of two-variable p-adic L-functions of congruent modular forms of different weights

Abstract

Vatsal (Duke Math J 98(2):397–419, 1999) proved that there are congruences between the p-adic L-functions (constructed by Mazur and Swinnerton-Dyer in Invent Math 25:1–61, 1974) of congruent modular forms of the same weight under some conditions. On the other hand, Kim (J Number Theory 144: 188–218, 2014), the second author, constructed two-variable p-adic L-functions of modular forms attached to imaginary quadratic fields generalizing Hida’s work (Invent Math 79:159–195, 1985), and the novelty of his construction was that it works whether p is an ordinary prime or not. In this paper, we prove congruences between the two-variable p-adic L-functions (of the second author) of congruent modular forms of different but congruent weights under some conditions when p is a nonordinary prime for the modular forms. This result generalizes the work of Emerton et al. (Invent Math 163(3): 523–580, 2006), who proved similar congruences between the p-adic L-functions of congruent modular forms of congruent weights when p is an ordinary prime.

Appendix (constructing the two-variable p-adic L-functions without any condition on the Hecke character \(\chi \))

We have two goals. First, we describe a way to remove some conditions that the second author assumed in [10] to construct the two-variable p-adic L-functions. However, the method we will describe is not suitable for obtaining congruences we study in this paper. So, second, we describe a construction of ${\mathcal {L}}_{f,\chi }(X,Y)$ slightly different from that of [10] based on a hypothesis which we will propose.

(The construction in [10] is somewhat different from that of this paper because [10] is not exactly suitable for studying the congruence between ${\mathcal {L}}_{f_2,\chi }$ and ${\mathcal {L}}_{f_k,\chi }$.)

First, we briefly explain the construction in [10] and then relax one of the conditions in it.

Let f(z) be an eigenform of level $\Gamma _1(pN)$, weight k, and character $\epsilon$. (The second author explained why we do not need to consider a seemingly more general case of the level $\Gamma _1(p^sN)$ in [10, Remark 1.2].)

Let $\chi$ be a finite (i.e., infinity type (0, 0)) Hecke character of K. As we saw in Sect.2 (and also in [10]), there is a distribution $\Phi _{k-1}^b(\chi , \omega , \cdot )$ for $\omega =\epsilon \chi _0^{-1} \chi _D^{-1}$ (simply denoted by $\Phi _{k-1}(\cdot )$) of the cosets of ${\bar{Z}} =\mathrm{Gal}(K_{\infty }/K)\cong \mathbb {Z}^2_p$ so that

$$\begin{aligned} \Phi _{k-1}({\mathfrak {c}})\in M_k(\Gamma _1(Mp^{i+j+2}); \epsilon ) \end{aligned}$$

for every ${\mathfrak {c}}\in {\bar{Z}}_{i,j}={\bar{Z}}/P_{i,j} \cong \mathbb {Z}/(p^i)\times \mathbb {Z}/(p^j)$. In [10], the second author assumed that $\chi \not =\lambda \circ N_{K/{\mathbb {Q}}}$ for any Dirichlet character $\lambda$ which ensures that $\Phi _{k-1}({\mathfrak {c}})$ is a cusp form. We will relax this condition.

Remark 3.25

• (a) We can always choose a character $\chi$ which does not factor through a Dirichlet character, and $N_{K/{\mathbb {Q}}}$. But doing so would exclude some interesting characters (for example, $\mathbf 1_K$, the trivial Hecke character of K).

• (b) We may assume $M=N$ by replacing f by some oldform ${\hat{f}}$ of level $\Gamma _1(Mp)$ stabilized from f.

The construction in [10] proceeds as follows:

We need to construct a linear map $l_f:S_k(\Gamma _1(pN))\rightarrow F$ for some finite extension $F/{\mathbb {Q}}$ so that

$$\begin{aligned} l_f(f)&\not =&0,\\ l_f(t\cdot g)= & {} a_t l_f(g) \end{aligned}$$

for any Hecke operator t and any $g\in S_k(\Gamma _1(pN))$, where $a_t$ is given by $t\cdot f=a_t f$ for some $a_t \in F$.

Then, the distribution $\mu _f$ (similar to Sect.refHabsburg) is given by

$$\begin{aligned} \mu _f({\mathfrak {c}})=l_f \left( \displaystyle \left( \frac{T_p}{a_p(f)}\right) ^{i+j+1} \Phi _{k-1}({\mathfrak {c}}) \right) \end{aligned}$$

for ${\mathfrak {c}} \in {\bar{Z}}_{i,j}$. So it all comes down to constructing the linear map $l_f$.

We presented one method of construction in Section 3.2. Alternatively, we can proceed as in [10] which we will demonstrate briefly now:

First, we note

$$\begin{aligned} S_k(\Gamma _1(pN))=\prod U(g_i) \end{aligned}$$

for distinct eigenforms $g_i$, where $U(g_i)$ is the set of elements of $S_k(\Gamma _1(pN))$ which have the same eigenvalues as $g_i$ for all but a finite number of the Hecke operators. Then, $U(g_i)$ is orthogonal to $U(g_j)$ for any $i\not =j$ with respect to the Peterson inner product. We may assume $g_1=f$.

The form f may or may not be a newform. If f is a newform, then $U(f)=\mathbb C\cdot f$.

If f is not a newform, then we can further decompose $U(f)=\prod _{i=1}^h \mathbb C\cdot f_i$ for some eigenforms $f_1,\ldots , f_h \in U(f)$.

More precisely, there is some form ${\tilde{f}}$ which is a newform for some level $\Gamma _1(N')$, $N'|Np$ so that the eigenforms $f_1,f_2,\ldots , f_h$ of level $\Gamma _1(pN)$ are stabilized from ${\tilde{f}}$, and form a basis of U(f).

We let

$$\begin{aligned} U=\mathbb C \cdot f_1,\quad V=\prod _{i= 2}^h \mathbb C\cdot f_i + \prod _{j\ge 2} U(g_j), \end{aligned}$$

then we have $S_k(\Gamma _1(pN))=U\oplus V$. Thus, for any $g \in S_k(\Gamma _1(pN))$,

$$\begin{aligned} g=g_1+g_2 \end{aligned}$$

for unique $g_1 \in U$, $g_2 \in V$ where $g_1 = x_g f_1=x_g f$ for some $x_g \in \mathbb C$. Then we define

$$\begin{aligned} l_f(g)=x_g. \end{aligned}$$

Now, we relax the condition on $\chi$ as follows: We do not assume anything on $\chi$ other than it being of finite type. In this case, $\Phi _{\chi }(\cdot )$ may not be a cusp form, so we need to find a way to decompose $M_k(\Gamma _1(pN))$ into orthogonal components as above. In fact, [7] has a solution.

Let $M_k(\Gamma _1(Np))$ denote the set of all modular forms of weight k and level $\Gamma _1(Np)$, and $N_k (\Gamma _1(Np))$ denote the set of the modular forms $g(z) \in M_k(\Gamma _1(Np))$ satisfying

$$\begin{aligned} \langle g(z), \, S_k(\Gamma _1(Np)) \rangle _{\Gamma _1(Np)} =0. \end{aligned}$$

(In other words, $N_k (\Gamma _1(Np))$ is the orthogonal complement of $S_k(\Gamma _1(Np))$ with respect to the Petersson Inner Product.) In fact, $N_k (\Gamma _1(Np))$ is generated by the Eisenstein series (see [19]). Since the set of cusp forms is invariant under the adjoint Hecke operators, $N_k (\Gamma _1(Np))$ is invariant under the Hecke algebra. Write $U_0 =N_k (\Gamma _1(Np))$ for convenience. Then, $M_k(\Gamma _1(Np))$ has the decomposition

$$\begin{aligned} M_k(\Gamma _1(Np))= & {} N_k (\Gamma _1(Np)) \oplus S_k (\Gamma _1(Np)) \\= & {} U_0+ \prod _{i\ge 1} U(g_i) \end{aligned}$$

(again, see [19]) where $S_k (\Gamma _1(Np))= \prod _{i \ge 1} U(g_i)$ as before. Similar to the above discussion, we can write $U(g_1)=\prod _{i=1}^h \mathbb C\cdot f_i$ for some eigenforms $f_i$, and assume $f_1=f$. Then we let

$$\begin{aligned} U=\mathbb C \cdot f_1,\quad V=U_0+\prod _{i= 2}^h \mathbb C\cdot f_i + \prod _{j\ge 2} U(g_j). \end{aligned}$$

For any $g \in M_k(\Gamma _1(pN))=U\oplus V$, we can write

$$\begin{aligned} g=g_1+g_2 \end{aligned}$$

for unique $g_1 \in U$, $g_2 \in V$ where $g_1 = x_g f_1=x_g f$ for some $x_g \in \mathbb C$. Then we define

$$\begin{aligned} l_f(g)=x_g. \end{aligned}$$

As explained earlier, $l_f$ induces a two-variable p-adic L-function.

However, this construction does not help us much when we want to study congruences between $l_{f_2}(\Phi _{1, \chi }(\cdot ))$ and $l_{f_k}(\Phi _{k-1, \chi }(\cdot ))$. For example, following the construction above, suppose we can write

$$\begin{aligned}&\Phi _1({\mathfrak {c}})=E_1+C_1, \\&\Phi _{k-1}({\mathfrak {c}})=E_{k-1}+C_{k-1}, \end{aligned}$$

where $E_1,E_{k-1}$ are some combinations of Eisenstein series and $C_1,C_{k-1}$ are cusp forms. But it is not clear that the congruence $\Phi _1({\mathfrak {c}}) \equiv \Phi _{k-1}({\mathfrak {c}}) (\mathrm{mod}{p^r})$ for some r implies $C_1 \equiv C_{k-1} (\mathrm{mod}{p^{r'}})$ for any $r'$ (because it is not clear that $E_1$ and $E_{k-1}$ are congruent modulo any power of p. In fact, it is not even clear that they are p-adically integral).

We can improvise a solution as follows:

Hypothesis 3.26

We assume

$$\begin{aligned} M_2(\Gamma _1(pN), {\mathcal {O}})_{{\mathfrak {m}}_{f_2}}= & {} S_2(\Gamma _1(pN), {\mathcal {O}})_{{\mathfrak {m}}_{f_2}}, \\ M_k(\Gamma _1(pN), {\mathcal {O}})_{{\mathfrak {m}}_{f_k}}= & {} S_k(\Gamma _1(pN), {\mathcal {O}})_{{\mathfrak {m}}_{f_k}}. \end{aligned}$$

(In other words, the Eisenstein series are 0 when localized by ${\mathfrak {m}}_{f_2}$ and ${\mathfrak {m}}_{f_k}$.) This eliminates the issue of noncusp forms at once. However, this assumption is probably difficult to verify. In any case, under this assumption, the main idea of this paper applies word by word.