WEIERSTRASS ZETA FUNCTIONS AND p-ADIC LINEAR RELATIONS

Abstract

We discuss the p-adic Weierstrass zeta functions associated with elliptic curves defined over the field of algebraic numbers and linear relations for their values in the p-adic domain. These results are extensions of the p-adic analogues of results given by Wüstholz in the complex domain [see A. Baker and G. Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs, 9 (Cambridge University Press, Cambridge, 2007), Theorem 6.3] and also generalise a result of Bertrand to higher dimensions [‘Sous-groupes à un paramètre p-adique de variétés de groupe’, Invent. Math. 40(2) (1977), 171–193].

1. Introduction

Let K be a subfield of the field of complex numbers $\mathbb C$ . Let E be an elliptic curve defined over K by the Weierstrass form

$$ \begin{align*}Y^2Z-4X^3+g_2XZ^2+g_3Z^3=0,\end{align*} $$

where $g_2, g_3$ are elements in K satisfying $g_2^3-27g_3^2\ne 0$ . Let $e_1$ and $e_2$ be two roots among the three (distinct) complex roots of the polynomial $4X^3-g_2X-g_3$ . Put ${\Lambda =\mathbb Z\omega ^*_1+\mathbb Z\omega ^*_2}$ with

$$ \begin{align*}\omega^*_1=\int_{e_1}^\infty\dfrac {dx}{\sqrt{4x^3-g_2x-g_3}} \quad \text{and}\quad \omega^*_2=\int_{e_2}^\infty\dfrac {dx}{\sqrt{4x^3-g_2x-g_3}}.\end{align*} $$

Then $\Lambda $ is a lattice in $\mathbb C$ . The elliptic function $\wp :\mathbb C\setminus \Lambda \rightarrow \mathbb C$ relative to $\Lambda $ is defined by

$$ \begin{align*}\wp(z)=\wp(z;\Lambda):=\dfrac 1 {z^2}+\sum_{\omega\in\Lambda\setminus\{0\}}\bigg(\dfrac 1 {(z-\omega)^2}-\dfrac 1 {\omega^2}\bigg).\end{align*} $$

This function is called the Weierstrass elliptic function associated with the elliptic curve E and $\Lambda $ is called the lattice of periods of $\wp $ (or the lattice associated with E). The Weierstrass zeta function associated with E (or relative to $\Lambda $ ) is the function ${\zeta :\mathbb C\setminus \Lambda \rightarrow \mathbb C}$ defined by

$$ \begin{align*}\zeta(z)=\zeta(z;\Lambda):=\dfrac 1 z+\sum_{\omega\in\Lambda\setminus\{0\}}\bigg(\dfrac 1 {z-\omega}+\dfrac 1 {\omega}+\dfrac{z}{\omega^2}\bigg).\end{align*} $$

The Weierstrass zeta function is related to the Weierstrass elliptic function by $\zeta '=-\wp $ and one can write the Laurent expansion at zero of $\zeta $ as

$$ \begin{align*}\zeta(z)=\dfrac 1 z-\sum_{k\ge 1}\mathcal {G}_{2k+2}(\Lambda)z^{2k+1},\end{align*} $$

where $\mathcal {G}_{2k+2}(\Lambda )$ is the Eisenstein series of weight $2k+2$ (with respect to the lattice  $\Lambda $ ). By induction, $\mathcal {G}_{2k+2}(\Lambda )$ can be represented as a polynomial in $g_2, g_3$ with rational coefficients (see [5, Ch. IV]). In other words,

$$ \begin{align*}\zeta(z)=\dfrac 1 z+\sum_{k\ge 1}\alpha_kz^{2k+1}\end{align*} $$

with $\alpha _k\in \mathbb Q[g_2,g_3]$ for all positive integers k.

Since $\wp $ is a periodic function, it follows that $\zeta $ is a quasiperiodic function, that is, for each $\omega \in \Lambda $ , there exists a complex number $\eta =\eta (\omega )$ satisfying ${\zeta (z+\omega )=\zeta (z)+\eta }$ for all $z\in \mathbb C\setminus \Lambda $ . The number $\eta $ is called a quasiperiod of the elliptic curve E. If $(\omega _1,\omega _2)$ is a pair of fundamental periods of $\Lambda $ (that is, $\omega _1$ and $\omega _2$ are complex numbers generating $\Lambda $ over $\mathbb Z$ ), one can show that $\eta (a\omega _1+b\omega _2)=a\eta _1+b\eta _2$ for any integers $a,b$ , where $\eta _1=\eta (\omega _1)$ and $\eta _2=\eta (\omega _2)$ . Furthermore, in the case when the ratio $\omega _2/\omega _1$ has positive imaginary part, we obtain the Legendre relation between the periods and the quasiperiods:

$$ \begin{align*}\omega_2\eta_1-\omega_1\eta_2=2\pi i.\end{align*} $$

Schneider was the first to give a transcendence result concerning linear relations between periods and quasiperiods, by showing that any nonvanishing linear combination of $\omega $ and $\eta $ over $\overline {\mathbb Q}$ is transcendental (see [12]). The result was extended by Coates to pairs of fundamental periods. He obtained a similar result for the numbers $2\pi i, \omega _1, \omega _2, \eta _1, \eta _2$ , where $(\omega _1,\omega _2)$ is a pair of fundamental periods (see [6]). Masser established the dimension of the vector space generated by $1, 2\pi i, \omega _1, \omega _2, \eta _1, \eta _2$ over $\overline {\mathbb Q}$ , proving that this dimension is either 4 if the elliptic curve E has complex multiplication, or 6 otherwise.

In the 1980s, Wüstholz formulated and proved a celebrated theorem in complex transcendental number theory which is called the analytic subgroup theorem (see [1] or [15]). The theorem states that an analytic subgroup defined over $\overline {\mathbb Q}$ of a commutative algebraic group defined over $\overline {\mathbb Q}$ contains a nontrivial algebraic point if and only if it contains a nontrivial algebraic subgroup defined over $\overline {\mathbb Q}$ . The analytic subgroup theorem has many significant consequences, some of which concern elliptic curves. In particular, Wüstholz himself used the theorem to deduce a result on linear relations for the values of the Weierstrass zeta function $\zeta $ at algebraic points of the Weierstrass elliptic function $\wp $ . Here, a complex number $u\in \mathbb C\setminus \Lambda $ is called an algebraic point of $\wp $ if $\wp (u)\in \overline {\mathbb Q}$ . Let $\mathrm {End}(E)$ denote the ring of endomorphisms of E. Then it is known that $K:=\mathrm {End}(E)\otimes _{\mathbb Z}\mathbb Q$ (the field of endomorphisms of E) is either $\mathbb Q$ or an imaginary quadratic field. The following theorem was given by Wüstholz (see [1, Theorem 6.3]).

Theorem 1.1. Let E be an elliptic curve defined over $\overline {\mathbb Q}$ and $\gamma _1,\ldots ,\gamma _n$ algebraic points of $\wp $ . Denote by W the vector space generated by $\gamma _1,\ldots ,\gamma _n$ over K and by V the vector space generated by $1, 2\pi i, \gamma _1,\ldots ,\gamma _n, \zeta (\gamma _1),\ldots ,\zeta (\gamma _n)$ over $\overline {\mathbb Q}$ . Then

$$ \begin{align*}\dim_{\overline{\mathbb Q}} V=2\dim_K W+2.\end{align*} $$

It is natural to extend this result to the p-adic case and the main goal of this paper is to establish an extension of the p-adic analogue of Theorem 1.1. To state it, let E be an elliptic curve given by

$$ \begin{align*}Y^2Z-4X^3+g_2XZ^2+g_3Z^3=0,\end{align*} $$

now defined over $\mathbb C_p$ (that is, $g_2, g_3\in \mathbb C_p$ ). Here, $\mathbb C_p$ denotes the completion of $\overline {\mathbb Q_p}$ with respect to the p-adic absolute value $|\cdot |_p$ as usual. Let $\wp _p$ be the (Lutz–Weil) p-adic elliptic function associated with the elliptic curve E (see [9, 14]). The function $\wp _p$ is analytic on the set $\mathscr {D}_p\setminus \{0\}$ , where $\mathscr {D}_p$ is the p-adic domain of E defined by

$$ \begin{align*}\mathscr D_p:= \{z\in\mathbb C_p : |1/4|_p\max\{|g_2|_p^{1/4}, |g_3|_p^{1/6}\}z\in B(r_p)\}\end{align*} $$

with $B(r_p)$ the set of all p-adic numbers x in $\mathbb C_p$ such that $|x|_p<r_p:=p^{-{1}/{(p-1)}}$ . As in the complex case, we say that a nonzero p-adic number $u\in \mathscr D_p$ is an algebraic point of $\wp _p$ if $\wp _p(u)\in \overline {\mathbb Q}$ . Let $\zeta _p$ be the p-adic Weierstrass zeta function (p-adic analogue of the Weierstrass zeta function $\zeta $ ) associated with E which is, by definition, the (unique) odd p-adic meromorphic function on $\mathscr D_p$ satisfying $\zeta _p'=-\wp _p$ . Let $\mathrm {Log}_p:\mathbb C_p\setminus \{0\}\rightarrow \mathbb C_p$ be the Iwasawa logarithm (see [11, Ch. 5, Section 4.5]). We now state our main theorem.

Theorem 1.2. Let E be an elliptic curve defined over $\overline {\mathbb Q}$ . Let $u_1,\ldots , u_l$ be nonzero algebraic numbers and $v_1,\ldots , v_n$ algebraic points of $\wp _p$ . Denote by U the vector space generated by $\mathrm {Log}_p(u_1),\ldots , \mathrm {Log}_p(u_l)$ over $\mathbb Q$ and by V the vector space generated by $v_1,\ldots ,v_n$ over the field K of endomorphisms of E. Then the dimension of the vector space W generated by $1, \mathrm {Log}_p(u_1),\ldots ,\mathrm {Log}_p(u_l), v_1,\ldots ,v_n, \zeta _p(v_1),\ldots ,\zeta _p(v_n)$ over $\overline {\mathbb Q}$ is determined by

$$ \begin{align*}\dim_{\overline{\mathbb Q}} W=1+\dim_{\mathbb Q}U+2\dim_KV.\end{align*} $$

In the case when $l=n=1$ , we deduce at once from Theorem 1.2 the following result which is an extension of a result given by Bertrand in 1977 (see [2, Proposition 1]).

Corollary 1.3. Let E be an elliptic curve defined over $\overline {\mathbb Q}$ . Let u be a nonzero algebraic number with $\mathrm {Log}_p(u)\ne 0$ and v an algebraic point of $\wp _p$ . Let $\alpha $ , $\beta $ and $\gamma $ be algebraic numbers not all zero. Then the number $\alpha \mathrm {Log}_p(u)+\beta v+\gamma \zeta _p(v)$ is transcendental.

2. Extensions of commutative algebraic groups

In this section, let K be a fixed algebraically closed field of characteristic $0$ . Let A and B be commutative algebraic groups defined over K. A commutative algebraic group C defined over K is called an extension of A by B if there is an exact sequence of commutative algebraic groups

To give an extension C of A by B is equivalent to giving a pair $(i,\pi )\in \mathrm {Hom}(B,C)\times \mathrm {Hom}(C,A)$ for which the above sequence is exact. Let

and

be extensions of commutative algebraic groups. A homomorphism between the above two extensions is a triple of homomorphisms $\varphi : C\rightarrow C', \alpha : A\rightarrow A', \beta : B\rightarrow B'$ of algebraic groups such that the diagram

commutes. Clearly, $\varphi $ is an isomorphism if and only if $\alpha $ and $\beta $ are isomorphisms. In the case $A=A', B=B'$ and $\alpha =\mathrm {id}_A, \beta =\mathrm {id}_B$ , we say that the two extensions C and $C'$ are equivalent if there is a homomorphism between them. The set of equivalence classes $[C]$ of extensions forms a commutative group $\mathrm {Ext}^1(A,B)$ with the neutral element $[A\times B]$ (via the Baer sum). We write C for its equivalence class $[C]$ by abuse of notation. The bi-functor $\mathrm {Ext}^1$ which assigns to the pair $(A,B)$ the group $\mathrm {Ext}^1(A,B)$ is contravariant in the first variable and covariant in the second one. This means that if $\alpha : A'\rightarrow A$ and $\beta : B\rightarrow B'$ are homomorphisms between commutative algebraic groups, then they induce homomorphisms $\alpha ^*:\mathrm {Ext}^1(A,B)\rightarrow \mathrm {Ext}^1(A', B)$ and $\beta _*:\mathrm {Ext}^1(A,B)\rightarrow \mathrm {Ext}^1(A, B')$ . The two homomorphisms $\alpha ^*$ and $\beta _*$ make the diagram

commute. Furthermore, $\mathrm {Ext}^1$ is additive in both variables, which implies that

$$ \begin{align*}\mathrm{Ext}^1(A_1\times A_2, B)=\mathrm{Ext}^1(A_1,B)\times \mathrm{Ext}^1(A_2,B)\end{align*} $$

and

$$ \begin{align*}\mathrm{Ext}^1(A, B_1\times B_2)=\mathrm{Ext}^1(A,B_1)\times \mathrm{Ext}^1(A,B_2).\end{align*} $$

For example, we describe the exponential map of G in the case where G is an extension of an elliptic curve by the additive group $\mathbb G_a$ defined over $\overline {\mathbb Q}$ as given in [4]. (We refer the reader to [7] for the general case.) Let E be an elliptic curve defined over $\overline {\mathbb Q}$ and let G be an extension of E by $\mathbb G_a$ . By compactification,

Denote by $0$ the identity element in E. The divisor $D=(\overline G-G)+3\pi ^*(0)$ is very ample for $\overline G$ and $l(D)=6$ . Hence, there is an embedding of $\overline G$ into $\mathbb P_5$ , and one can express the exponential map of G in terms of the Weierstrass elliptic and zeta functions $\wp (z), \zeta (z)$ associated with E. One can identify the Lie algebra $\mathrm {Lie}(G(\mathbb C))$ with $\mathbb C^2$ and the exponential map of G is expressed by

$$ \begin{align*}\exp_{G(\mathbb C)}(z,t)=(1:\wp(z):\wp'(z):f_1(z,t):f_2(z,t):f_3(z,t)) \quad \mbox{for } z\not\in \Lambda\end{align*} $$

and $\exp (z,t)=(0:0:1:0:0:t+b\eta (z))$ for $z\in \Lambda $ , where

$$ \begin{align*}f_1(z,t)=t+b\zeta(z), f_2(z,t)=\wp(z)f_1(z,t)+\dfrac b 2\wp'(z), f_3(z,t)=\wp'(z)f_1(z,t)+2b\wp^2(z)\end{align*} $$

for some algebraic number b.

3. Analytic representation of exponential maps

In this section, we discuss the analytic representation of the complex and p-adic exponential maps of a commutative algebraic group defined over $\overline {\mathbb Q}$ (with respect to a fixed basis for its Lie algebra). Let G be a commutative algebraic group defined over $\overline {\mathbb Q}$ of positive dimension n. It is known that, by a compactification constructed by Serre, there is an embedding defined over $\overline {\mathbb Q}$ from G into the projective space $\mathbb P^N$ with projective coordinates $X_0,\ldots , X_N$ for some positive integer N (see [13]). One can now describe the exponential maps of G (over $\mathbb C$ and $\mathbb C_p$ ) by analytic functions as follows. Let $\overline G$ denote the Zariski closure of G in $\mathbb P^N$ and let $G_0$ be the open affine subset defined by $\overline G\cap \{X_0\neq 0\}$ . Then the affine algebra $\Gamma (G_0, O_{\overline G})$ of $G_0$ is generated over $\overline {\mathbb Q}$ by $\xi _i=X_i/X_0$ (the affine coordinates on $G_0$ ) for $i=1,\ldots ,N$ , and we write it as $\overline {\mathbb Q}[\xi _1,\ldots ,\xi _N]$ . It is known that any element in the Lie algebra $\mathrm {Lie}(G)$ of G maps $\overline {\mathbb Q}[\xi _1,\ldots , \xi _N]$ into itself. In particular, for each $D\in \mathrm {Lie}(G)$ , there exist polynomials $P_{1,D},\ldots , P_{N,D}$ in N variables with algebraic coefficients such that

$$ \begin{align*}D\xi_i=P_{i,D}(\xi_1,\ldots,\xi_N)\quad\mbox{for } i=1,\ldots, N.\end{align*} $$

Let v be a place of $\overline {\mathbb Q}$ . Then there is a natural embedding from $\overline {\mathbb Q}$ into $C_v$ , where $C_v=\mathbb C$ if v is infinite, and $C_v=\mathbb C_p$ if v is finite and lies above p. The set $G(C_v)$ is a v-adic Lie group whose Lie algebra $\mathrm {Lie}(G(C_v))=\mathrm {Lie}(G)\otimes _{\overline {\mathbb Q}}C_v$ , and it is known that the v-adic exponential map $\exp _{G(C_v)}$ of the Lie group $G(C_v)$ is a local diffeomorphism defined on a subgroup $G_v$ of $\mathrm {Lie}(G(C_v))$ (see [3]). From now on, we fix a basis $D_1,\ldots , D_n$ for the $\overline {\mathbb Q}$ -vector space $\mathrm {Lie}(G)$ which is also a basis for the $C_v$ -vector space $\mathrm {Lie}(G(C_v))$ (by the identifications $D_i=D_i\otimes 1$ for $i=1,\ldots ,N$ ). Let $\delta _1,\ldots ,\delta _n$ denote the canonical basis of $\mathrm {Lie}(C_v^n)$ , that is, $\partial _ix_j=\delta _{ij}$ for $i=1,\ldots ,n$ and for $j=1,\ldots ,N$ , where $\delta _{ij}$ is Kronecker’s delta and $x_1,\ldots ,x_n$ are the coordinate functions of $C_v^n$ . There exists an isomorphism $\phi : C_v^n\rightarrow \mathrm {Lie}(G(C_v))$ with the property that the differential of the composition map $\exp _{G(C_v)}\circ \phi $ satisfies

$$ \begin{align*}d(\exp_{G(C_v)}\circ \phi)(\partial_i)=D_i\quad\mbox{for } i=1,\ldots,n.\end{align*} $$

Put $f_{i,v}=\xi _i\circ \exp _{G(C_v)}\circ \phi $ for $i=1,\ldots , N$ . The functions $f_{1,v},\ldots , f_{N,v}$ are analytic on a neighbourhood $\mathcal C_v$ of the origin in $C_v^n$ , and the system $\{f_{1,v},\ldots , f_{N,v}\}$ is called the (normalised) analytic representation of the exponential map $\exp _{G(C_v)}$ (with respect to D). By convention, for each $i\in \{1,\ldots , N\}$ , we write $f_{i,p}$ for $f_{i,v}$ and $\mathcal C_p$ for $\mathcal C_v$ if $C_v=\mathbb C_p$ , and we write $f_{i}$ for $f_{i,v}$ and $\mathcal C$ for $\mathcal C_v$ if $C_v=\mathbb C$ . (Note that in the complex case, the functions $f_1,\ldots , f_N$ can be extended as meromorphic functions on the whole space $\mathbb C^n$ .) For $i=1,\ldots , N$ and $j=1,\ldots , n$ ,

$$ \begin{align*} \partial_j(f_{i,v})&=\partial_j(\xi_i\circ\exp_{G(C_v)}\circ\phi)=(d(\exp_{G(C_v)}\circ \phi)(\partial_i)\xi_i)\circ\exp_{G(C_v)}\circ\phi \\ &=(D_j\xi_i)\circ\exp_{G(C_v)}\circ\phi=P_{i,D_j}(\xi_1,\ldots,\xi_N)\circ\exp_{G(C_v)}\circ\phi=P_{i,D_j}(f_{1,v},\ldots,f_{N,v}). \end{align*} $$

By induction, one can show that for $j=1,\ldots , N$ and for nonnegative integers $i_1,\ldots , i_n$ , there exists a polynomial $P_{i_1,\ldots ,i_n, j}$ in N variables with coefficients in $\overline {\mathbb Q}$ such that

$$ \begin{align*}(\partial_1^{i_1}\cdots \partial_n^{i_n})f_{j,v}=P_{i_1,\ldots,i_n, j}(f_{1,v},\ldots, f_{N,v}).\end{align*} $$

Since $\exp _{G(C_v)}(0)=e\in G(\overline {\mathbb Q})$ (where e denotes the identity element of G), it follows that $f_i(0)=f_{i,p}(0)\in \overline {\mathbb Q}$ for $i=1,\ldots ,N$ . Using the Taylor expansions of $f_{1,v},\ldots , f_{N,v}$ at $0$ , we get the following proposition.

Proposition 3.1. There exist formal power series $F_1,\ldots , F_N\in \overline {\mathbb Q}[[X_1,\ldots , X_N]]$ converging both in $\mathcal C$ and $\mathcal C_p$ such that

$$ \begin{align*}f_{1}(x)=F_1(x),\ldots, f_{N}(x)=F_N(x) \quad\mbox{for all } x\in\mathcal C\end{align*} $$

and

$$ \begin{align*}f_{1,p}(x)=F_1(x),\ldots, f_{N,p}(x)=F_N(x) \quad\mbox{for all } x\in\mathcal C_p.\end{align*} $$

4. Proof of the main theorem

This section is devoted to the proof of Theorem 1.2 which follows that of Theorem 1.1 with some extensions.

Proof of Theorem 1.2.

Without loss of generality, we may assume that the elements $\mathrm {Log}_p(u_1),\ldots , \mathrm {Log}_p(u_l)$ are linearly independent over $\mathbb Q$ and the elements $v_1,\ldots , v_n$ are linearly independent over K, that is, $\dim _{\mathbb Q}U=l$ and $\dim _KV=n$ . It is clear that there exists a positive integer r sufficiently large for which $w_i:=u_i^{p^r}\in B(r_p)$ for all $i=1,\ldots , l$ . Let $\log _p$ denote the p-adic logarithm function. Then

$$ \begin{align*}\log_p(w_i)=\mathrm{Log}_p(w_i)=\mathrm{Log}_p(u_i^{p^r})=p^r\mathrm{Log}_p(u_i) \quad\mbox{for } i=1,\ldots,l.\end{align*} $$

We have to show that the elements

$$ \begin{align*}1, \log_p(w_1),\ldots,\log_p(w_l), v_1,\ldots, v_n, \zeta_p(v_1),\ldots,\zeta_p(v_n)\end{align*} $$

are linearly independent over $\overline {\mathbb Q}$ . Suppose that this is not true. Then there exists a nonzero linear form L in $l+2n+1$ variables $T_{0}, T_1,\ldots ,T_l, T_1',\ldots , T_n', T_1",\ldots , T_n"$ with coefficients in $\overline {\mathbb Q}$ such that L vanishes on $1, \log _p(w_1),\ldots ,\log _p(w_l), v_1,\ldots , v_n, \zeta _p(v_1)$ , $\ldots ,\zeta _p(v_n)$ . We write L in the form $L=L_0+L'+L"$ , where $L_0=\alpha T_{0}+\beta _1 T_1+\cdots +\beta _lT_l$ with $\alpha , \beta _1,\ldots ,\beta _l\in \overline {\mathbb Q}$ and where $L',L"$ are linear forms in $T_1',\ldots , T_n'$ and $T_1",\ldots , T_n"$ , respectively. Let $G\in \mathrm {Ext}^1(\mathbb G^l_m\times E^n, \mathbb G_a)$ be the extension of $\mathbb G^l_m\times E^n$ by $\mathbb G_a$ determined by $L"$ . The components of the complex exponential map $\exp _{G(\mathbb C)}$ of G are give by the functions

$$ \begin{align*}x_{0}+L"(\zeta(y_1),\ldots,\zeta(y_n)), e^{x_1},\ldots, e^{x_l}, \wp(y_1),\wp'(y_1),\ldots, \wp(y_n),\wp'(y_n)\end{align*} $$

for complex variables $x_0, x_1,\ldots , x_l, y_1,\ldots , y_n$ . By Proposition 3.1, the corresponding components of the p-adic exponential map $\exp _{G(\mathbb C_p)}$ are given by the functions

$$ \begin{align*}z_{0}+L"(\zeta_p(t_1),\ldots,\zeta_p(t_n)), e_p(z_1),\ldots,e_p(z_l), \wp_p(t_1),\wp'(t_1),\ldots, \wp_p(t_n),\wp'_p(t_n)\end{align*} $$

for p-adic variables $z_0, z_1,\ldots , z_l, t_1,\ldots , t_n$ , where $e_p$ denotes the p-adic exponential function. Consider the point

$$ \begin{align*}\epsilon=(\beta_1 \log_p(w_1)+\cdots+\beta_l\log_p(w_l)+L'(v_1,\ldots,v_n), \log_p(w_1),\ldots,\log_p(w_l), v_1,\ldots, v_n).\end{align*} $$

Then the point $\gamma :=\exp _{G(\mathbb C_p)}(\epsilon )$ is

$$ \begin{align*} \begin{split} (\beta_1\log_p(w_1)+\cdots+\beta_l\log_p&(w_l)+L'(v_1,\ldots,v_n)+L"(\zeta_p(v_1),\ldots,\zeta_p(v_n)), \\ & w_1,\ldots,w_l, \wp_p(v_1),\wp'_p(v_1),\ldots, \wp_p(v_n),\wp_p'(v_n)). \end{split} \end{align*} $$

Since

$$ \begin{align*}L(1, \log_p(w_1),\ldots,\log_p(w_l), v_1,\ldots, v_n, \zeta_p(v_1),\ldots,\zeta_p(v_n))=0,\end{align*} $$

it follows that

$$ \begin{align*}\beta_1 \log_p(w_1)+\cdots+\beta_l\log_p(w_l)+L'(v_1,\ldots,v_n)+L"(\zeta_p(v_1),\ldots,\zeta_p(v_n))=-\alpha\in\overline{\mathbb Q}.\end{align*} $$

In particular, this means that the point $\gamma $ is an algebraic point of G. Let $\log _{G(\mathbb C_p)}$ be the p-adic logarithm map of G and let S be the $\overline {\mathbb Q}$ -vector subspace of $\mathrm {Lie}(G)$ (which is identified with $\overline {\mathbb Q}^{l+n+1}$ ) given by

$$ \begin{align*}S=\{(s_0,s_1,\ldots,s_{l+n})\in\overline{\mathbb Q}^{l+n+1}: s_0-\beta_1s_1-\cdots-\beta_ls_l-L'(s_{l+1},\ldots,s_{l+n})=0\}.\end{align*} $$

We see that

$$ \begin{align*}\log_{G(\mathbb C_p)}(\gamma)=\log_{G(\mathbb C_p)}(\exp_{G(\mathbb C_p)}(\epsilon))=\epsilon\in S\otimes_{\overline{\mathbb Q}}\mathbb C_p.\end{align*} $$

Thanks to the p-adic analytic subgroup theorem (see [8] or [10]), there exists a nontrivial connected algebraic subgroup H of G defined over $\overline {\mathbb Q}$ such that $\gamma \in H(\overline {\mathbb Q})$ and $\mathrm {Lie}(H)\subseteq S$ . Let $\pi $ be the composition of the homomorphism $G\rightarrow \mathbb G_m^l\times E^n$ and the canonical projection $\mathbb G^l_m\times E^n\rightarrow E^n$ . Then the algebraic subgroup $\mathcal E:=\pi (H)$ is isogenous (over $\overline {\mathbb Q}$ ) to $E^m$ with $m\le n$ . This gives a corresponding element $p: E^n\rightarrow \mathcal E\hookrightarrow E^n$ in $\mathrm {End}(E^n)$ . Note that $\pi :G\rightarrow E^n$ induces the differential $d\pi $ from the Lie algebra of G to that of $E^n$ and the algebra of endomorphisms $\mathrm {End}(E^n)\otimes _{\mathbb Z} \mathbb Q$ is identified with the matrix algebra $M_n(K)$ . This means that the endomorphism $\mathrm {id}_{E^n}-p$ can be written as an $n\times n$ matrix with entries in K. Furthermore, since $\gamma \in H$ , one has

$$ \begin{align*}\epsilon=\log_{G(\mathbb C_p)}(\gamma)=\log_{H(\mathbb C_p)}(\gamma)\in\mathrm{Lie}(H)\otimes_{\mathbb Q}\mathbb C_p.\end{align*} $$

It follows that the point $(v_1,\ldots , v_n)=d\pi (\epsilon )\in \mathrm {Lie}(\mathcal E)$ which turns out to be the kernel of the endomorphism given by the above matrix. However, the elements $v_1,\ldots , v_n$ are linearly independent over K, so that this matrix must be trivial. In other words, $p=\mathrm {id}_{E^n}$ , that is, $\mathcal E=E^n$ .

Next, we see that $G\cong \mathbb G^l_m\times G_0$ , where $G_0\in \mathrm {Ext}^1(E^n,\mathbb G_a)$ since $\mathrm {Ext}^1(\mathbb G_m,\mathbb G_a)$ is trivial (in fact, it is known more generally that the group extension of linear algebraic groups is trivial). Hence, without loss of generality, we may assume that the algebraic numbers $\beta _1, \ldots , \beta _l$ are not all zero (since, if not, one can take the quotient of G by the multiplicative group $\mathbb G_m$ , and we are in a simpler case with $G_0$ ). The intersection of H with $\mathbb G_a\times \mathbb G^l_m$ is an algebraic subgroup of $\mathbb G_a\times \mathbb G_m^l$ , and therefore has the form $H_a\times H_m$ , where $H_a$ and $H_m$ are (connected) algebraic subgroups of $\mathbb G_a$ and $\mathbb G_m^l$ , respectively (see [1, Proposition 4.3]). This leads to

$$ \begin{align*}\mathrm{Lie}(H_a)\times \mathrm{Lie}(H_m)=&~\mathrm{Lie}(H_a\times H_m) \\ =&~\mathrm{Lie}(H)\cap (\mathrm{Lie}(\mathbb G_a)\times \mathrm{Lie}(\mathbb G_m^l))= \mathrm{Lie}(H)\cap (\overline{\mathbb Q}\times \overline{\mathbb Q}^l). \end{align*} $$

If $H_m$ is a proper algebraic subgroup of the torus $\mathbb G_m^l$ , it follows from [1, Lemma 4.4] that the Lie algebra $\mathrm {Lie}(H_m)$ is given by $L_1=\cdots =L_d=0$ , where $d=l-\dim H_m\ge 1$ and $L_1,\ldots , L_d$ are nonzero linear forms in n variables with integer coefficients. In particular, this means that $\log _p(w_1),\ldots ,\log _p(w_l)$ are linearly dependent over $\mathbb Q$ , or equivalently, $\mathrm {Log}_p(u_1),\ldots , \mathrm {Log}_p(u_l)$ are linearly dependent over $\mathbb Q$ . This contradiction shows that $H_m=\mathbb G_m^l$ and then $H_a$ must be trivial (since $\dim H\le \dim _{\overline {\mathbb Q}}S=n+l$ ). This enables us to conclude that $\beta _1s_1+\cdots +\beta _ls_l=0$ for all $s_1,\ldots , s_l\in \overline {\mathbb Q}$ and this happens if and only if $\beta _1=\cdots =\beta _l=0$ , which is a contradiction. The theorem is proved.

As in the complex case, it is also possible to slightly extend the main theorem to the case of several p-adic Weierstrass zeta functions as follows. Let $E_1,\ldots , E_n$ be elliptic curves defined over $\overline {\mathbb Q}$ . For each $i\in \{1,\ldots ,n\}$ , denote by $\wp _{p,i}$ and $\zeta _{p,i}$ the p-adic elliptic function and the p-adic Weierstrass zeta function associated with the elliptic curve $E_i$ , respectively. Let $v_i$ be an algebraic point of $\wp _{p,i}$ for $i=1,\ldots , n$ . Let $I_\nu \ (\nu =1,\ldots ,k)$ be maximal sets of indices such that $E_i$ are pairwise isogenous (over $\overline {\mathbb Q}$ ) for all $i\in I_\nu $ . Fix an element $E^{(\nu )}$ in the set $\{E_j : j\in I_\nu \}$ . The field of endomorphisms of $E^{(\nu )}$ is the same as that of $E_j$ for any $j\in I_\nu $ , and we denote it by $K_\nu $ . Let $V_\nu $ be the vector space generated by the set $\{v_j : j\in I_\nu \}$ over $K_\nu $ . Then we obtain the following theorem which is an extension of the p-adic analogue of [1, Theorem 6.4].

Theorem 4.1. Let $u_1,\ldots ,u_l$ be nonzero algebraic numbers and U the vector space generated by $\mathrm {Log}_p(u_1),\ldots ,\mathrm {Log}_p(u_l)$ over $\mathbb Q$ . Then the dimension of the vector space W generated by $1, \mathrm {Log}_p(u_1),\ldots ,\mathrm {Log}_p(u_l), v_1,\ldots , v_n, \zeta _{p,1}(v_1),\ldots ,\zeta _{p,n}(v_n)$ over $\overline {\mathbb Q}$ is determined by

$$ \begin{align*}\dim_{\overline{\mathbb Q}}W=1+\dim_{\mathbb Q}U+2(\dim_{K_1}V_1+\cdots+\dim_{K_k}V_k).\end{align*} $$