# ρ — Adic Analogues of Ramanujan Type Formulas for 1/π

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## Abstract

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**MSC**11G07; 11G15; 11F11; 44A20

## 1. Introduction

## 2. Statement of Results

**Theorem 1.**For $d\in \{2,3,4,6\}$, let ${\lambda}_{d}\in \overline{\mathbb{Q}}$ such that $\mathbb{Q}\left({\lambda}_{d}\right)$ is totally real, the elliptic curve ${E}_{d}\left({\lambda}_{d}\right)$ has complex multiplication, and $|{\lambda}_{d}|<1$ for an embedding of ${\lambda}_{d}$ to $\mathbb{C}$. For each prime p that is unramified in $\mathbb{Q}\left(\sqrt{1-{\lambda}_{d}}\right)$ and coprime to the discriminant of ${E}_{d}\left({\lambda}_{d}\right)$ such that $a,{\lambda}_{d}$ can be embedded in ${\mathbb{Z}}_{p}^{*}$ (and we fix such embeddings), then

**Theorem 2.**With notation and assumptions as in Theorem 1,

## 3. Ramanujan Type Formula for $1/\pi $

#### 3.1. Hypergeometric Formulas

**Lemma 3.**For x such that both hand sides converge,

#### 3.2. Picard–Fuchs Equation

**Lemma 4.**Let $f\left(\lambda \right)$ be defined by

#### 3.3. Modular Forms and Singular Values of Modular Forms

**Proposition 5.**For each imaginary quadratic field K, there is a number ${\Omega}_{K}\in {\mathbb{C}}^{*}$ such that for all meromorphic modular forms F of weight k with algebraic coefficients, and all $\tau \in K\cap \mathcal{H}$,

**Lemma 6.**Let K be an imaginary quadratic field and F a weight k meromorphic modular form with algebraic coefficients. Then there is a number ${\Omega}_{K}\in {\mathbb{C}}^{*}$, depending on K, such that for any $\tau \in K\cap \mathcal{H}$, $F\left(\tau \right){\Omega}_{K}^{-k}$ and ${\partial}_{k}F\left(\tau \right){\Omega}_{K}^{-k-2}$ are algebraic.

**Lemma 7.**The monodromy group Γ for the Picard–Fuchs equation $P{F}_{\lambda}$ from Equation (3.6) is isomorphic to a level 2 subgroup of $S{L}_{2}\left(\mathbb{Z}\right)$.

**Lemma 8.**Both

#### 3.4. Formulas for $1/\pi $ in Terms of Periods

**Lemma 9.**Let $\tau \in K$ be such that $w:={f}_{\lambda}\left(\tau \right){\Omega}_{K}^{-1}\ne 0$. Then there is a unique algebraic number ${c}^{\prime}$ such that

**Corollary 10.**The function $p\left(\lambda \right)$ describing the variation of the period integrals is equal to the product of π with the solution to the Picard–Fuchs equation, up to multiplication by an algebraic number;

**Proposition 11**(Borweins and Chudnovskys [3,4]) There exists $a\in \overline{\mathbb{Q}}$ such that

## 4. The Arithmetic of the Elliptic Curves

#### 4.1. Expansions of the Invariant Differentials at Infinity

**Lemma 12.**Let $p\ge 5$ be a prime. For each family of curves ${\tilde{E}}_{d}\left(t\right):{y}^{2}+Ay=B,$ listed in the introduction, with $A={a}_{1}x+{a}_{3}$, $B={x}^{3}+{a}_{2}{x}^{2}+{a}_{4}x+{a}_{6}$, let $z=\frac{-x}{y}$ be the local uniformizer. Expand

- For $d=2$: $A=0$, $B=x(x-1)(x-t)$.$$\begin{array}{ccc}\hfill {H}_{2,p-1}\left(t\right)& =& {\phantom{\rule{0.166667em}{0ex}}}_{2}{F}_{1}\left[\begin{array}{cc}\frac{1-p}{2}& \frac{1-p}{2}\\ & 1\end{array}\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}t\right]\hfill \\ & \equiv & {\phantom{\rule{0.166667em}{0ex}}}_{2}{F}_{1}{\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ & 1\end{array}\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}t\right]}_{(p-1)}\phantom{\rule{10.0pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}p)\hfill \end{array}$$
- For $d=3$: $A=x+\frac{t}{27}$, $B={x}^{3}$. ${H}_{3,p-1}\left(t\right)$ is the coefficient of ${x}^{p-1}$ in$$\sum _{\begin{array}{c}s+j=p-1\\ j\le s\end{array}}\left(\genfrac{}{}{0pt}{}{s}{j}\right){B}^{j}{A}^{s-j}$$By a similar argument, we obtain$$\begin{array}{ccc}\hfill {H}_{3,p-1}\left(t\right)& =& \sum _{j=0}^{\frac{p-1}{3}}\left(\genfrac{}{}{0pt}{}{p-1-j}{j}\right)\left(\genfrac{}{}{0pt}{}{p-1-2j}{p-1-3j}\right){\left(\frac{t}{27}\right)}^{j}\hfill \\ & =& \phantom{\rule{0.166667em}{0ex}}{}_{3}{F}_{2}\left[\begin{array}{ccc}\frac{1-p}{3}& \frac{2-p}{3}& \frac{3-p}{3}\\ & 1& 1-p\end{array}\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}t\right]\hfill \\ & \equiv & {\phantom{\rule{0.166667em}{0ex}}}_{2}{F}_{1}{\left[\begin{array}{cc}\frac{1}{3}& \frac{2}{3}\\ & 1\end{array}\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}t\right]}_{(p-1)}\phantom{\rule{10.0pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}p)\hfill \end{array}$$
- For $d=4$: $A=0$, $B=x({x}^{2}+x+\frac{t}{4})$.$$\begin{array}{ccc}\hfill {H}_{4,p-1}& =& \sum _{i=0}^{\frac{p-1}{4}}\left(\genfrac{}{}{0pt}{}{\frac{p-1}{2}}{j,j,\frac{p-1}{2}-2j}\right){\left(\frac{t}{4}\right)}^{j}\hfill \\ & =& {\phantom{\rule{0.166667em}{0ex}}}_{2}{F}_{1}{\left[\begin{array}{cc}\frac{1-p}{4}& \frac{3-p}{4}\\ & 1\end{array}\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}t\right]}_{(p-1)}\phantom{\rule{10.0pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}p)\hfill \end{array}$$
- For $d=6$: $A=x$, $B={x}^{3}-\frac{t}{432}$.$$\begin{array}{ccc}\hfill {H}_{6,p-1}\left(t\right)& \equiv & \sum _{j=0}^{\frac{p-1}{6}}\left(\genfrac{}{}{0pt}{}{p-1-3j}{3j}\right)\left(\genfrac{}{}{0pt}{}{3j}{j}\right){\left(\frac{-t}{432}\right)}^{j}\hfill \\ & \equiv & \phantom{\rule{0.166667em}{0ex}}{}_{6}{F}_{5}\left[\begin{array}{cccccc}\frac{1-p}{6}& \frac{2-p}{6}& \frac{3-p}{6}& \frac{4-p}{6}& \frac{5-p}{6}& \frac{6-p}{6}\\ & \frac{1-p}{3}& \frac{2-p}{3}& \frac{3-p}{3}& 1& \frac{1}{2}\end{array}\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}t\right]\hfill \\ & \equiv & {\phantom{\rule{0.166667em}{0ex}}}_{2}{F}_{1}{\left[\begin{array}{cc}\frac{1}{6}& \frac{5}{6}\\ & 1\end{array}\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}t\right]}_{(p-1)}\phantom{\rule{10.0pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}p)\hfill \end{array}$$

**Lemma 13.**For $d=2,3,4,6$, and $p\ge 5$ prime,

**Corollary 14.**For $d=2,3,4,6$, and $p\ge 5$ prime,

#### 4.2. Geometric Interpretation of Lemma 13

#### 4.3. Interpretation in Terms of Galois Representations

**Lemma 15.**For $\tilde{\rho}$ defined above, the determinant of $\tilde{\rho}\left({\mathrm{Fr}}_{v}\right)$ is p and its trace is determined up to a ± sign.

## 5. Proof of Theorems 1 and 2

**Results of Atkin, Swinnerton-Dyer and Katz**

**Proposition 16.**With notation and assumptions as above, if under the choice of the local uniformizer there exists a degree p (resp. ${p}^{2}$) Frobenius lifting Φ that commutes with the induced action of R on ${H}_{DR}^{1}(\widehat{E}/A,\left(p\right))$ in the ordinary (resp. supersingular case) then

**Lemma 17.**Let $d=\{2,3,4,6\}$. Then for any odd prime p coprime to d,

**Lemma 18.**Let $d\in \{2,3,4,6\}$, p be an odd prime coprime to d, and $t\in {\mathbb{Z}}_{p}$. Then

## Acknowledgements

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**MDPI and ACS Style**

Chisholm, S.; Deines, A.; Long, L.; Nebe, G.; Swisher, H.
*ρ *— Adic Analogues of Ramanujan Type Formulas for 1/π. *Mathematics* **2013**, *1*, 9-30.
https://doi.org/10.3390/math1010009

**AMA Style**

Chisholm S, Deines A, Long L, Nebe G, Swisher H.
*ρ *— Adic Analogues of Ramanujan Type Formulas for 1/π. *Mathematics*. 2013; 1(1):9-30.
https://doi.org/10.3390/math1010009

**Chicago/Turabian Style**

Chisholm, Sarah, Alyson Deines, Ling Long, Gabriele Nebe, and Holly Swisher.
2013. "*ρ *— Adic Analogues of Ramanujan Type Formulas for 1/π" *Mathematics* 1, no. 1: 9-30.
https://doi.org/10.3390/math1010009