数理解析研究所講究録
第 1806 巻 2012 年 16-21

16Transformations of -values$L$

Wadim Zudilin*
School of Mathematical and Physical Sciences,
The University of Newcastle, Callaghan, NSW 2308, Australia

Apri12012

Abstract

In our recent work with M. Rogers on resolving some Boyd’s conjectures
on two-variate Mahler measures, a new analytical machinery was introduced towrite the values $L(E, 2)$ of -series of elliptic curves as periods in the sense ofKontsevich and Zagier. Here we outline, in slightly more general settings, thenovelty of our method with Rogers, and provide a simple illustrative example.
$L$

Throughout the note we keep the notation
for from the upper half-plane${\rm Re}\tau>0$ , so that $|q|<1$ .
Our basic constructor of modular forms and functions is$q=e^{2\pi i\tau}$$\tau$

Dedekind’s eta-function$\eta(\tau):=q^{1/24}\prod_{m=1}^{\infty}(1-q^{m})=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{(6n+1)^{2}/24}$

with is modular involution$\eta(-1/\tau)=\sqrt{-i\tau}\eta(\tau)$(1).

$:=\eta(k\tau)$ for short.
We also set
We first describe a part of the general machinery from our joint works [6, 7] with
M. Rogers on an example of computing the value $L(E_{32},2)$ of the -series associatedwith a conductor 32 elliptic curve. It is known [3] that the corresponding cusp form in, so that $L(E_{32}, s)=L(f_{32}, s)$ . We choose the conductor 32
this case iscase here because it is not discussed in [6, 7].
Note the (Lambert series) expansion$\eta_{k}$$L-$$f_{32}(\tau)$$:=\eta_{4}^{2}\eta_{8}^{2}$$\frac{\eta_{8}^{4}}{\eta_{4}^{2}}=\sum_{m\geq 1}(\frac{-4}{m})\frac{q^{m}}{1-q^{2m}}=m,n\geq 1\sum_{nodd}(\frac{-4}{m})q^{mn}$,(2)*This work is supported by Australian Research Council grant $DP$ 110104419. The text is loosely
based on my talk “Mahler measures and -series of elliptic curves” at the conference “Analytic number theory–related multiple aspects of arithmetic functions” (Research Institute for Mathematical
Sciences, Kyoto University, Japan, ctober 31-November 2, 2011).
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$O$

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where
and ifis the quadratic residue character modulo 4. In notation
is odd, we can write (2) as$( \frac{-4}{m})$

$0$

$n$

$\frac{\eta_{8}^{4}}{\eta_{4}^{2}}=\sum_{m_{)}n\geq 1}a(m)b(n)q^{mn}$

,

where

$a(m):=( \frac{-4}{m})$,$\delta_{2|n}=1$

if 2 $|n$$b(n):=1-\delta_{2|n}.$Then$f_{32}(it)= \frac{\eta_{8}^{4}}{\eta_{4}^{2}}\frac{\eta_{4}^{4}}{\eta_{8}^{2}}|_{\tau=it}=\frac{\eta_{8}^{4}}{\eta_{4}^{2}}|_{\tau=it}\cdot\frac{1}{2t}\frac{\eta_{8}^{4}}{\eta_{4}^{2}}|_{\tau=i/(32t)}$$= \frac{1}{2t}\sum_{m_{1},n_{1}\geq 1}a(m_{1})b(n_{1})e^{-2\pi m_{1}n_{1}}t\sum_{m_{2},n_{2}\geq 1}b(m_{2})a(n_{2})e^{-2\pi m2n_{2}/(32t)},$where $t>0$ and the modular involution (1) was used.
Now,$L(E_{32},2)=L(f_{32},2)= \int_{0}^{1}f_{32}\log q\frac{dq}{q}=-4\pi^{2}\int_{0}^{\infty}f_{32}(it)tdt$$=-2 \pi^{2}\int_{0}^{\infty}\sum_{2m_{1},n_{1},m2n\geq 1},a(m_{1})b(n_{1})b(m_{2})a(n_{2})$$\cross\exp(-2\pi(m_{1}n_{1}t+\frac{m_{2}n_{2}}{32t}))dt$$=-2 \pi^{2}\sum_{m_{1},n_{1},m_{2}n_{2}\geq 1},a(m_{1})b(n_{1})b(m_{2})a(n_{2})$$\cross\int_{0}^{\infty}\exp(-2\pi(m_{1}n_{1}t+\frac{m_{2}n_{2}}{32t}))dt.$

Here comes the crucial transformation of purely analytical origin: we make the changeof variable $t=n_{2}u/n_{1}$ . It does not change the form of the integrand but affects thedifferential, and we obtain$L(E_{32},2)=-2 \pi^{2}\sum_{1m_{1},n,m_{2}n_{2}\geq 1},\frac{a(m_{1})b(n_{1})b(m_{2})a(n_{2})n_{2}}{n_{1}}$$\cross\int_{0}^{\infty}\exp(-2\pi(m_{1}n_{2}u+\frac{m_{2}n_{1}}{32u}))du$$=-2 \pi^{2}\int_{0}^{\infty}\sum_{1mn\geq 1}a(m_{1})a(n_{2})n_{2}e^{-2\pi m_{1}n2u}$$\cross\sum_{m2,n_{1}\geq 1}\frac{b(m_{2})b(n_{1})}{n_{1}}e^{-2\pi m_{2}n_{1}/(32u)}du.$What are the resulting series in the product? The first one corresponds to$\sum_{m,n\geq 1}a(m)a(n)nq^{mn}=\sum_{m,n\geq 1}(\frac{-4}{mn})nq^{mn}=\sum_{n\geq 1}n(\frac{-4}{n})\frac{nq^{n}}{1+q^{2n}}=\frac{\eta_{2}^{4}\eta_{8}^{4}}{\eta_{4}^{4}},$

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while the second one is$\sum_{m,n\geq 1}\frac{b(m)b(n)}{n}q^{mn}=\sum_{m,n\geq 1}\frac{q^{mn}}{n}-\frac{q^{(2m)n}}{n}-\frac{q^{m(2n)}}{2n}+\frac{q^{(2m)(2n)}}{2n}$$= \frac{1}{2}\sum_{m,n\geq 1}\frac{2q^{mn}-3q^{2mn}+q^{4mn}}{n}$$=- \frac{1}{2}\log\prod_{m\geq 1}\frac{(1-q^{m})^{2}(1-q^{4m})}{(1-q^{2m})^{3}}=-\frac{1}{2}\log\frac{\eta_{1}^{2}\eta_{4}}{\eta_{2}^{3}},$hence$L(E_{32},2)= \pi^{2}\int_{0}^{\infty}\frac{\eta_{2}^{4}\eta_{8}^{4}}{\eta_{4}^{4}}|_{\tau=iu}\cdot\log\frac{\eta_{1}^{2}\eta_{4}}{\eta_{2}^{3}}|_{\tau=i/(32u)}du.$Applying the involution (1) to the eta quotient under the logarithm $sign$ we obtain$L(E_{32},2)= \pi^{2}\int_{0}^{\infty}\frac{\eta_{2}^{4}\eta_{8}^{4}}{\eta_{4}^{4}}\log\frac{\sqrt{2}\eta_{8}\eta_{32}^{2}}{\eta_{16}^{3}}du\tau=iu.$

Now comes the modular magic: assisted with Ramanujan’s knowledge [1] we choose, which ranges from 1 to whena particular modular function$(0, i\infty)$ , and verify that$x(\tau)$$0$$:=\eta_{2}^{4}\eta_{8}^{2}/\eta_{4}^{6}$$\frac{1}{2\pi i}\frac{xdx}{2\sqrt{1-x^{4}}}=-\frac{\eta_{2}^{4}\eta_{8}^{4}}{\eta_{4}^{4}}d\tau$and$\mathcal{T}\in$$( \frac{\sqrt{2}\eta_{8}\eta_{32}^{2}}{\eta_{16}^{3}})^{2}=\frac{1-x}{1+x}.$

Thus,$L(E_{32},2)= \frac{\pi}{8}\int_{0}^{1}\frac{x}{\sqrt{1-x^{4}}}\log\frac{1+x}{1-x}dx.$

The result is a period in the sense of [2], and as such it can be compared withseveral other objects like values of generalized hypergeometric functions or even Mahler
measures [4, 5]. This however involves a different set of routines which we do not touch
here.
into aTo summarize, in our evaluation of $L(E, 2)=L(f, 2)$ we first split
product of two Eisenstein series of weight 1 and at the end we arrive at a product of
two Eisenstein(-like) series
andof weights 2 and , respectively, so that
$L(f, 2)=cL(g_{2}g_{0},1)$ for some algebraic constant . The latter object is doomed to beis, up to aa period as
is a logarithm of a modular function, whilemodular function multiple, the differential of a modular function, and finally any two
modular functions are tied up by an algebraic relation over $Q.$
The method however can be formalized to even more general settings, and it is thisextension which we attempt to outline below.
$f(\tau)$$g_{2}(\tau)$$0$$g_{0}(\tau)$$c$$g_{0}(\tau)$$2\pi ig_{2}(\mathcal{T})d\tau$

For two bounded sequences $a(m),$ $b(n)$ , we refer to an expression of the form
$g_{k}( \tau)=a+\sum_{m,n\geq 1}a(m)b(n)n^{k-1}q^{mn}, q:=e^{2\pi i\tau}$

,

(3)

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as to an Eisenstein-like series of weight , especially in the case whenis a modular
form of certain level, that is, when it transforms sufficiently‘nice’ under $\tau\mapsto-1/(N\tau)$for some positive integer $N$ . This automatically happens whenis indeed an Eisenstein series $(for$ example, $when a(m)=1$ and $b(n)$ is a Dirichlet character modulo $N$$:=g_{k}(-1/(N\tau))(\sqrt{-N}\tau)^{-k}$of designated parity, $b(-1)=(-1)^{k})$ , in which caseis again an Eisenstein series. It is worth mentioning that the above notion has perfect sense in case $k\leq 0$ as well. Indeed, $mo$ dular units, or week modular forms ofweight , that are the logarithms of modular functions are examples of Eisenstein-like. Also, for $k\leq 0$ examples are given by Eichler integrals, the $(1-k)$ thseries-derivatives of holomorphic Eisenstein series of weight $2-k$ , a consequence of thefamous lemma of Hecke [8, Section 5].
Suppose we are interested in the -value $L(f, k_{0})$ of a cusp formof weight$k=k_{1}+k_{2}$ which can be represented as a product (in general, as a linear combinationof several products) of two Eisenstein(-like) series, where the firstandone vanishes at infinity $(a=9k_{1}(i\infty)=0 in$ (3) and the second one vanishes at zero
. (The vanishing happens because the product is a cusp form!) In reality,
we need the series
to be Eisenstein-like:$k$$g_{k}(\tau)$$g_{k}(\tau)$$\hat{g}_{k}(\tau)$$0$$g_{0}(\tau)$$\tau$$L$$f(\tau)$$g_{k_{1}}(\tau)$$\hat{g}_{k_{2}}(\tau)$$)$$(\hat{g}_{k_{2}}(iO)=0)$$g_{k_{2}}(\tau)$$:=\hat{g}_{k_{2}}(-1/(N\tau))(\sqrt{-N}\tau)^{-k_{2}}$$g_{k_{1}}( \tau)=\sum_{m,n\geq 1}a_{1}(m)b_{1}(n)n^{k_{1}-1}q^{mn}$and$g_{k_{2}}( \tau)=\sum_{m,n\geq 1}a_{2}(m)b_{2}(n)n^{k_{2}-1}q^{mn}.$

We have$L(f, k_{0})=L(g_{k_{1}} \hat{g}_{k_{2}}, k_{0})=\frac{1}{(k_{0}-1)!}\int_{0}^{1}g_{k_{1}}\hat{g}_{k_{2}}\log^{k_{0}-1}q\frac{dq}{q}$$= \frac{(-1)^{k_{0}-1}(2\pi)^{k_{0}}}{(k_{0}-1)!}\int_{0}^{\infty}g_{k_{1}}(it)\hat{g}_{k_{2}}(it)t^{k_{0}-1}dt$$= \frac{(-1)^{k_{0}-1}(2\pi)^{k_{0}}}{(k_{0}-1)!N^{k_{2}/2}}\int_{0}^{\infty}g_{k_{1}}$( ) $g_{k_{2}}(i/(Nt))t^{k_{0}-k_{2}-1}dt$$it$$= \frac{(-1)^{k_{0}-1}(2\pi)^{k_{0}}}{(k_{0}-1)!N^{k_{2}/2}}\int_{0}^{\infty}\sum_{m_{1},n_{1}\geq 1}a_{1}(m_{1})b_{1}(n_{1})n_{1}^{k_{1}-1}e^{-2\pi m_{1}n_{1}t}$$\cross\sum_{m_{2},n_{2}\geq 1}a_{2}(m_{2})b_{2}(n_{2})n_{2}^{k_{2}-1}e^{-2\pi m_{2}n2/(Nt)}t^{k_{0}-k_{2}-1}dt$$= \frac{(-1)^{k_{0}-1}(2\pi)^{k_{0}}}{(k_{0}-1)!N^{k_{2}/2}}\sum_{m_{1},n_{1},m_{2}n_{2}\geq 1},a_{1}(m_{1})b_{1}(n_{1})a_{2}(m_{2})b_{2}(n_{2})n_{1}^{k_{1}-1}n_{2}^{k_{2}-1}$;$\cross\int_{0}^{\infty}\exp(-2\pi(m_{1}n_{1}t+\frac{m_{2}n_{2}}{Nt}))t^{k_{0}-k_{2}-1}dt$

the interchange of integration and summation is legitimate because of the exponential
decrease of the integrand at the endpoints. After performing the change of variable20$t=n_{2}u/n_{1}$and interchanging back summation and integration we obtain$L(f, k_{0})= \frac{(-1)^{k_{0}-1}(2\pi)^{k_{0}}}{(k_{0}-1)!N^{k_{2}/2}}\sum_{m_{1},n_{1},m_{2},n_{2}\geq 1}a_{1}(m_{1})b_{1}(n_{1})a_{2}(m_{2})b_{2}(n_{2})n_{1}^{k_{1}+k_{2}-k_{0}-1}n_{2}^{k_{0}-1}$$\cross\int_{0}^{\infty}\exp(-2\pi(m_{1}n_{2}u+\frac{m_{2}n_{1}}{Nu}))u^{k_{0}-k_{2}-1}du$$= \frac{(-1)^{k_{0}-1}(2\pi)^{k_{0}}}{(k_{0}-1)!N^{k_{2}/2}}\int_{0}^{\infty}\sum_{m_{1},n_{2}\geq 1}a_{1}(m_{1})b_{2}(n_{2})n_{2^{012}}^{k-1}e^{-2\pi mnu}$$\cross\sum_{2mn\geq 1}a_{2}(m_{2})b_{1}(n_{1})n_{1}^{k_{1}+k_{2}-k_{0}-1}e^{-2\pi m_{2}n_{1}/(Nu)}u^{k_{0}-k_{2}-1}du$$= \frac{(-1)^{k_{0}-1}(2\pi)^{k_{0}}}{(k_{0}-1)!N^{k_{2}/2}}\int_{0}^{\infty}g_{k_{0}}(iu)g_{k_{1}+k_{2}-k_{0}}(i/(Nu))u^{k_{0}-k_{2}-1}du.$Assuming a modular transformation of the Eisenstein-like series
under, we can realize the resulting integral as, whereis algebraic (plus extra terms whenis an Eichler integral). Altematively,iftransforms under the involution, we perform the transformation and switch
to the variable $v=1/(Nu)$ to arrive at. In both cases weobtain an identity which relates the starting -value $L(f, k_{0})$ to a different -value’ ofa modular-like object of the same weight.
The case $k_{1}=k_{2}=1$ and $k_{0}=2$ , discussed in [6, 7] and in our example above,
allows one to reduce the -values to periods. In our future work [9] we plan to addresssome examples with $k_{0}>2.$$g_{k_{1}+k_{2}-k_{0}}(\tau)$$\tau\mapsto-1/(N\tau)$$c\pi^{k_{0}-k_{1}}L(g_{k_{0}}\hat{g}_{k_{1}+k_{2}-k_{0}}, k_{1})$$g_{k_{1}+k_{2}-k_{0}}(\tau)$$c$$g_{k_{0}}(\tau)$$c\pi^{k_{0}-k_{1}}L(\hat{g}_{k_{0}}g_{k_{1}+k_{2}-k_{0}}, k_{1})$

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‘ $L$

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Acknowledgements. am thankful to the organizers of the RIMS conference “Analytic number theory– related multiple aspects of arithmetic functions” (Kyoto University, Japan, October 31-November 2, 2011) represented by Takumi Noda for invitation
to give a talk at the meeting. Special thanks go to my host Yasuo Ohno and his teamfrom the Kinki University (Osaka); they made my stay in Japan both culturally andscientifically enjoyable.
I am indebted to Anton Mellit and Mat Rogers for fruitful conversations on thesubject, and to Don Zagier for his encouragement to isolate the transformation partfrom [6, 7].
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