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Recently the elliptic curve $E:y^2+y=x^3-x^2$ of conductor $11$ (which appears in my answer) became my favourite elliptic over $\bf Q$ because the associated modular form $$ F=q\prod_{n>0}(1-q^n)^2(1-q^{11n})^2 $$ is such a nice "$\eta$-product". (This modular form is also associated to the isogenous elliptic curve $y^2+y=x^3-x^2-10x-20$ which appears in Franz's question.)

Question. Are there other elliptic curves over $\bf Q$ which have a simple minimal equation and whose associated modular form is a nice $\eta$-product or even a nice $\eta$-quotient?

I know two references which might have a bearing on the question

--- Koike's article on McKay's conjecture

and

--- p.18 of Ono's Web of modularity on $\eta$-quotients.

Can someone provide a partial or exhaustive list of such nice pairs $(E,F)$ ?

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Actually Kilford has a great paper on generating spaces of modular forms using linear combinations of Dedekind Eta-functions (arxiv.org/abs/math/0701478). They have a less elegant form though. –  Eugene May 13 '12 at 6:56
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3 Answers

up vote 15 down vote accepted

There is an exhaustive list in the paper [Y. Martin and K. Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math Soc. 125 (1997), no. 11, 3169--3176]. Suppose that $E_N$ is an elliptic curve of conductor $N$, then the corresponding $L$-series is assigned to the eta product $$ \eta(a\tau)\eta(ab\tau)\eta(ac\tau)\eta(abc\tau), $$ where $a+ab+ac+abc=24$, $a,b,c\in\mathbb Z$, for the following values of $N$ and $(b,c)$: $$ \begin{align*} N &\quad (b,c)\cr 11 &\quad (1,11)\cr 14 &\quad (2,7)\cr 15 &\quad (3,5)\cr 20 &\quad (1,5)\cr 24 &\quad (2,3)\cr 27 &\quad (1,3)\cr 32 &\quad (1,2)\cr 36 &\quad (1,1)\cr \end{align*} $$ It is probably more exciting that all these elliptic curves and their $L$-series at $s=2$ appear in Boyd's conjectures on Mahler's measure. For a nice review of this story see [M.D. Rogers, Hypergeometric formulas for lattice sums and Mahler measures, arXiv:0806.3590] and the original paper [D.W. Boyd, Mahler's measure and special values of $L$-functions, Experiment. Math. 7 (1998) 37--82].

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How wonderful! Many thanks for the references. It is surprising that the Martin-Ono paper is so recent... –  Chandan Singh Dalawat Jul 23 '10 at 9:32
    
You are welcome, Chandan. A lot of interesting nontrivial identities for eta products are given by Michael Somos (see, for example, cis.csuohio.edu/~somos/retaprod.pdf). They are also discussed in the Zagier's lectures in 1-2-3 of Modular Forms. –  Wadim Zudilin Jul 23 '10 at 9:44
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This won't fit into a comment box so here it is :

I came across another small nugget in Kilford's book (p.101) this afternoon. Let $k>0$, $N>0$ be integers such that $k.(N+1)=24$, namely one of the pairs

$(12,1)$, $(8,2)$, $(6,3)$, $(4,5)$, $(3,7)$, $(2,11)$, or $(1,23)$.

Then $(\eta(q)\eta(q^N))^k$ is in $S_k(\Gamma_0(N))$ unless $k=1$ or $k=3$, in which case it is in $S_k(\Gamma_0(N),({{}\over N}))$. This implies in particular that the form

$q\prod_{n>0}(1-q^n)(1-q^{23n})$

which appears in Emerton's answer belongs to $S_1(\Gamma_0(23),({{}\over 23}))$.

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This result is also in Koblitz and is used by Kilford to show that every modular form of even weight of $\Gamma(2)$ is generated by eta-quotients. –  Eugene May 13 '12 at 6:58
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Today I came across the following theorem of Mersmann: There are precisely 14 primitive eta-products which are holomorphic modular forms of weight $\frac{1}{2}$, namely...

He also proves a conjecture of Zagier to the effect that there are essentially only finitely many such products of any given weight.

I learnt these facts from Zagier's contribution to the The 1-2-3 of modular forms (which happens to have been metioned by Wadim Zudilin).

Addendum (2011/02/10) Came across an advertisement for the book Eta products and theta series identities by Günter Köhler.

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