Elliptic curves, eta-quotients and hypergeometric functions

Abstract

Abstract. The well-known fact that all elliptic curves are modular, proven by Wiles, Taylor, Breuil, Conrad and Diamond, leaves open the question whether there exists a 'nice' representation of the modular form associated to each elliptic curve. Here we provide explicit representations of the modular forms associated to certain Legendre form elliptic curves 2 E 1 (λ) as linear combinations of quotients of Dedekind's eta-function. We also give congruences for some of the modular forms' coefficients in terms of Gaussian hypergeometric functions.

On the Mahler Measure of 1+X+1/X+Y +1/Y

“…127, Entry 7.30]. The identification of L(E 33 , 2) in terms of F (1,11) and F (3,11), follows from integrating the associated cusp form [17].…”

Eta-quotients of Prime or Semiprime Level and Elliptic Curves

“…It is natural to ask when elliptic curves are associated to modular forms that can be written as linear combinationa of eta-quotients. Recently, Pathakjee, RosnBrick, and Toong [22] demonstrated four such examples, utilizing spaces of cusp forms which are spanned by eta-quotients. Further work on when spaces of modular forms are spanned by eta-quotients has been done by Rouse and Webb [23], Arnold-Roksandich, James, and Keaton [3], and Kilford [10], for example.…”

“…if (p, q) ≡ (5, 5) (mod 24). However, neither (22) nor (23) have nonnegative integer solutions which satisfy (21), which is a contradiction. Thus we have shown that there are no eta-quotients in M 2 (Γ 1 (pq)) for (p, q) ≡ (1, 5), (5, 1), (5, 5) (mod 24) and p, q > 5.…”

“…However, how to find results like these may not be immediately apparent. Our approach, which is inspired by work of Pathakjee, RosnBrick, and Yoong [22], utilizes results from Section 4 and has the potential to generate many new examples. We note that many of the steps require the aid of mathematical software to be practical.…”

Eisenstein Series, Eta Quotients and Their Applications in Number Theory

“…and Ono[49] listed all the weight 2 newforms that are eta quotients, and gave their corresponding strong Weil curves. There are such eta quotients only for M 2 (Γ 0 (N )) with N = 11, 14, 15, 20, 24, 27, 32, 36, 48, 64, 80, 144.Recently in[59], newforms in M 2 (Γ 0 (N )) with N = 33, 40, 42, 70 are given as linear combinations of eta quotients. In[2], congruences between Fourier coefficients of some newforms is given.…”