All modular forms of weight 2 can be expressed by Eisenstein series
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RESEARCH
All modular forms of weight 2 can be expressed by Eisenstein series Martin Raum∗ and Jiacheng Xia * Correspondence:
[email protected] http://raum-brothers.eu/martin Institutionen för Matematiska vetenskaper, Chalmers tekniska högskola och Göteborgs Universitet, 412 96 Göteborg, Sweden The first author was partially supported by Vetenskapsrådet Grant 2015-04139.
Abstract We show that every elliptic modular form of integral weight greater than 1 can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central L-values present in all previous work. For weights greater than 2, we refine our result further, showing that linear combinations of products of exactly two cusp expansions of Eisenstein series suffice. Keywords: Central values of L-functions, Vector-valued Hecke operators, Products of Eisenstein series Mathematics Subject Classification: Primary 11F11, Secondary 11F67, 11F25
1 Introduction Kohnen–Zagier proved in their work on periods of modular forms [18] that every modular form of level 1 can be written as a linear combination of products of at most two Eisenstein series. Their insight provides a precise connection between the resulting expressions for cuspidal Hecke eigenforms and the special values of the associated L-functions. This connection also appeared in subsequent work by Borisov–Gunnells [4–6], who investigated specific modular forms associated with toric varieties, Kohnen–Martin [17], the first named author [23], and Dickson–Neururer [11], who investigated the case of higher levels. The nonvanishing of specific L-values was crucial in all cases. For levels that are square-free away from at most two primes, Dickson–Neururer obtain a characterization of weight 2 newforms that can be expressed as a linear combination of products of at most two Eisenstein series for the congruence subgroup 1 (N ). These are exactly those newforms whose central L-values do not vanish. In particular, results for newforms of weight 2 whose central L-values vanish are not included in any of the cited papers. The condition on the central L-value for weight 2 newforms is a severe restriction in light of the Birch–Swinnerton–Dyer Conjecture which relates it to the rank of the Mordell– Weil groups of elliptic curves. For instance, if a newform f of weight 2 has rational Fourier coefficients and negative Atkin–Lehner eigenvalue, it corresponds to an elliptic curve over Q with Mordell–Weil-rank at least 1 by work of Gross–Zagier [14]; See [2,3,13] for a discussion of and results on distributions of ranks of elliptic curves. However, the case of vanishing central L-values of weight 2 newforms is excluded from all available statements
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