Eichler-shimura Theory for Mock Modular Forms
نویسندگان
چکیده
We use mock modular forms to compute generating functions for the critical values of modular L-functions, and we answer a generalized form of a question of Kohnen and Zagier by deriving the “extra relation” that is satisfied by even periods of weakly holomorphic cusp forms. To obtain these results we derive an Eichler-Shimura theory for weakly holomorphic modular forms and mock modular forms. This includes two “Eichler-Shimura isomorphisms”, a “multiplicity two” Hecke theory, a correspondence between mock modular periods and classical periods, and a “Haberland-type” formula which expresses Petersson’s inner product and a related antisymmetric inner product on M ! k in terms of periods.
منابع مشابه
Cycle Integrals of the J-function and Mock Modular Forms
In this paper we construct certain mock modular forms of weight 1/2 whose Fourier coefficients are cycle integrals of the modular j-function and whose shadows are weakly holomorphic forms of weight 3/2. As an application we construct through a Shimura-type lift a holomorphic function that transforms with a rational period function having poles at certain real quadratic integers. This function y...
متن کاملPeriods of Eisenstein Series and Some Applications
Bringmann, Guerzhoy, Ono, and the author recently developed an Eichler-Shimura theory for weakly holomorphic modular forms by studying associated period polynomials. Here we include the Eisenstein series in this theory. Classically, the periods of a weight k (k ≥ 4 will always denote an even integer) cusp form f = ∑ l≥1 af (l)q l (q := e with τ = x+ iy ∈ C with y > 0) on Γ := SL2(Z) are given by
متن کاملExamples of abelian surfaces with everywhere good reduction
We describe several explicit examples of simple abelian surfaces over real quadratic fields with real multiplication and everywhere good reduction. These examples provide evidence for the Eichler–Shimura conjecture for Hilbert modular forms over a real quadratic field. Several of the examples also support a conjecture of Brumer and Kramer on abelian varieties associated to Siegel modular forms ...
متن کاملExplicit Formulas for Hecke Operators on Cusp Forms, Dedekind Symbols and Period Polynomials
Let Sw+2 be the vector space of cusp forms of weight w + 2 on the full modular group, and let S∗ w+2 denote its dual space. Periods of cusp forms can be regarded as elements of S∗ w+2. The Eichler-Shimura isomorphism theorem asserts that odd (or even) periods span S w+2 . However, periods are not linearly independent; in fact, they satisfy the Eichler-Shimura relations. This leads to a natural ...
متن کاملExplicit Computation of Cusp Forms via Hecke Action on Cohomology and Its Complexity
Abstract. In the literature, the standard approach to finding bases of spaces of modular forms is via modular symbols and the homology of modular curves. By using the Eichler-Shimura isomorphism, a work by Wang shows how one can use a cohomological viewpoint to determine bases of spaces of cusp forms on Γ0(N) of weight k ≥ 2 and character χ. It is interesting to look at the complexity of this a...
متن کامل