Shifted convolution \(L\)-series values for elliptic curves. (English) Zbl 1441.11170
Summary: Using explicit constructions of the Weierstrass mock modular form and Eisenstein series coefficients, we obtain closed formulas for the generating functions of values of shifted convolution \(L\)-functions associated to certain elliptic curves. These identities provide a surprising relation between weight 2 newforms and shifted convolution \(L\)-values when the underlying elliptic curve has modular degree 1 with conductor \(N\) such that \(\text{genus}(X_0(N))=1\).
MSC:
| 11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |
| 11F37 | Forms of half-integer weight; nonholomorphic modular forms |
Keywords:
elliptic curve; modular form; \(L\)-series values; holomorphic projection; Eisenstein series; modularity theorem; complex multiplication.References:
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