Number Theory Group at Tsinghua University

We present a version of projective bundle theorem in MW-motives (resp. Chow-Witt rings), which says that (ℙ(E)) is determined by (X) and (X × ℙ²) for smooth quasi-projective schemes X and vector bundles E over X with odd rank. If the rank of E is even, the theorem is still true under a new kind of orientability, which we call it by projective orientability.

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A classical result of Eichler, Shimura and Manin asserts that the map that assigns to a cusp form f its period polynomial r_f is a Hecke equivariant map. We propose a generalization of this result to a setting where r_f is replaced by a family of rational function of N variables equipped with the action of GL_N(ℤ). For this purpose, we develop a theory of Hecke operators for the elliptic cocycle recently introduced by Charollois. In particular, when f is an eigenform, the corresponding rational function is also an eigenvector respect to Hecke operator for GL_N. Finally, we give some examples for Eisenstein series and the Ramanujan Delta function.

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Let Y(N) be the modular curve of level N and E(N) be the universal elliptic curve over Y(N). Beilinson (1986) defined the Eisenstein symbol in the motivic cohomology of E_k(N) and the work of Deninger-Scholl (1989) shows the Petersson inner product of its regulator gives us special L-values. In this talk I will present how to relate the modular regulator with L-value of quasi-modular forms by using Lanphier’s formula and Rogers-Zudilin method.

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Meeting ID: 916 5344 6007
Password: 8Ma4ed