# Integral Elements in K-Theory and Products of Modular Curves

@article{Scholl2000IntegralEI, title={Integral Elements in K-Theory and Products of Modular Curves}, author={Anthony J. Scholl}, journal={arXiv: Number Theory}, year={2000}, pages={467-489} }

In the first part of this paper we use de Jong’s method of alterations to contruct unconditionally ‘integral’ subspaces of motivic cohomology (with rational coefficients) for Chow motives over local and global fields. In the second part, we investigate the integrality of the elements constructed by Beilinson in the motivic cohomology of the product of two modular curves, completing the discussion in section 6 of his paper [1].

## 41 Citations

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## References

SHOWING 1-10 OF 23 REFERENCES

Arithmetic moduli of elliptic curves

- Mathematics
- 1985

This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova"…

Motives for modular forms

- Mathematics
- 1990

Let M be a pure motive over a number field F of rank n with coefficients in T ⊂ C; M may be thought of as a direct factor of the cohomology of a smooth projective variety X over F cut out by an…

On the image of p-adic regulators

- Mathematics
- 1997

Let V denote a complete discrete valuation ring with a fraction eld K of characteristic 0 and a perfect residue eld k of positive characteristic p, let V0 = W (k) denote the ring of Witt vectors with…

Algebraic cycles and values of L-functions.

- Mathematics
- 1984

Let X be a smooth projective algebraic variety of dimension d over a number field k, and let n ̂ 0 be an integer, /-adic cohomology in degree /i, H(X^, Ot), is a representation space for Gal (Jc/k)…

On the semi-simplicity of the
$U_p$-operator on modular forms

- Mathematics
- 1996

For and positive integers, let denote the -vector space of cuspidal modular forms of level and weight . This vector space is equipped with the usual Hecke operators , . If we need to consider several…

On modular units

- Mathematics
- 1989

Introduction. In [6], Kubert and Lang describe the group of integral modular units on Γ(n) (“units over Z” in their terminology), and in particular determine its rank. Their method is based on…

The theorem of Riemann-Roch

- Mathematics
- 1995

The classical theory of algebraic number-fields, as described above in Chapter V, rests upon the fact that such fields have a non-empty set of places, the infinite ones, singled out by intrinsic…

Higher regulators and values of L-functions

- Mathematics
- 1985

In the work conjectures are formulated regarding the value of L-functions of motives and some computations are presented corroborating them.

Modular forms and algebraic K-theory

- Mathematics
- 1992

© Société mathématique de France, 1992, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les…

Opérations En K-Théorie Algébrique

- MathematicsCanadian Journal of Mathematics
- 1985

C'est pour étendre le théorème de Riemann-Roch à un morphisme projectif arbitraire que Grothendieck a introduit le groupe K(X) (noté aujourd'hui K 0(X)), construit à l'aide des -modules localement…