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Rankin-Selberg Euler systems and p-adic interpolation · arXivDesk

1405.3079May 13, 2014This paper has been withdrawn, as it is superseded by the two newer papers "Rankin-Eisenstein classes for modular forms" (arXiv:1501.03289) and "Rankin-Eisenstein classes and explicit reciprocity laws" (arXiv:1503.02888)

Rankin-Selberg Euler systems and p-adic interpolation

Guido Kings, David Loeffler, Sarah Livia Zerbes

Number Theory

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Abstract

We construct motivic cohomology classes attached to Rankin--Selberg convolutions of modular forms of weights $\ge 2$ , show that these vary analytically in p-adic families, and relate their image under the p-adic regulator map to values of L-functions. As consequences, we prove new cases of Perrin-Riou's conjecture on motivic L-values; we prove finiteness results for Tate--Shafarevich groups for twists of elliptic curves by dihedral Artin characters; and we prove one inclusion in the Iwasawa main conjecture for a single modular form over an imaginary quadratic field.

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We construct motivic cohomology classes attached to Rankin--Selberg convolutions of modular forms of weights ≥ 2, show that these vary analytically in p-adic families, and relate their image under the p-adic regulator map to values of L-functions. As consequences, we prove new cases of Perrin-Riou's conjecture on motivic L-values; we prove finiteness results for Tate--Shafarevich groups for twists of elliptic curves by dihedral Artin characters; and we prove one inclusion in the Iwasawa main conjecture for a single modular form over an imaginary quadratic field.

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Claims

The paper addresses: Rankin-Selberg Euler systems and p-adic interpolation.
We construct motivic cohomology classes attached to Rankin--Selberg convolutions of modular forms of weights ≥ 2, show that these vary analytically in p-adic families, and relate their image under the p-adic regulator map to values of L-functions.

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Author comment: This paper has been withdrawn, as it is superseded by the two newer papers "Rankin-Eisenstein classes for modular forms" (arXiv:1501.03289) and "Rankin-Eisenstein classes and explicit reciprocity laws" (arXiv:1503.02888)