21 research outputs found

    Many pp-adic odd zeta values are irrational

    Full text link
    For any prime pp and ε>0\varepsilon>0 we prove that for any sufficiently large positive odd integer ss at least (cpε)slogs(c_p-\varepsilon) \sqrt{\frac{s}{\log s}} of the pp-adic zeta values ζp(3),ζp(5),,ζp(s)\zeta_p(3),\zeta_p(5),\dots,\zeta_p(s) are irrational. The constant cpc_p is positive and does only depend on pp. This result establishes a pp-adic version of the elimination technique used by Fischler--Sprang--Zudilin and Lai--Yu to prove a similar result on classical zeta values. The main difficulty consists in proving the non-vanishing of the resulting linear forms. We overcome this problem by using a new irrationality criterion.Comment: 35 page

    Eisenstein series via the Poincaré bundle and applications

    Get PDF
    We give a new purely algebraic construction of real-analytic Eisenstein series via the Poincaré bundle. Building on this, we explicity describe the realization of the elliptic polylogarithm in algebraic de Rham cohomology as well as in syntomic cohomology. This generalizes results of Scheider, Bannai-Kobayashi-Tsuji and Bannai-Kings. The syntomic polylogarithm plays an important role for studying particular cases of the p-adic Beilinson conjectures. A further application relates the Amice transform of a certain p-adic theta function of the Poincaré bundle to Katz' two-variable p-adic Eisenstein measure. This gives a more conceptional construction of Katz' p-adic Eisenstein measure and provides a direct bridge between p-adic theta functions and p-adic modular forms

    EISENSTEIN–KRONECKER SERIES VIA THE POINCARÉ BUNDLE

    Get PDF
    A classical construction of Katz gives a purely algebraic construction of Eisenstein-Kronecker series using the Gau beta-Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and p-adic properties of real-analytic Eisenstein series. In the first part of this paper we provide an alternative algebraic construction of Eisenstein-Kronecker series via the Poincare bundle. Building on this, we give in the second part a new conceptional construction of Katz' two-variable p-adic Eisenstein measure through p-adic theta functions of the Poincare bundle

    Integral Comparison of Monsky-Washnitzer and overconvergent de Rham-Witt cohomology

    Get PDF
    The goal of this small note is to extend a result by Christopher Davis and David Zureick-Brown on the comparison between integral Monsky-Washnitzer cohomology and overconvergent de~Rham-Witt cohomology for a smooth variety over a perfect field of positive characteristic p to all cohomological degrees independent of the dimension of the base or the prime number p. Le but de ce travail est de prolonger un r\'esultat de Christopher Davis et David Zureick-Brown concernant la comparaison entre la cohomologie de Monsky-Washnitzer enti\`ere et la cohomologie de de~Rham-Witt surconvergente d'une vari\'et\'e lisse sur un coprs parfait de charact\'eristique positive p \`a tous les degr\'es cohomologiques ind\'epnedent de la dimension de base et du nombre premier p

    Simultaneous Kummer congruences and E\mathbb{E}_\infty-orientations of KO and tmf

    Full text link
    Building on results of M. Ando, M.J. Hopkins and C. Rezk, we show the existence of uncountably many E\mathbb{E}_\infty-String orientations of real K-theory KO and of topological modular forms tmf, generalizing the A^\hat{A}- (resp. the Witten) genus. Furthermore, the obstruction to lifting an E\mathbb{E}_\infty-String orientations from KO to tmf is identified with a classical Iwasawa-theoretic condition. The common key to all these results is a precise understanding of the classical Kummer congruences, imposed for all primes simultaneously. This result is of independent arithmetic interest.Comment: final versio
    corecore