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A classical construction of Katz gives a purely algebraic construction of Eisenstein–Kronecker series using the Gauß–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 Poincaré 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 Poincaré bundle.

Building upon ideas of the second and third authors, we prove that at least$2^{(1-\\unicode[STIX]{x1D700})(\\log s)/(\\text{log}\\log s)}$values of the Riemann zeta function at odd integers between 3 and$s$are irrational, where$\\unicode[STIX]{x1D700}$is any positive real number and$s$is large enough in terms of$\\unicode[STIX]{x1D700}$. This lower bound is asymptotically larger than any power of$\\log s$; it improves on the bound$(1-\\unicode[STIX]{x1D700})(\\log s)/(1+\\log 2)$that follows from the Ball–Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.

We give an explicit description of the syntomic elliptic polylogarithm on the universal elliptic curve over the ordinary locus of the modular curve in terms of certain p -adic analytic moment functions associated to Katz' two-variable p -adic Eisenstein measure. The present work generalizes previous results of Bannai-Kobayashi-Tsuji and Bannai-Kings on the syntomic Eisenstein classes.

In this work we use elementary methods to discuss the question of the minimal number of points with bad reduction over \\pl for elliptic curves E/k(T) which are non-constant, respectively have non-constant j-invariant.

In this work we use elementary methods to discuss the question of the minimal number of points with bad reduction over $P_k^1$ for elliptic curves E/k(T) which are non-constant, respectively have non-constant j-invariant.

In this paper, we describe the algebraic de Rham realization of the elliptic polylogarithm for arbitrary families of elliptic curves in terms of the Poincaré bundle. Our work builds on previous work of Scheider and generalizes results of Bannai-Kobayashi-Tsuji and Scheider. As an application, we compute the de Rham Eisenstein classes explicitly in terms of certain algebraic Eisenstein series.

A well known result of B. Mazur gives a lower bound for the Krull dimension of the universal deformation ring associated to an absolutely irreducible residual representation in terms of the group cohomology of the adjoint representation. The question about equality - at least in the Galois case - also goes back to B. Mazur. In the general case the question about equality is the subject of Gouv\\^{e}a's \"Dimension conjecture\". In this note we provide a counterexample to this conjecture. More precisely, we construct an absolutely irreducible residual representation with smooth universal deformation ring of strict greater Krull dimension as expected.

A classical construction of Katz gives a purely algebraic construction of Eisenstein--Kronecker series using the Gau\\ss--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 Poincaré 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 Poincaré bundle.

In this survey, we review the known results on the algebraicity of critical values of Hecke \\(L\\)-functions and explain the new developments in \\cite{Kings-Sprang}.

The goal of this paper is to give a numerical criterion for an open question in \\(p\\)-adic Fourier theory. Let \\(F\\) be a finite extension of \\(\\mathbf{Q}_p\\). Schneider and Teitelbaum defined and studied the character variety \\(\\mathfrak{X}\\), which is a rigid analytic curve over \\(F\\) that parameterizes the set of locally \\(F\\)-analytic characters \\(\\lambda : (o_F,+) \\to (\\mathbf{C}_p^\\times,\\times)\\). Determining the structure of the ring \\(\\Lambda_F(\\mathfrak{X})\\) of bounded-by-one functions on \\(\\mathfrak{X}\\) defined over \\(F\\) seems like a difficult question. Using the Katz isomorphism, we prove that if \\(F= \\mathbf{Q}_{p^2}\\), then \\(\\Lambda_F(\\mathfrak{X}) = o_F [\\![o_F]\\!]\\) if and only if the \\(o_F\\)-module of integer-valued polynomials on \\(o_F\\) is generated by a certain explicit set. Some computations in SageMath indicate that this seems to be the case.