70 research outputs found
Supercongruences for sporadic sequences
We prove two-term supercongruences for generalizations of recently discovered
sporadic sequences of Cooper. We also discuss recent progress and future
directions concerning other types of supercongruences.Comment: 16 pages, to appear in Proceedings of the Edinburgh Mathematical
Societ
Interpolated sequences and critical -values of modular forms
Recently, Zagier expressed an interpolated version of the Ap\'ery numbers for
in terms of a critical -value of a modular form of weight 4. We
extend this evaluation in two directions. We first prove that interpolations of
Zagier's six sporadic sequences are essentially critical -values of modular
forms of weight 3. We then establish an infinite family of evaluations between
interpolations of leading coefficients of Brown's cellular integrals and
critical -values of modular forms of odd weight.Comment: 23 pages, to appear in Proceedings for the KMPB conference: Elliptic
Integrals, Elliptic Functions and Modular Forms in Quantum Field Theor
Supercongruences and Complex Multiplication
We study congruences involving truncated hypergeometric series of the
form_rF_{r-1}(1/2,...,1/2;1,...,1;\lambda)_{(mp^s-1)/2} =
\sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r \lambda^k where p is a prime and m, s, r
are positive integers. These truncated hypergeometric series are related to the
arithmetic of a family of algebraic varieties and exhibit Atkin and
Swinnerton-Dyer type congruences. In particular, when r=3, they are related to
K3 surfaces. For special values of \lambda, with s=1 and r=3, our congruences
are stronger than what can be predicted by the theory of formal groups because
of the presence of elliptic curves with complex multiplications. They
generalize a conjecture made by Rodriguez-Villegas for the \lambda=1 case and
confirm some other supercongruence conjectures at special values of \lambda.Comment: 19 page
Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant
Let be the triangle with vertices (1,0), (0,1), (1,1). We study certain
integrals over , one of which was computed by Euler. We give expressions for
them both as a linear combination of multiple zeta values, and as a polynomial
in single zeta values. We obtain asymptotic expansions of the integrals, and of
sums of certain multiple zeta values with constant weight. We also give related
expressions for Euler's constant. In the final section, we evaluate more
general integrals -- one is a Chen (Drinfeld-Kontsevich) iterated integral --
over some polytopes that are higher-dimensional analogs of . This leads to a
relation between certain multiple polylogarithm values and multiple zeta
values.Comment: 19 pages, to appear in Mat Zametki. Ver 2.: Added Remark 3 on a Chen
(Drinfeld-Kontsevich) iterated integral; simplified Proposition 2; gave
reference for (19); corrected [16]; fixed typ
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