For a given elliptic curve $E$ defined over the rationals, we study density of primes $p$ satisfying $\mathrm{gcd}(\#E(\mathbb{F}_p),p-1)=1$ and give conjectural value for this with strong heuristic evidence most curves, in an appropriate sense.

In this paper, we mainly prove the following conjectures of Z.-H. Sun \cite{SH2}: Let $p>3$ be a prime. If $p\equiv1\pmod3$ and $p=x^2+3y^2$, then have $$ \sum_{k=0}^{p-1}\frac{D_k}{4^k}\equiv\sum_{k=0}^{p-1}\frac{D_k}{16^k}\equiv4x^2-2p-\frac{p^2}{4x^2}\pmod{p^3}, if $p\equiv2\pmod3$, \sum_{k=0}^{p-1}\frac{D_k}{4^k}\equiv-2\sum_{k=0}^{p-1}\frac{D_k}{16^k}\equiv\frac{p^2}2\binom{\frac{p-1}2}{\f...

In an article by Sasaki and Yoshida (2000), we encountered Schottky groups of genus 2 as monodromy groups of the hypergeometric equation with purely imaginary exponents. In this paper we study automorphic functions for these Schottky groups, and give a conjectural infinite product formula for the elliptic modular function λ.

We generalize multivariate hook product formulae for P -partitions. We use Macdonald symmetric functions to prove a (q, t)-deformation of Gansner’s hook product formula for the generating functions of reverse (shifted) plane partitions. For a d-complete poset, we present a conjectural (q, t)-deformation of Peterson–Proctor’s hook product formula.

We continue studying the properties of the multiplicative structure on valuations. We prove a new version of the hard Lefschetz theorem for even translation invariant continuous valuations, and discuss problems of integral geometry staying behind these properties. Then we formulate a conjectural analogue of this result for odd valuations.

We survey recent developments in Hodge theory which are closely tied to families of CY varieties, including Mumford-Tate groups and boundary components, as well as limits of normal functions and generalized Abel-Jacobi maps. While many of the techniques are representation-theoretic rather than motivic, emphasis is placed throughout on the (known and conjectural) arithmetic properties accruing t...

We study the Harish-Chandra Schwartz space of an adelic quotient G(F )\G(A). We state a conjectural spectral decomposition of it in terms of parabolic induction. We verify a cuspidal version of this conjecture under additional hypotheses on the group G, which are known to be satisfied for G = GLn.

We prove the conjectural relations between Mahler measures and L-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions for L-values of CM elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family (1 +X)(1 + Y )(X + Y )− αXY , α ∈ R.