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We introduce a non-linear injective transformation τ from the set of non-vanishing normalized Hausdorff moment sequences to the set of normalized Stieltjes moment sequences by the formula$T$[($a$_{$n$})_{$n$=1}^{∞}]_{$n$}= 1/$a$_{1}...$a$_{$n$}. Special cases of this transformation have appeared in various papers on exponential functionals of Lévy processes, partly motivated by mathematical finance. We give several examples of moment sequences arising from the transformation and provide the corresponding measures, some of which are related to$q$-series.

In this paper we prove the \pm-main conjecture formulated by Kobayashi for elliptic curves with supersingular reduction at p such that a_p = 0. We also prove as a corollary the p-part of the BSD formula at supersingular primes when the analytic rank is 0.

We study smooth toroidal compactifications of Siegel varieties thoroughly from the viewpoints of mixed Hodge theory and K\"ahler-Einstein metric. We observe that any cusp of a Siegel space can be identified as a set of certain weight one polarized mixed Hodge structures. We then study the infinity boundary divisors of toroidal compactifications, and obtain a global volume form formula of an arbitrary smooth Siegel variety $\sA_{g,\Gamma}(g>1)$ with a smooth toroidal compactification $\overline{\sA}_{g,\Gamma}$ such that $D_\infty:=\overline{\sA}_{g,\Gamma}\setminus \sA_{g,\Gamma}$ is normal crossing. We use this volume form formula to show that the unique group-invariant K\"ahler-Einstein metric on $\sA_{g,\Gamma}$ endows some restraint combinatorial conditions for all smooth toroidal compactifications of $\sA_{g,\Gamma}.$ Again using the volume form formula, we study the asymptotic behaviour of logarithmical canonical line bundle on any smooth toroidal compactification of $\sA_{g,\Gamma}$ carefully and we obtain that the logarithmical canonical bundle degenerate sharply even though it is big and numerically effective.

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