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Register a new userArchimedean period relations and period relations for Rankin-Selberg convolutions
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We prove the Archimedean period relations for Rankin-Selberg convolutions for $mathrm{GL}(n)times mathrm{GL}(n-1)$. This implies the period relations for critical values of the Rankin-Selberg L-functions for $mathrm{GL}(n)times mathrm{GL}(n-1)$.
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This article is to understand the critical values of $L$-functions $L(s,Piotimes chi)$ and to establish the relation of the relevant global periods at the critical places. Here $Pi$ is an irreducible regular algebraic cuspidal automorphic representation of $mathrm{GL}_{2n}(mathbb A)$ of symplectic type and $chi$ is a finite order automorphic character of $mathrm{GL}_1(mathbb A)$, with $mathbb A$ is the ring of adeles of a number field $mathrm k$.
We complete the proof of Proposition 5.3 of [GJR04].
In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that if the central critical value of the Rankin-Selberg $L$-function does not vanish, then the Bloch-Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch-Kato Selmer group constructed from certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanishing of the central critical first derivative of the Rankin-Selberg $L$-function, then the Bloch-Kato Selmer group is of rank one.
In this paper we confirm the existence of period-luminosity (PL) and period-luminosity-colour (PLC) relations at maximum light for O and C Mira variables in the LMC. We demonstrate that in the J and H bands the maximum light PL relations have a significantly smaller dispersion than their counterparts at mean light, while the K band and bolometric PL relations have a dispersion comparable to that at mean light. In the J, H and K bands the fitted PL relations for the O Miras are found to have smaller dispersion than those for the C Miras, at both mean and maximum light, while the converse is true for the relations based on bolometric magnitudes. The inclusion of a non-zero log period term is found to be highly significant in all cases except that of the C Miras in the J band, for which the data are found to be consistent with having constant absolute magnitude. This suggests the possibility of employing C Miras as standard candles. We suggest both a theoretical justification for the existence of Mira PL relations at maximum light and a possible explanation of why these relations should have a smaller dispersion than at mean light. The existence of such maximum light relations offers the possibility of extending the range and improving the accuracy of the Mira distance scale to Galactic globular clusters and to other galaxies.
Asymptotic Giant Branch variables are found to obey period-luminosity relations in the mid-IR similar to those seen at K_S (2.14 microns), even at 24 microns where emission from circumstellar dust is expected to be dominant. Their loci in the M, logP diagrams are essentially the same for the LMC and for NGC6522 in spite of different ages and metallicities. There is no systematic trend of slope with wavelength. The offsets of the apparent magnitude vs. logP relations imply a difference between the two fields of 3.8 in distance modulus. The colours of the variables confirm that a principal period with log P > 1.75 is a necessary condition for detectable mass-loss. At the longest observed wavelength, 24 microns, many semi-regular variables have dust shells comparable in luminosity to those around Miras. There is a clear bifurcation in LMC colour-magnitude diagrams involving 24 micron magnitudes.
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