Bounds for Rankin--Selberg integrals and quantum unique ergodicity for powerful levels

43 pages; various minor corrections (many thanks to the referee) and improvements in clarity and exposition. To appear in JAMS43 pages; various minor corrections (many thanks to the referee) and improvements in clarity and exposition. To appear in JAMSLet f be a classical holomorphic newform of level q and even weight k. We show that the pushforward to the full level modular curve of the mass of f equidistributes as qk -> infinity. This generalizes known results in the case that q is squarefree. We obtain a power savings in the rate of equidistribution as q becomes sufficiently "powerful" (far away from being squarefree), and in particular in the "depth aspect" as q traverses the powers of a fixed prime. We compare the difficulty of such equidistribution problems to that of corresponding subconvexity problems by deriving explicit extensions of Watson's formula to certain triple product integrals involving forms of non-squarefree level. By a theorem of Ichino and a lemma of Michel--Venkatesh, this amounts to a detailed study of Rankin--Selberg integrals int|f|^2 E attached to newforms f of arbitrary level and Eisenstein series E of full level. We find that the local factors of such integrals participate in many amusing analogies with global L-functions. For instance, we observe that the mass equidistribution conjecture with a power savings in the depth aspect is equivalent to the union of a global subconvexity bound and what we call a "local subconvexity bound"; a consequence of our local calculations is what we call a "local Lindelof hypothesis"

Local and global Maass relations

30 pages, minor changes. This is a slightly expanded version of the article that has been submitted by us to a journal. A preprint of the shorter version can be downloaded from the webpage of any of the authorsWe characterize the irreducible, admissible, spherical representations of GSp(4,F) (where F is a p-adic field) that occur in certain CAP representations in terms of relations satisfied by their spherical vector in a special Bessel model. These local relations are analogous to the Maass relations satisfied by the Fourier coefficients of Siegel modular forms of degree 2 in the image of the Saito-Kurokawa lifting. We show how the classical Maass relations can be deduced from the local relations in a representation theoretic way, without recourse to the construction of Saito-Kurokawa lifts in terms of Fourier coefficients of half-integral weight modular forms or Jacobi forms. As an additional application of our methods, we give a new characterization of Saito-Kurokawa lifts involving a certain average of Fourier coefficients

Integrality and cuspidality of pullbacks of nearly holomorphic Siegel Eisenstein series

26 pages26 pagesWe study nearly holomorphic Siegel Eisenstein series of general levels and characters on $\mathbb{H}_{2n}$, the Siegel upper half space of degree $2n$. We prove that the Fourier coefficients of these Eisenstein series (once suitably normalized) lie in the ring of integers of $\mathbb{Q}_p$ for all sufficiently large primes $p$. We also prove that the pullbacks of these Eisenstein series to $\mathbb{H}_n \times \mathbb{H}_n$ are cuspidal under certain assumptions

Lowest weight modules of Sp_4(R) and nearly holomorphic Siegel modular forms

We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair (G,K), where G=Sp_4(R) and K is its maximal compact subgroup. In particular, we determine K-types and composition series, and write down explicit differential operators that navigate all the highest weight vectors of such a module starting from the unique lowest-weight vector. By rewriting these operators in classical language, we show that the automorphic forms on G that correspond to the highest weight vectors are exactly those that arise from nearly holomorphic vector-valued Siegel modular forms of degree 2. Further, by explicating the algebraic structure of the relevant space of n-finite automorphic forms, we are able to prove a structure theorem for the space of nearly holomorphic vector-valued Siegel modular forms of (arbitrary) weight $det^\ell$ sym^m with respect to an arbitrary congruence subgroup of Sp_4(Q). We show that the cuspidal part of this space is the direct sum of subspaces obtained by applying explicit differential operators to holomorphic vector-valued cusp forms of weight $det^{\ell'} sym^{m'}$ with $(\ell', m')$ varying over a certain set. The structure theorem for the space of all modular forms is similar, except that we may now have an additional component coming from certain nearly holomorphic forms of weight $det^{3}sym^{m'}$ that cannot be obtained from holomorphic forms. As an application of our structure theorem, we prove several arithmetic results concerning nearly holomorphic modular forms that improve previously known results in that direction

An explicit construction of non-tempered cusp forms on O(1,8n+1)O(1,8n+1)

We explicitly construct cusp forms on the orthogonal group of signature $(1,8n+1)$ for an arbitrary natural number $n$ as liftings from Maass cusp forms of level one. In our previous works, the fundamental tool to show the automorphy of the lifting was the converse theorem by Maass. In this paper, we use the Fourier expansion of the theta lifts by Borcherds instead. We also study cuspidal representations generated by such cusp forms and show that they are irreducible and that all of their non-archimedean local components are non-tempered while the archimedean component is tempered, if the Maass cusp forms are Hecke eigenforms. The standard $L$-functions of the cusp forms are proved to be products of symmetric square $L$-functions of the Hecke-eigen Maass cusp forms with shifted Riemann zeta functions

Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level

We formulate an explicit refinement of B\"ocherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of L-functions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan-Gross-Prasad conjecture for Bessel periods as proposed by Yifeng Liu. We note several consequences of our conjecture to arithmetic and analytic properties of L-functions and Fourier coefficients of Siegel modular forms

Biophysical investigations on the active site of brain hexokinase

Replacement of Mg (II), the natural activator of brain hexokinase (EC 2.7.1.1) by paramagnetic Mn (II) without affecting the physiological properties of the enzyme, has rendered brain hexokinase accessible to investigations by magnetic resonance methods. Based on such studies, a site on the enzyme, where Mn (II) binds directly with high affinity has been identified and characterized in detail. Use of β,γ-bidentate Cr (III) ATP as an exchange-inert analogue for Mn (II) ATP has shown that Mn (II) binding directly to the enzyme has no catalytic role but another Mn (II) ion binding simultaneously and independently to the enzyme through the nucleotide bridge participates in enzyme function. However, using this direct binding Mn (II) ion and a covalently bound spin label as paramagnetic probes a beginning has been made in mapping the ligand binding sites of the enzyme. Ultra-violet difference spectroscopy has revealed the presence of at least two glucose 6-phosphate locations on the enzyme one of which presumably is the high affinity regulatory site modulated by substrate glucose. Elution behaviour of the enzyme on a phosphocellulose column suggests that glucose induces a specific phosphate site on the enzyme to which the phosphate bearing regulatory ligands of the enzyme may bind

Urbanization and Green Spaces—A Study on Jnana Bharathi Campus, Bangalore University

Global warming is amongst the most alarming problems of the new era. Carbon emission is evidently the strongest fundamental factor for global warming. So increasing carbon emission is one of today’s major concerns, which is well addressed in the Kyoto Protocol. Trees are amongst the most significant elements of any landscape, because of both biomass and diversity, and their key role in ecosystem dynamics is well known. Trees absorb the atmospheric carbon dioxide and act as a carbon sink, since 50 % of biomass is carbon itself and the importance of carbon sequestration in forest areas is already accepted, and well documented. With this background, a carbon sequestration potential study was carried out in Jnana Bharathi campus, Bangalore University using the Quadrat method. The total geographical area is about 449.74 ha with a rich vegetation sector and the total amount of both above ground carbon (AGC) and below ground carbon (BGC) was estimated as an average of 54.8 t/ha. The total amount of carbon dioxide assimilated into the vegetation in terms of both above ground and below ground biomass was estimated as an average of 200.9 t/ha. Urbanization and habitat fragmentation seem to be increasing worldwide, substantiated by a case study in Bangalore City. The analysis revealed that increase in built-up area at the city level was by about 164.62 km2, while the vegetation and water bodies decreased by about 285.72 and 7.2 km2 respectively. However, Bangalore University, Jnana Bharathi campus attains a good vegetation cover and is seen as one of the ‘green lungs’ of Bangalore city