Volume 33, Issue 1

Recently, the problem of bounding the sup norms of L2{L^{2}}-normalized cuspidal automorphic newforms ϕ on GL2{\mathrm{GL}_{2}} in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character χ of ϕ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general χ. If the level N is a square, our result reduces to ∥ϕ∥∞≪N14+ϵ,\|\phi\|_{\infty}\ll N^{\frac{1}{4}+\epsilon}, at least under the Ramanujan Conjecture. In particular, when χ has conductor N , this improves upon the previous best known bound ∥ϕ∥∞≪N12+ϵ{\|\phi\|_{\infty}\ll N^{\frac{1}{2}+\epsilon}} in this setup (due to [A. Saha, Hybrid sup-norm bounds for Maass newforms of powerful level, Algebra Number Theory 11 2017, 1009–1045]) and matches a lower bound due to [N. Templier, Large values of modular forms, Camb. J. Math. 2 2014, 1, 91–116], thus our result is essentially optimal in this case.

Let ℛ{\mathcal{R}} be a finite valuation ring of order qr{q^{r}}. In this paper, we prove that for any quadratic polynomial f⁢(x,y,z)∈ℛ⁢[x,y,z]{f(x,y,z)\in\mathcal{R}[x,y,z]} that is of the form a⁢x⁢y+R⁢(x)+S⁢(y)+T⁢(z){axy+R(x)+S(y)+T(z)} for some one-variable polynomials R,S,T{R,S,T}, we have |f⁢(A,B,C)|≫min⁡{qr,|A|⁢|B|⁢|C|q2⁢r-1}|f(A,B,C)|\gg\min\biggl{\{}q^{r},\frac{|A||B||C|}{q^{2r-1}}\bigg{\}} for any A,B,C⊂ℛ{A,B,C\subset\mathcal{R}}. We also study the sum-product type problems over finite valuation ring ℛ{\mathcal{R}}. More precisely, we show that for any A⊂ℛ{A\subset\mathcal{R}} with |A|≫qr-13{|A|\gg q^{r-\frac{1}{3}}} then max⁡{|A⁢A|,|Ad+Ad|}{\max\{|AA|,|A^{d}+A^{d}|\}}, max⁡{|A+A|,|A2+A2|}{\max\{|A+A|,|A^{2}+A^{2}|\}}, max⁡{|A-A|,|A⁢A+A⁢A|}≫|A|23⁢qr3{\max\{|A-A|,|AA+AA|\}\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}}, and |f⁢(A)+A|≫|A|23⁢qr3{|f(A)+A|\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}} for any one variable quadratic polynomial f .

We extend the definition of order zero maps to the setting of pro-C*{C^{*}}-algebras and generalize structure theorems of order zero maps between C*{C^{*}}-algebras to strongly bounded order zero maps between pro-C∗{C^{\ast}}-algebras. An application to tensor products is included.

In this paper, we study hybrid subconvexity bounds for class group L -functions associated to quadratic extensions K/ℚ{K/\mathbb{Q}} (real or imaginary). Our proof relies on relating the class group L -functions to Eisenstein series evaluated at Heegner points using formulas due to Hecke. The main technical contribution is the uniform sup norm bound for Eisenstein series E⁢(z,1/2+i⁢t)≪εy1/2⁢(|t|+1)1/3+ε{E(z,1/2+it)\ll_{\varepsilon}y^{1/2}(\lvert t\rvert+1)^{1/3+\varepsilon}}, y≫1{y\gg 1}, extending work of Blomer and Titchmarsh. Finally, we propose a uniform version of the sup norm conjecture for Eisenstein series.

We establish explicit Ichino’s formulae for the central values of the triple product L -functions with emphasis on the calculations for the real place. The key ingredient for our computations is Proposition 8 which generalizes a result in [P. Michel and A. Venkatesh, The subconvexity problem for GL2{\rm GL}_{2}, Publ. Math. Inst. Hautes Études Sci. 111 2010, 171–271]. As an application we prove the optimal upper bound of a sum of restricted L2{L^{2}}-norms of the L2{L^{2}}-normalized newforms on certain quadratic extensions with prime level and bounded spectral parameter following the methods in [V. Blomer, On the 4-norm of an automorphic form, J. Eur. Math. Soc. (JEMS) 15 2013, 5, 1825–1852].

We study the generalised Iwasawa invariants of ℤpd{\mathbb{Z}_{p}^{d}}-extensions of a fixed number field K . Based on an inequality between ranks of finitely generated torsion ℤp⁢[[T1,…,Td]]{\mathbb{Z}_{p}[\kern-2.133957pt[T_{1},\dots,T_{d}]\kern-2.133957pt]}-modules and their corresponding elementary modules, we prove that these invariants are locally maximal with respect to a suitable topology on the set of ℤpd{\mathbb{Z}_{p}^{d}}-extensions of K , i.e., that the generalised Iwasawa invariants of a ℤpd{\mathbb{Z}_{p}^{d}}-extension 𝕂{\mathbb{K}} of K bound the invariants of all ℤpd{\mathbb{Z}_{p}^{d}}-extensions of K in an open neighbourhood of 𝕂{\mathbb{K}}. Moreover, we prove an asymptotic growth formula for the class numbers of the intermediate fields in certain ℤp2{\mathbb{Z}_{p}^{2}}-extensions, which improves former results of Cuoco and Monsky. We also briefly discuss the impact of generalised Iwasawa invariants on the global boundedness of Iwasawa λ-invariants.

Moderate deviation principles (MDPs) for random walks on covering graphs with groups of polynomial volume growth are discussed in a geometric point of view. They deal with any intermediate spatial scalings between those of laws of large numbers and those of central limit theorems. The corresponding rate functions are given by quadratic forms determined by the Albanese metric associated with the given random walks. We apply MDPs to establish laws of the iterated logarithm on the covering graphs by characterizing the set of all limit points of the normalized random walks.

We develop a version of Ekedahl’s geometric sieve for integral quadratic forms of rank at least five. As one ranges over the zeros of such quadratic forms, we use the sieve to compute the density of coprime values of polynomials, and furthermore, to address a question about local solubility in families of varieties parameterised by the zeros.

We consider iterated integrals of log⁡ζ⁢(s){\log\zeta(s)} on certain vertical and horizontal lines. Here, the function ζ⁢(s){\zeta(s)} is the Riemann zeta-function. It is a well-known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of ∫0tlog⁡ζ⁢(12+i⁢t′)⁢𝑑t′{\int_{0}^{t}\log\zeta(\frac{1}{2}+it^{\prime})\,dt^{\prime}} under the Riemann Hypothesis. Moreover, we show that, for any m≥2{m\geq 2}, the denseness of the values of an m -times iterated integral on the critical line is equivalent to the Riemann Hypothesis.

We classify commutative algebraic monoid structures on normal affine surfaces over an algebraically closed field of characteristic zero. The answer is given in two languages: comultiplications and Cox coordinates. The result follows from a more general classification of commutative monoid structures of rank 0, n-1n-1 or 𝑛 on a normal affine variety of dimension 𝑛.

Let Gn{G_{n}} denote either the group SO⁢(2⁢n+1,F){\mathrm{SO}(2n+1,F)} or Sp⁢(2⁢n,F){\mathrm{Sp}(2n,F)} over a non-archimedean local field of characteristic different than two. We study parabolically induced representations of the form 〈Δ〉⋊σ{\langle\Delta\rangle\rtimes\sigma}, where 〈Δ〉{\langle\Delta\rangle} denotes the Zelevinsky segment representation of the general linear group attached to the segment Δ, and σ denotes a discrete series representation of Gn{G_{n}}. We determine the composition series of 〈Δ〉⋊σ{\langle\Delta\rangle\rtimes\sigma} in the case when Δ=[νa⁢ρ,νb⁢ρ]{\Delta=[\nu^{a}\rho,\nu^{b}\rho]} where a is half-integral.

In this paper, we prove a genuine analogue of the Wiener Tauberian theorem for Lp,1⁢(G){L^{p,1}(G)} (1≤p<2{1\leq p<2}), with G=SL⁢(2,ℝ){G=\mathrm{SL}(2,\mathbb{R})}.

In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander–Reiten quiver of the subcategories is obtained only by the Auslander–Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, when Λ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the 2×2{2\times 2} lower triangular matrix algebra over Λ and the stable Auslander algebra of Λ can be estimated by the category of modules over the stable Cohen–Macaulay Auslander algebra of Λ.

The spectrum of the Laplacian operator on the positive theta line bundle over the quasi-torus reduces to eigenvalues π⁢ℓ{\pi\ell}, ℓ=0,1,…{\ell=0,1,\ldots{}}, which are called Landau levels. This paper discusses the coherent state transform for each eigenspace associated with a Landau level. We construct a unitary transform valid for each eigenspace. A concrete form of the inverse formula for the proposed transform is also obtained.

We prove a result that enables us to calculate the rational homotopy of a wide class of spaces by the theory of minimal models.