scholarly journals Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity

2020 ◽  
Vol 156 (11) ◽  
pp. 2368-2398
Author(s):  
Yueke Hu ◽  
Abhishek Saha
Keyword(s):  
Quadratic Field ◽  
Theta Lift ◽  
Amplification Method ◽  
Matrix Coefficients ◽  
Arithmetic Surfaces ◽  
Local Bound ◽  
Associated Matrix ◽  
Norm Bound ◽  
Stationary Phase Analysis ◽  
Level Aspect

We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$, where $D$ is an indefinite quaternion division algebra over ${\mathbb {Q}}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times \theta _\chi )$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $\theta _\chi$ is an (essentially fixed) automorphic form on $\textrm {GL}_2$ obtained as the theta lift of a Hecke character $\chi$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.

scholarly journals Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces

2019 ◽  
Vol 376 (1-2) ◽  
pp. 609-644 ◽  
Author(s):  
Abhishek Saha
Keyword(s):  
Number Theory ◽  
Automorphic Forms ◽  
Upper Bounds ◽  
Automorphic Representations ◽  
Arithmetic Surfaces ◽  
Local Bound ◽  
Minimal Vectors ◽  
Special Case ◽  
Level Aspect ◽  
Quaternion Division Algebra

Abstract Let D be an indefinite quaternion division algebra over $${{\mathbb {Q}}}$$Q. We approach the problem of bounding the sup-norms of automorphic forms $$\phi $$ϕ on $$D^\times ({{\mathbb {A}}})$$D×(A) that belong to irreducible automorphic representations and transform via characters of unit groups of orders of D. We obtain a non-trivial upper bound for $$\Vert \phi \Vert _\infty $$‖ϕ‖∞ in the level aspect that is valid for arbitrary orders. This generalizes and strengthens previously known upper bounds for $$\Vert \phi \Vert _\infty $$‖ϕ‖∞ in the setting of newforms for Eichler orders. In the special case when the index of the order in a maximal order is a squarefull integer N, our result specializes to $$\Vert \phi \Vert _\infty \ll _{\pi _\infty , \epsilon } N^{1/3 + \epsilon } \Vert \phi \Vert _2$$‖ϕ‖∞≪π∞,ϵN1/3+ϵ‖ϕ‖2. A key application of our result is to automorphic forms $$\phi $$ϕ which correspond at the ramified primes to either minimal vectors, in the sense of Hu et al. (Commun Math Helv, to appear) or p-adic microlocal lifts, in the sense of Nelson in “Microlocal lifts and and quantum unique ergodicity on $$\mathrm{GL}_2({{\mathbb {Q}}}_{p})$$GL2(Qp)” (Algebra Number Theory 12(9):2033–2064, 2018). For such forms, our bound specializes to $$\Vert \phi \Vert _\infty \ll _{ \epsilon } C^{\frac{1}{6} + \epsilon }\Vert \phi \Vert _2$$‖ϕ‖∞≪ϵC16+ϵ‖ϕ‖2 where C is the conductor of the representation $$\pi $$π generated by $$\phi $$ϕ. This improves upon the previously known local bound$$\Vert \phi \Vert _\infty \ll _{\lambda , \epsilon } C^{\frac{1}{4} + \epsilon }\Vert \phi \Vert _2$$‖ϕ‖∞≪λ,ϵC14+ϵ‖ϕ‖2 in these cases.

scholarly journals Approximated least-squares solutions of a generalized Sylvester-transpose matrix equation via gradient-descent iterative algorithm

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Adisorn Kittisopaporn ◽  
Pattrawut Chansangiam
Keyword(s):  
Least Squares ◽  
Iterative Algorithm ◽  
Gradient Descent ◽  
Main Idea ◽  
Full Column Rank ◽  
Minimum Error ◽  
Matrix Coefficients ◽  
Special Cases ◽  
Efficiency And Effectiveness ◽  
Associated Matrix

AbstractThis paper proposes an effective gradient-descent iterative algorithm for solving a generalized Sylvester-transpose equation with rectangular matrix coefficients. The algorithm is applicable for the equation and its interesting special cases when the associated matrix has full column-rank. The main idea of the algorithm is to have a minimum error at each iteration. The algorithm produces a sequence of approximated solutions converging to either the unique solution, or the unique least-squares solution when the problem has no solution. The convergence analysis points out that the algorithm converges fast for a small condition number of the associated matrix. Numerical examples demonstrate the efficiency and effectiveness of the algorithm compared to renowned and recent iterative methods.

scholarly journals Iwasawa Theory of Elliptic Modular Forms Over Imaginary Quadratic Fields at Non-ordinary Primes

2019 ◽  
Author(s):  
Kâzım Büyükboduk ◽  
Antonio Lei
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quadratic Field ◽  
Euler System ◽  
Base Change ◽  
Iwasawa Theory ◽  
Imaginary Quadratic Field ◽  
Selmer Groups ◽  
Imaginary Quadratic Fields ◽  
Elliptic Modular Form

AbstractThis article is a continuation of our previous work [7] on the Iwasawa theory of an elliptic modular form over an imaginary quadratic field $K$, where the modular form in question was assumed to be ordinary at a fixed odd prime $p$. We formulate integral Iwasawa main conjectures at non-ordinary primes $p$ for suitable twists of the base change of a newform $f$ to an imaginary quadratic field $K$ where $p$ splits, over the cyclotomic ${\mathbb{Z}}_p$-extension, the anticyclotomic ${\mathbb{Z}}_p$-extensions (in both the definite and the indefinite cases) as well as the ${\mathbb{Z}}_p^2$-extension of $K$. In order to do so, we define Kobayashi–Sprung-style signed Coleman maps, which we use to introduce doubly signed Selmer groups. In the same spirit, we construct signed (integral) Beilinson–Flach elements (out of the collection of unbounded Beilinson–Flach elements of Loeffler–Zerbes), which we use to define doubly signed $p$-adic $L$-functions. The main conjecture then relates these two sets of objects. Furthermore, we show that the integral Beilinson–Flach elements form a locally restricted Euler system, which in turn allow us to deduce (under certain technical assumptions) one inclusion in each one of the four main conjectures we formulate here (which may be turned into equalities in favorable circumstances).

scholarly journals Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms

2018 ◽  
Vol 30 (4) ◽  
pp. 887-913 ◽  
Author(s):  
Kâzım Büyükboduk ◽  
Antonio Lei
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quadratic Field ◽  
Iwasawa Theory ◽  
Imaginary Quadratic Field ◽  
Main Conjecture ◽  
Elliptic Modular Form

Abstract This is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic {\mathbb{Z}_{p}} -tower of an imaginary quadratic field K where the prime p splits completely. Our goal in this portion is to prove the Iwasawa main conjecture for suitable twists of f assuming that f is p-ordinary, both in the definite and indefinite setups simultaneously, via an analysis of Beilinson–Flach elements.

scholarly journals Ramanujan’s function k(τ)=r(τ)r 2(2τ) and its modularity

2020 ◽  
Vol 18 (1) ◽  
pp. 1727-1741
Author(s):  
Yoonjin Lee ◽  
Yoon Kyung Park
Keyword(s):  
Continued Fraction ◽  
Quadratic Field ◽  
Singular Values ◽  
Modular Function ◽  
Imaginary Quadratic Field ◽  
Modular Equations ◽  
Positive Rational Number ◽  
Congruence Relations ◽  
Symmetry Relations ◽  
Ray Class Field

Abstract We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

scholarly journals SARS-CoV-2 RNA Detection with Duplex-Specific Nuclease Signal Amplification

Micromachines ◽  
2021 ◽  
Vol 12 (2) ◽  
pp. 197
Author(s):  
Meiqing Liu ◽  
Haoran Li ◽  
Yanwei Jia ◽  
Pui-In Mak ◽  
Rui P. Martins
Keyword(s):  
Signal Amplification ◽  
Incubation Temperature ◽  
Dna Probes ◽  
Detection Sensitivity ◽  
N Gene ◽  
Rna Detection ◽  
Gold Standard Method ◽  
Complementary Method ◽  
Amplification Method ◽  
Virus Rna

The emergence of the novel severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), a zoonotic pathogen, has led to the outbreak of coronavirus disease 2019 (COVID-19) pandemic and brought serious threats to public health worldwide. The gold standard method for SARS-CoV-2 detection requires both reverse transcription (RT) of the virus RNA to cDNA and then polymerase chain reaction (PCR) for the cDNA amplification, which involves multiple enzymes, multiple reactions and a complicated assay optimization process. Here, we developed a duplex-specific nuclease (DSN)-based signal amplification method for SARS-CoV-2 detection directly from the virus RNA utilizing two specific DNA probes. These specific DNA probes can hybridize to the target RNA at different locations in the nucleocapsid protein gene (N gene) of SARS-CoV-2 to form a DNA/RNA heteroduplex. DSN cleaves the DNA probe to release fluorescence, while leaving the RNA strand intact to be bound to another available probe molecule for further cleavage and fluorescent signal amplification. The optimized DSN amount, incubation temperature and incubation time were investigated in this work. Proof-of-principle SARS-CoV-2 detection was demonstrated with a detection sensitivity of 500 pM virus RNA. This simple, rapid, and direct RNA detection method is expected to provide a complementary method for the detection of viruses mutated at the PCR primer-binding regions for a more precise detection.

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