Regularized theta lift and formulas of Katok–Sarnak type

2012 ◽  
Vol 24 (6) ◽  
Author(s):  
Roland Matthes ◽  
Yoshinori Mizuno
Keyword(s):  
Theta Lift

scholarly journals On a theta lift related to the Shintani lift

2018 ◽  
Vol 328 ◽  
pp. 858-889 ◽  
Author(s):  
Claudia Alfes-Neumann ◽  
Markus Schwagenscheidt
Keyword(s):  
Theta Lift

scholarly journals On two arithmetic theta lifts

2018 ◽  
Vol 154 (10) ◽  
pp. 2090-2149 ◽  
Author(s):  
Stephan Ehlen ◽  
Siddarth Sankaran
Keyword(s):  
Modular Forms ◽  
Theta Functions ◽  
Chow Group ◽  
Shimura Varieties ◽  
Green Functions ◽  
Theta Lift ◽  
Moderate Growth ◽  
Generating Series ◽  
Cm Points ◽  
The Difference

Our aim is to clarify the relationship between Kudla’s and Bruinier’s Green functions attached to special cycles on Shimura varieties of orthogonal and unitary type, which play a key role in the arithmetic geometry of these cycles in the context of Kudla’s program. In particular, we show that the generating series obtained by taking the differences of the two families of Green functions is a non-holomorphic modular form and has trivial (cuspidal) holomorphic projection. Along the way, we construct a section of the Maaß lowering operator for moderate growth forms valued in the Weil representation using a regularized theta lift, which may be of independent interest, as it in particular has applications to mock modular forms. We also consider arithmetic-geometric applications to integral models of $U(n,1)$ Shimura varieties. Each family of Green functions gives rise to a formal arithmetic theta function, valued in an arithmetic Chow group, that is conjectured to be modular; our main result is the modularity of the difference of the two arithmetic theta functions. Finally, we relate the arithmetic heights of the special cycles to special derivatives of Eisenstein series, as predicted by Kudla’s conjecture, and describe a refinement of a theorem of Bruinier, Howard and Yang on arithmetic intersections against CM points.

scholarly journals Completions and algebraic formulas for the coefficients of Ramanujan’s mock theta functions

2020 ◽  
Author(s):  
David Klein ◽  
Jennifer Kupka
Keyword(s):  
Theta Functions ◽  
Mock Theta Functions ◽  
Maass Forms ◽  
Theta Lift ◽  
Singular Moduli

Abstract We present completions of mock theta functions to harmonic weak Maass forms of weight $$\nicefrac {1}{2}$$ 1 2 and algebraic formulas for the coefficients of mock theta functions. We give several harmonic weak Maass forms of weight $$\nicefrac {1}{2}$$ 1 2 that have mock theta functions as their holomorphic part. Using these harmonic weak Maass forms and the Millson theta lift, we compute finite algebraic formulas for the coefficients of the appearing mock theta functions in terms of traces of singular moduli.

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.

TWISTED BORCHERDS PRODUCTS ON HILBERT MODULAR SURFACES AND THE REGULARIZED THETA LIFT

2010 ◽  
Vol 06 (07) ◽  
pp. 1473-1489 ◽  
Author(s):  
STEPHAN EHLEN
Keyword(s):  
Modular Forms ◽  
Infinite Product ◽  
Hilbert Modular Forms ◽  
Quadratic Number Field ◽  
Weakly Holomorphic Modular Forms ◽  
Theta Lift ◽  
Hilbert Modular Group ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Holomorphic Modular Forms

We construct a lifting from weakly holomorphic modular forms of weight 0 for SL 2(ℤ) with integral Fourier coefficients to meromorphic Hilbert modular forms of weight 0 for the full Hilbert modular group of a real quadratic number field with an infinite product expansion and a divisor given by a linear combination of twisted Hirzebruch–Zagier divisors. The construction uses the singular theta lifting by considering a suitable twist of a Siegel theta function. We generalize the work by Bruinier and Yang who showed the existence of the lifting for prime discriminants using a different approach.

On adelic automorphic forms with respect to a quadratic extension

2000 ◽  
Vol 62 (1) ◽  
pp. 29-43 ◽  
Author(s):  
Ze-Li Dou
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Quadratic Extension ◽  
Algebraic Number Field ◽  
Hilbert Modular Forms ◽  
Arithmetic Properties ◽  
Theta Lift ◽  
Hilbert Modular ◽  
Algebraic Foundation ◽  
Totally Real

Let E/F be a totally real quadratic extension of a totally real algebraic number field. The author has in an earlier paper considered automorphic forms defined with respect to a quaternion algebra BE over E and a theta lift from such quaternionic forms to Hilbert modular forms over F. In this paper we construct adelic forms in the same setting, and derive explicit formulas concerning the action of Hecke operators. These formulas give an algebraic foundation for further investigations, in explicit form, of the arithmetic properties of the adelic forms and of the associated zeta and L-functions.

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