scholarly journals A symplectic restriction problem

2021 ◽  
Author(s):  
Valentin Blomer ◽  
Andrew Corbett
Keyword(s):  
Asymptotic Formula ◽  
Modular Form ◽  
Half Space ◽  
Imaginary Part ◽  
Dimensional Subspace ◽  
Siegel Modular Form ◽  
3 Dimensional ◽  
Relative Trace Formula ◽  
Lindelöf Hypothesis ◽  
Restriction Problem

AbstractWe investigate the norm of a degree 2 Siegel modular form of asymptotically large weight whose argument is restricted to the 3-dimensional subspace of its imaginary part. On average over Saito–Kurokawa lifts an asymptotic formula is established that is consistent with the mass equidistribution conjecture on the Siegel upper half space as well as the Lindelöf hypothesis for the corresponding Koecher–Maaß series. The ingredients include a new relative trace formula for pairs of Heegner periods.

scholarly journals On mod p singular modular forms

2016 ◽  
Vol 28 (6) ◽  
Author(s):  
Siegfried Böcherer ◽  
Toshiyuki Kikuta
Keyword(s):  
Modular Form ◽  
Number Field ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Siegel Modular Form ◽  
Mod P

AbstractWe show that a Siegel modular form with integral Fourier coefficients in a number field

scholarly journals Explicit computations of Serre’s obstruction for genus-3 curves and application to optimal curves

2010 ◽  
Vol 13 ◽  
pp. 192-207 ◽  
Author(s):  
Christophe Ritzenthaler
Keyword(s):  
Modular Form ◽  
Elliptic Curve ◽  
Complex Multiplication ◽  
Numerical Results ◽  
Siegel Modular Form ◽  
Square Root ◽  
The Third

AbstractLetkbe a field of characteristic other than 2. There can be an obstruction to a principally polarized abelian threefold (A,a) overk, which is a Jacobian over$\bar {k}$, being a Jacobian over k; this can be computed in terms of the rationality of the square root of the value of a certain Siegel modular form. We show how to do this explicitly for principally polarized abelian threefolds which are the third power of an elliptic curve with complex multiplication. We use our numerical results to prove or refute the existence of some optimal curves of genus 3.

scholarly journals Siegel modular forms and theta series attached to quaternion algebras II

1997 ◽  
Vol 147 ◽  
pp. 71-106 ◽  
Author(s):  
S. Böcherer ◽  
R. Schulze-Pillot
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quaternion Algebra ◽  
Automorphic Forms ◽  
Multiplicative Group ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Quaternion Algebras

AbstractWe continue our study of Yoshida’s lifting, which associates to a pair of automorphic forms on the adelic multiplicative group of a quaternion algebra a Siegel modular form of degree 2. We consider here the case that the automorphic forms on the quaternion algebra correspond to modular forms of arbitrary even weights and square free levels; in particular we obtain a construction of Siegel modular forms of weight 3 attached to a pair of elliptic modular forms of weights 2 and 4.

scholarly journals Restriction of Siegel modular forms to modular curves

2002 ◽  
Vol 65 (2) ◽  
pp. 239-252 ◽  
Author(s):  
Cris Poor ◽  
David S. Yuen
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Congruence Subgroup ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Modular Curves ◽  
Linear Relations

We study homomorphisms form the ring of Siegel modular forms of a given degree to the ring of elliptic modular forms for a congruence subgroup. These homomorphisms essentially arise from the restriction of Siegel modular forms to modular curves. These homomorphisms give rise to linear relations among the Fourier coefficients of a Siegel modular form. We use this technique to prove that dim .

A p-ADIC LIMIT OF SIEGEL–EISENSTEIN SERIES OF PRIME LEVEL q

2008 ◽  
Vol 04 (05) ◽  
pp. 735-746 ◽  
Author(s):  
YOSHINORI MIZUNO
Keyword(s):  
Modular Form ◽  
Eisenstein Series ◽  
Simple Formula ◽  
Fourier Coefficients ◽  
Siegel Modular Form

We show that a p-adic limit of a Siegel–Eisenstein series of prime level q becomes a Siegel modular form of level pq. This paper contains a simple formula for Fourier coefficients of a Siegel–Eisenstein series of degree two and prime levels.

The Dirichlet Problem on the Heisenberg Group III: Harmonic Measure of a certain Half-Space

1986 ◽  
Vol 38 (3) ◽  
pp. 666-671 ◽  
Author(s):  
Bernard Gaveau ◽  
Jacques Vauthier
Keyword(s):  
Heisenberg Group ◽  
Half Space ◽  
Harmonic Measure ◽  
Short Note ◽  
Type Equation ◽  
Probabilistic Argument ◽  
Group Iii ◽  
3 Dimensional ◽  
Degenerate Hypergeometric Function ◽  
Made In

0. Introduction. In this short note we give an explicit computation of the harmonic measure of a half space x > 0 in the 3-dimensional Heisenberg group in terms of a degenerate hypergeometric function. A probabilistic argument reduces the whole problem to a Hermite-type equation on a half line, that we can solve in terms of the function G(l/4, 1/2; x2).A preliminary attempt to compute this kernel was done in [1] p. 107 and, cited by Huber [4]. Unfortunately a small mistake was made in [1] and the problem was still open until now. The first author is very grateful to Prof. Huber for having pointed out the weak argument of [1]. Since that time, other harmonic measures and even Green functions have been explicitly computed (see [2]).

scholarly journals A BÖCHERER-TYPE CONJECTURE FOR PARAMODULAR FORMS

2011 ◽  
Vol 07 (05) ◽  
pp. 1395-1411 ◽  
Author(s):  
NATHAN C. RYAN ◽  
GONZALO TORNARÍA
Keyword(s):  
Modular Form ◽  
Siegel Modular Form ◽  
Numerical Evidence ◽  
Quadratic Twists ◽  
Central Value

In the 1980s Böcherer formulated a conjecture relating the central value of the quadratic twists of the spinor L-function attached to a Siegel modular form F to the coefficients of F. He proved the conjecture when F is a Saito–Kurokawa lift. Later Kohnen and Kuß gave numerical evidence for the conjecture in the case when F is a rational eigenform that is not a Saito–Kurokawa lift. In this paper we develop a conjecture relating the central value of the quadratic twists of the spinor L-function attached to a paramodular form and the coefficients of the form. We prove the conjecture in the case when the form is a Gritsenko lift and provide numerical evidence when it is not a lift.

scholarly journals TWO CONDITIONAL RESULTS ABOUT PRIMES IN SHORT INTERVALS

2011 ◽  
Vol 07 (07) ◽  
pp. 1753-1759
Author(s):  
DANILO BAZZANELLA
Keyword(s):  
Asymptotic Formula ◽  
Short Intervals ◽  
Lindelöf Hypothesis

In 1937, Ingham proved that ψ(x + xθ) - ψ(x) ~ xθ for x → ∞, under the assumption of the Lindelöf hypothesis for θ > 1/2. In this paper we examine how the above asymptotic formula holds by assuming in turn two different heuristic hypotheses. It must be stressed that both the hypotheses are implied by the Lindelöf hypothesis.

scholarly journals K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space IV: The structure of the invariant

2020 ◽  
Vol 156 (10) ◽  
pp. 1965-2019
Author(s):  
Shouhei Ma ◽  
Ken-Ichi Yoshikawa
Keyword(s):  
Modular Form ◽  
Moduli Space ◽  
Automorphic Forms ◽  
K3 Surfaces ◽  
Automorphic Function ◽  
Siegel Modular Form ◽  
Analytic Torsion ◽  
Invent Math ◽  
Torsion Invariant

AbstractYoshikawa in [Invent. Math. 156 (2004), 53–117] introduces a holomorphic torsion invariant of $K3$ surfaces with involution. In this paper we completely determine its structure as an automorphic function on the moduli space of such $K3$ surfaces. On every component of the moduli space, it is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. We also introduce its twisted version. We prove its modularity and a certain uniqueness of the modular form corresponding to the twisted holomorphic torsion invariant. This is used to study an equivariant analogue of Borcherds’ conjecture.

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