Pullbacks of Eisenstein series, Hecke-Jacobi theory and automorphic 𝐿-functions

Heim, Bernhard

doi:10.1090/pspum/066.2/1703760

Cited by 25 publications

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“…where we have denoted s 12 = s * 1 + s 2 = x 1 + x 2 + i(y 2 − y 1 ) and used the expressions (22) and (28).…”

Section: The Operator That Yields the Riemann Zerosmentioning

confidence: 99%

On Strategies Towards the Riemann Hypothesis: Fractal Supersymmetric Qm and a Trace Formula

Castro

2007

Int. J. Geom. Methods Mod. Phys.

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The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn = 1/2+iλn. An improvement of our previous construction to prove the RH is presented by implementing the Hilbert-Polya proposal and furnishing the Fractal Supersymmetric Quantum Mechanical (SUSY-QM) model whose spectrum reproduces the imaginary parts of the zeta zeros. We model the fractal fluctuations of the smooth Wu-Sprung potential ( that capture the average level density of zeros ) by recurring to a weighted superposition of Weierstrass functions p W (x, p, D) and where the summation has to be performed over all primes p in order to recapture the connection between the distribution of zeta zeros and prime numbers. We proceed next with the construction of a smooth version of the fractal QM wave equation by writing an ordinary Schroedinger equation whose fluctuating potential (relative to the smooth Wu-Sprung potential) has the same functional form as the fluctuating part of the level density of zeros. The second approach to prove the RH relies on the existence of a continuous family of scaling-like operators involving the Gauss-Jacobi theta series. An explicit trace formula related to a superposition of eigenfunctions of these scaling-like operators is defined. If the trace relation is satisfied this could be another test of the Riemann Hypothesis

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“…where we have denoted s 12 = s * 1 + s 2 = x 1 + x 2 + i(y 2 − y 1 ) and used the expressions (22) and (28).…”

Section: The Operator That Yields the Riemann Zerosmentioning

confidence: 99%

On Strategies Towards the Riemann Hypothesis: Fractal Supersymmetric Qm and a Trace Formula

Castro

2007

Int. J. Geom. Methods Mod. Phys.

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“…Then (a) A(F, g; k) is algebraic, and (b) A(F, g; k) σ = A(F σ , g σ ; k) for an automorphism σ of ‫.ރ‬ The completely unramified case M = 1 and N = 1 of this theorem was already known in [Heim 1999] and [Böcherer and Heim 2006], which used a very different integral representation from the one in this paper. Also, just the algebraicity part of this theorem (that is, part (a)) has been proved for the rightmost critical value (corresponding to k = 1) in various settings earlier by Furusawa [1993], Pitale and Schmidt [2009b] and the author [Saha 2009].…”

Section: Introductionmentioning

confidence: 82%

Pullbacks of Eisenstein series from GU(3,3) and criticalL-values for GSp(4) ×GL(2)

Saha¹

2010

Pacific J. Math.

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“…Garrett [Ga87] computed the scalar product of the restricted Eisenstein series with three elliptic cusp forms. A detailed analysis and combination of the two papers of Garrett (see also [He99]) leads to the complete pullback formula. We obtain:…”

Section: Proof Of the Arithmetic Trace Formulamentioning

confidence: 98%

“…These numbers are essentially critical values of the triple L-function in the sense of Deligne. They had been first explicitely determined by Garrett [Ga87] (see also Mizumoto [Mi97], page 192, and Heim [He99], page 236, for the explicit value of the constants and further explanation):…”

Section: Proof Of the Arithmetic Trace Formulamentioning

confidence: 99%

A Trace Formula of Special Values of Automorphic L-Functions

Heim

2012

Multiple Dirichlet Series, L-Functions and Automorphic Forms

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Abstract. Deligne introduced the concept of special values of automorphic L-functions. The arithmetic properties of these L-functions play a fundamental role in modern number theory. In this paper we prove a trace formula which relates special values of the Hecke, Rankin, and the central value of the Garrett triple L-function attached to primitive newforms. This type of trace formula is new and involves special values in the convergent and non-convergent domain of the underlying L-functions.

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Pullbacks of Eisenstein series, Hecke-Jacobi theory and automorphic 𝐿-functions

Cited by 25 publications

References 0 publications

On Strategies Towards the Riemann Hypothesis: Fractal Supersymmetric Qm and a Trace Formula

On Strategies Towards the Riemann Hypothesis: Fractal Supersymmetric Qm and a Trace Formula

Pullbacks of Eisenstein series from GU(3,3) and criticalL-values for GSp(4) ×GL(2)

A Trace Formula of Special Values of Automorphic L-Functions

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