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Sturm's operator acting on vector valued $K$ ‐types

Kathrin Maurischat

Mathematische Nachrichten > 293 > 6 > 1136 - 1155

We define Sturm's operator for vector valued Siegel modular forms obtaining an explicit description of their holomorphic projection in case of large absolute weight. However, for small absolute weight, Sturm's operator produces phantom terms in addition. This confirms our earlier results for scalar Siegel modular forms.

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Contributions to algebraic number theory from India : From independence to late 90’s

Dipendra Prasad

Indian Journal of Pure and Applied Mathematics > 2019 > 50 > 3 > 779-793

There was a conference organised at the Institite of Mathematical Sciences, Chennai in 1997 on the occasion of the 50th anniversary of India’s Independence to review contributions by Indians to various branches of mathematics since Independence. This is a written account of the lecture I gave then on some of the contributions by Indian mathematicians to Algebraic number theory from India’s independence...

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One-line formula for automorphic differential operators on Siegel modular forms

Tomoyoshi Ibukiyama

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg > 2019 > 89 > 1 > 17-43

We consider the Siegel upper half space $$H_{2m}$$ H 2 m of degree 2m and a subset $$H_m\times H_m$$ H m × H m of $$H_{2m}$$ H 2 m consisting of two $$m\times m$$ m × m diagonal block matrices. We consider two actions of $$Sp(m,{\mathbb R})\times Sp(m,{\mathbb R}) \subset Sp(2m,{\mathbb R})$$ S p ( m , R ) × S p ( m , R ) ⊂ S p ( 2 m , R ) , one is the...

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Functional equations of real analytic Jacobi Eisenstein series

Shin-ichiro Mizumoto

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg > 2019 > 89 > 1 > 55-75

We prove the existence of meromorphic continuation and the functional equation of the real analytic Jacobi Eisenstein series of degree m and matrix index T in case T is a kernel form.

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Sturm bounds for Siegel modular forms of degree 2 and odd weights

Toshiyuki Kikuta, Sho Takemori

Mathematische Zeitschrift > 2019 > 291 > 3-4 > 1419-1434

We correct the proof of the theorem in the previous paper presented by Kikuta, which concerns Sturm bounds for Siegel modular forms of degree 2 and of even weights modulo a prime number dividing $$2\cdot 3$$ 2 · 3 . We give also Sturm bounds for them of odd weights for any prime numbers, and we prove their sharpness. The results cover the case where Fourier coefficients are algebraic numbers.

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Explicitly realizing average Siegel theta series as linear combinations of Eisenstein series

Lynne H. Walling

The Ramanujan Journal > 2018 > 47 > 3 > 475-499

We find nice representatives for the 0-dimensional cusps of the degree n Siegel upper half-space under the action of $$\Gamma _0(\mathcal N )$$ Γ0(N) . To each of these, we attach a Siegel Eisenstein series, and then we make explicit a result of Siegel, realizing any integral weight average Siegel theta series of arbitrary level $$\mathcal N $$ N and Dirichlet character $$\chi _{_L}$$ χL modulo...

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Some relations on Fourier coefficients of degree 2 Siegel forms of arbitrary level

Lynne H. Walling

Journal of Number Theory > 2017 > 180 > C > 349-359

We extend some recent work of D. McCarthy, proving relations among some Fourier coefficients of a degree 2 Siegel modular form F with arbitrary level and character, provided there are some primes p so that F is an eigenform for the Hecke operators T(p) and T1(p2).

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Improved bounds for Fourier coefficients of Siegel modular forms

Kathrin Bringmann

Journal of Mathematical Analysis and Applications > 2017 > 451 > 2 > 1123-1132

In this paper we improve existing bounds for Siegel modular forms of small weight.

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p-Adic Admissible Measures Attached to Siegel Modular Forms of Arbitrary Genus

Anh Tuan Do

Vietnam Journal of Mathematics > 2017 > 45 > 4 > 695-711

Let p be a prime number and f ∈ S n l ( Γ 0 ( N ) , ψ ) $f\in {S_{n}^{l}}({\Gamma }_{0}(N), \psi )$ be a Siegel cusp eigenform of genus n. We consider the standard zeta function D^(Np)(f,s,χ), which takes algebraic values at critical points after normalization. We construct two h-admissible measures μ⁺ and μ⁻ for certain h = [4ord_p(α₀(p))] + 1 explained in the Main Theorem with...

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Maass Spezialschar of level N

Bernhard Heim

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg > 2017 > 87 > 1 > 181-195

In this paper the image of the Saito–Kurokawa lift of level N with Dirichlet character is studied. We give a new characterization of this so called Maass Spezialschar of level N by symmetries involving Hecke operators related to $$\Gamma _0(N)$$ Γ 0 ( N ) . We finally obtain for all prime numbers p local Maass relations. This generalizes known results for level $$N=1$$ N = 1 ...

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Multiplicative relations for Fourier coefficients of degree 2 Siegel eigenforms

Dermot McCarthy

Journal of Number Theory > 2017 > 170 > C > 263-281

We prove multiplicative relations between certain Fourier coefficients of degree 2 Siegel eigenforms. These relations are analogous to those for elliptic eigenforms. We also provide two sets of formulas for the eigenvalues of degree 2 Siegel eigenforms. The first evaluates the eigenvalues in terms of the form's Fourier coefficients, in the case a(I)≠0. The second expresses the eigenvalues of index...

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Almost holomorphic Poincaré series corresponding to products of harmonic Siegel–Maass forms

Kathrin Bringmann, Olav K. Richter, Martin Westerholt-Raum

Research in the Mathematical Sciences > 2016 > 3 > 1 > 1-16

We investigate Poincaré series, where we average products of terms of Fourier series of real-analytic Siegel modular forms. There are some (trivial) special cases for which the products of terms of Fourier series of elliptic modular forms and harmonic Maass forms are almost holomorphic, in which case the corresponding Poincaré series are almost holomorphic as well. In general, this is not the case...

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A Dirichlet series attached to three Siegel modular forms

Shin-ichiro Mizumoto

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg > 2017 > 87 > 1 > 113-134

For three Siegel modular forms of degrees $$n_1, n_2$$ n 1 , n 2 and $$n_1+n_2-1$$ n 1 + n 2 - 1 respectively, we attach a certain Dirichlet series which has a meromorphic continuation to the whole complex plane and satisfies a functional equation. We also show some algebraicity property of its special values. For the proof we use a Rankin-Selberg type integral involving...

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Non-vanishing of Koecher-Maass series attached to Siegel cusp forms on the real line

Winfried Kohnen

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg > 2017 > 87 > 1 > 39-41

Communicated by Jens Funke.

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Rankin–Cohen brackets on Siegel modular forms and special values of certain Dirichlet series

Abhash Kumar Jha, Brundaban Sahu

The Ramanujan Journal > 2017 > 44 > 1 > 63-73

Given a fixed Siegel cusp form of genus two, we consider a family of linear maps between the spaces of Siegel cusp forms of genus two by using the Rankin–Cohen brackets and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Siegel cusp forms of genus two constructed using this method involve special values of certain...

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Rankin’s Lemma of Higher Genus and Explicit Formulas for Hecke Operators

Alexei Panchishkin, Kirill Vankov

Progress in Mathematics > Algebra, Arithmetic, and Geometry > 533-554

Summary We develop explicit formulas for Hecke operators of higher genus in terms of spherical coordinates. Applications are given to summation of various generating series with coefficients in local Hecke algebras and in a tensor product of such algebras. In particular, we formulate and prove Rankin’s lemma in genus two. An application to a holomorphic lifting from GSp₂ ×GSp₂ to GSp₄ is given using...

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Cuspidality and the growth of Fourier coefficients: small weights

Siegfried Böcherer, Soumya Das

Mathematische Zeitschrift > 2016 > 283 > 1-2 > 539-553

We characterize Siegel cusp forms in the space of Siegel modular forms of small weight $$k \ge n+4$$ k ≥ n + 4 on the congruence subgroups $$\Gamma ^n_0(N)$$ Γ 0 n ( N ) of any degree n and any level N, by a suitable growth of their Fourier coefficients (e.g., by the well known Hecke bound) at any one of the cusps. For this, we use the formalism of Jacobi forms and the ‘Witt-operator’...

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Cuspidality and the growth of Fourier coefficients of modular forms : A survey

Soumya Das

Indian Journal of Pure and Applied Mathematics > 2016 > 47 > 1 > 9-22

In this paper, we present a survey of the recent results on the characterization of the cuspidality of classical modular forms on various groups by a suitable growth of their Fourier coefficients.

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Mixed-level Saito–Kurokawa liftings

Jim Brown, Dania Zantout

The Ramanujan Journal > 2016 > 39 > 2 > 247-257

In this article we construct Saito–Kurokawa lifts of mixed level. These are constructed via representation theoretic arguments originally used by Schmidt to construct congruence level and paramodular Saito–Kurokawa lifts.

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Notes on theta series for Niemeier lattices

Shoyu Nagaoka, Sho Takemori

The Ramanujan Journal > 2017 > 42 > 2 > 385-400

Some explicit expressions are given for the theta series of Niemeier lattices. As an application, we present some of their congruence relations.

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Search results

Sturm's operator acting on vector valued $K$ ‐types

Contributions to algebraic number theory from India : From independence to late 90’s

One-line formula for automorphic differential operators on Siegel modular forms

Functional equations of real analytic Jacobi Eisenstein series

Sturm bounds for Siegel modular forms of degree 2 and odd weights

Explicitly realizing average Siegel theta series as linear combinations of Eisenstein series

Some relations on Fourier coefficients of degree 2 Siegel forms of arbitrary level

Improved bounds for Fourier coefficients of Siegel modular forms

p-Adic Admissible Measures Attached to Siegel Modular Forms of Arbitrary Genus

Maass Spezialschar of level N

Multiplicative relations for Fourier coefficients of degree 2 Siegel eigenforms

Almost holomorphic Poincaré series corresponding to products of harmonic Siegel–Maass forms

A Dirichlet series attached to three Siegel modular forms

Non-vanishing of Koecher-Maass series attached to Siegel cusp forms on the real line

Rankin–Cohen brackets on Siegel modular forms and special values of certain Dirichlet series

Rankin’s Lemma of Higher Genus and Explicit Formulas for Hecke Operators

Cuspidality and the growth of Fourier coefficients: small weights

Cuspidality and the growth of Fourier coefficients of modular forms : A survey

Mixed-level Saito–Kurokawa liftings

Notes on theta series for Niemeier lattices

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