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We prove Ibukiyama’s conjectures on Siegel modular forms of half-integral weight and of degree 2 by using Arthur’s multiplicity formula on the split odd special orthogonal group $${\text {SO}}_5$$ and Gan–Ichino’s multiplicity formula on the metaplectic group $${\text {Mp}}_4$$ . In the proof, the representation theory of the Jacobi groups also plays an important role.

We prove that if $F$ is a non-zero (possibly non-cuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many non-zero Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and thus fundamental) discriminant. The proof uses an induction argument in the setting of vector-valued modular forms. In an Appendix, as an application of a variant of our result and building upon the work of A. Pollack, we show how to obtain an unconditional proof of the functional equation of the spinor $L$-function of a holomorphic cuspidal Siegel eigenform of degree $3$., Comment: 52 pages, expanded with more explanations, corrections. An Appendix has been included, which however will not appear in the journal version

We study several aspects of nonvanishing Fourier coefficients of elliptic modular forms $$\bmod \, \ell $$ , partially answering a question of Bellaiche-Soundararajan concerning the asymptotic formula for the count of the number of Fourier coefficients upto x which do not vanish $$\bmod \, \ell $$ . We also propose a precise conjecture as a possible answer to this question. Further, we prove several results related to the nonvanishing of arithmetically interesting (e.g., primitive or fundamental) Fourier coefficients $$\bmod \, \ell $$ of a Siegel modular form with integral algebraic Fourier coefficients provided $$\ell $$ is large enough. We also make some efforts to make this “largeness” of $$\ell $$ effective.

Notable results on the special values of $L$-functions of Siegel modular forms were obtained by J. Sturm in the case when the degree $n$ is even and the weight $k$ is an integer. In this paper we extend this method to half-integer weights $k$ and arbitrary degree $n$, determining the algebraic field in which they lie. This method hinges on the Rankin-Selberg method; our extension of this is aided by the theory of half-integral modular forms developed by G. Shimura. In the second half, an analogue of P. B. Garrett's conjecture is proved in this setting, a result that is of independent interest but that bears direct applications to our first results. It determines exactly how the decomposition of modular forms into cusp forms and Eisenstein series preserves algebraicity and, ultimately, the full range of special values.

Motivated by recent advances in Donaldson-Thomas theory, four-dimensional $$ \mathcal{N} $$ N = 4 string-string duality is examined in a reduced rank theory on a less studied BPS sector. In particular we identify candidate partition functions of “untwisted” quarter-BPS dyons in the heterotic ℤ2 CHL model by studying the associated chiral genus two partition function, based on the M-theory lift of string webs argument by Dabholkar and Gaiotto. This yields meromorphic Siegel modular forms for the Iwahori subgroup B(2) ⊂ Sp4(ℤ) which generate BPS indices for dyons with untwisted sector electric charge, in contrast to twisted sector dyons counted by a multiplicative lift of twisted-twining elliptic genera known from Mathieu moonshine. The new partition functions are shown to satisfy the expected constraints coming from wall-crossing and S-duality symmetry as well as the black hole entropy based on the Gauss-Bonnet term in the effective action. In these aspects our analysis confirms and extends work of Banerjee, Sen and Srivastava, which only addressed a subset of the untwisted sector dyons considered here. Our results are also compared with recently conjectured formulae of Bryan and Oberdieck for the partition functions of primitive DT invariants of the CHL orbifold X = (K3 × T2)/ℤ2, as suggested by string duality with type IIA theory on X.

Yoshikawa 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.

Correction to my paper on the poles of standard L-functions attached to Siegel modular forms.

We construct non-geometric string compactifications by using the F-theory dual of the heterotic string compactified on a two-torus with two Wilson line parameters, together with a close connection between modular forms and the equations for certain K3 surfaces of Picard rank $16$. We construct explicit Weierstrass models for all inequivalent Jacobian elliptic fibrations supported on this family of K3 surfaces and express their parameters in terms of modular forms generalizing Siegel modular forms. In this way, we find a complete list of all dual non-geometric compactifications obtained by the partial higgsing of the heterotic string gauge algebra using two Wilson line parameters., 22 pages. arXiv admin note: substantial text overlap with arXiv:1908.09578, arXiv:1806.07460

We consider four-dimensional dyonic single-center BPS black holes in the $N=2$ STU model of Sen and Vafa. By working in a region of moduli space where the real part of two of the three complex scalars $S, T, U$ are taken to be large, we evaluate the quantum entropy function for these BPS black holes. In this regime, the subleading corrections point to a microstate counting formula partly based on a Siegel modular form of weight two. This is supplemented by another modular object that takes into account the dependence on $Y^0$, a complex scalar field belonging to one of the four off-shell vector multiplets of the underlying supergravity theory. We also observe interesting connections to the rational Calogero model and to formal deformation of a Poisson algebra, and suggest a string web picture of our counting proposal., Comment: 72 pages; v2: matches published version

ith respect to the analytic‐algebraic dichotomy, the theory of Siegel modular forms of half‐integral weight is lopsided; the analytic theory is strong, whereas the algebraic lags behind. In this paper, we capitalise on this to establish the fundamental object needed for the analytic side of the Iwasawa main conjecture — the p ‐adic L ‐function obtained by interpolating the complex L ‐function at special values. This is achieved through the Rankin–Selberg method and the explicit Fourier expansion of non‐holomorphic Siegel Eisenstein series. The construction of the p ‐stabilisation in this setting is also of independent interest.

We define Sturm's operator on 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.

We present five open problems in the theory of vertex rings. They cover a variety of different areas of research where vertex rings have been, or are threatening to be, relevant. They have also been chosen because I personally find them interesting, and because I think each of them has a chance (the title of the paper notwithstanding!) of being solved. In each case we give some explanatory background and motivation, sometimes including proofs of special cases. Beyond vertex rings per se, the topics covered include connections to real Lie theory, formal group laws, modular linear differential equations, Pierce bundles, and genus 2 Siegel modular forms and the Moonshine Module.

A CHL model is the quotient of $\mathrm{K3} \times E$ by an order $N$ automorphism which acts symplectically on the K3 surface and acts by shifting by an $N$-torsion point on the elliptic curve $E$. We conjecture that the primitive Donaldson-Thomas partition function of elliptic CHL models is a Siegel modular form, namely the Borcherds lift of the corresponding twisted-twined elliptic genera which appear in Mathieu moonshine. The conjecture matches predictions of string theory by David, Jatkar and Sen. We use the topological vertex to prove several base cases of the conjecture. Via a degeneration to $\mathrm{K3} \times \mathbb{P}^1$ we also express the DT partition functions as a twisted trace of an operator on Fock space. This yields further computational evidence. An extension of the conjecture to non-geometric CHL models is discussed. We consider CHL models of order $N=2$ in detail. We conjecture a formula for the Donaldson-Thomas invariants of all order two CHL models in all curve classes. The conjecture is formulated in terms of two Siegel modular forms. One of them, a Siegel form for the Iwahori subgroup, has to our knowledge not yet appeared in physics. This discrepancy is discussed in an appendix with Sheldon Katz.

We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair (G,K), where G=Sp_4(R) and K is its maximal compact subgroup. In particular, we determine K-types and composition series, and write down explicit differential operators that navigate all the highest weight vectors of such a module starting from the unique lowest-weight vector. By rewriting these operators in classical language, we show that the automorphic forms on G that correspond to the highest weight vectors are exactly those that arise from nearly holomorphic vector-valued Siegel modular forms of degree 2. Further, by explicating the algebraic structure of the relevant space of n-finite automorphic forms, we are able to prove a structure theorem for the space of nearly holomorphic vector-valued Siegel modular forms of (arbitrary) weight $det^\ell$ sym^m with respect to an arbitrary congruence subgroup of Sp_4(Q). We show that the cuspidal part of this space is the direct sum of subspaces obtained by applying explicit differential operators to holomorphic vector-valued cusp forms of weight $det^{\ell'} sym^{m'}$ with $(\ell', m')$ varying over a certain set. The structure theorem for the space of all modular forms is similar, except that we may now have an additional component coming from certain nearly holomorphic forms of weight $det^{3}sym^{m'}$ that cannot be obtained from holomorphic forms. As an application of our structure theorem, we prove several arithmetic results concerning nearly holomorphic modular forms that improve previously known results in that direction., 51 pages. Accepted version, to appear in Kyoto J. Math

We prove (under certain assumptions) the irreducibility of the limit $$\sigma _2$$ σ 2 of a sequence of irreducible essentially self-dual Galois representations $$\sigma _k: G_{{\mathbf {Q}}} \rightarrow {{\,\mathrm{GL}\,}}_4(\overline{{\mathbf {Q}}}_p)$$ σ k : G Q → GL 4 ( Q ¯ p ) (as k approaches 2 in a p-adic sense) which mod p reduce (after semi-simplifying) to $$1 \oplus \rho \oplus \chi $$ 1 ⊕ ρ ⊕ χ with $$\rho $$ ρ irreducible, two-dimensional of determinant $$\chi $$ χ , where $$\chi $$ χ is the mod p cyclotomic character. More precisely, we assume that $$\sigma _k$$ σ k are crystalline (with a particular choice of weights) and Siegel-ordinary at p. Such representations arise in the study of p-adic families of Siegel modular forms and properties of their limits as $$k\rightarrow 2$$ k → 2 appear to be important in the context of the Paramodular Conjecture. The result is deduced from the finiteness of two Selmer groups whose order is controlled by p-adic L-values of an elliptic modular form (giving rise to $$\rho $$ ρ ) which we assume are non-zero.

We determine the structure of the ring of Siegel modular forms of degree 2 in characteristic 3., Comment: 10 pages; minor revision. To appear in Math. Zeitschrift

In this work, we use the Rankin–Selberg method to obtain results on the analytic properties of the standard [Formula: see text]-function attached to vector-valued Siegel modular forms. In particular we provide a detailed description of its possible poles and obtain a non-vanishing result of the twisted [Formula: see text]-function beyond the usual range of absolute convergence. Our results include also the case of metaplectic Siegel modular forms. We remark that these results were known in this generality only in the case of scalar weight Siegel modular forms. As an interesting by-product of our work we establish the cuspidality of some theta series.

It is known via work of Duke and Ghate that there are only finitely many pairs of full level, degree one eigenforms f and g whose product fg is also an eigenform. We prove a partial generalization of this theorem for degree two Siegel modular forms. Namely, we show that there is only one pair of eigenforms F and G such that FG is a non-cuspidal eigenform. In the case that FG is a cuspform, we provide necessary conditions for FG to be an eigenform, give one example, and conjecture that is the only example.

We consider two dimensional $\mathcal{N}=(4,4)$ superconformal field theories in the moduli space of symmetric orbifolds of K3. We complete a classification of the discrete groups of symmetries of these models, conditional to a series of assumptions and with certain restrictions. Furthermore, we provide a partial classification of the set of twining genera, encoding the action of a discrete symmetry $g$ on a space of supersymmetric states in these models. These results suggest the existence of a number of surprising identities between seemingly different Borcherds products, representing Siegel modular forms of degree two and level $N>1$. We also provide a critical review of various properties of the moduli space of these superconformal field theories, including the groups of dualities, the set of singular models and the locus of symmetric orbifold points, and describe some puzzles related to our (lack of) understanding of these properties., 54 pages; v3: various points clarified; appendix E added; matches with published version

Статья посвящена памяти Георгия Вороного. Описываются новые избранные результаты о рядах Эйзенштейна, о (мотивных), (p-адических), (кратных) значениях (круговых) дзета и L-функций, и их приложения, полученные ниже перечисляемыми авторами, а также элементарное введение в эти результаты. Дан краткий обзор новых результатов о (мотивных), (p-адических), (кратных) значениях (круговых) дзета функциях, L-функциях и рядах Эйзенштейна. Статья ориентирована на избранные задачи и не является исчерпывающей. Начало статьи содержит краткое изложение результатов о числах Бернулли, связанных с исследованиями Георгия Вороного. Результаты о кратных значениях дзета функций были представлены Д. Загиром, П. Делинем и А. Гончаровым, А. Гончаровым, Ф. Брауном, К. Глэносом (Glanois) и другими. С. Унвер ("Unver) исследовал кратные p-адические дзета-значения глубины два. Таннакиева интерпретация кратных p-адических дзета-значений дана Х. Фурушо. Краткая история и связи между группами Галуа, фундаментальными группами, мотивами и арифметическими функциями представлены в докладе Ю. Ихара. Результаты о кратных дзета-значениях, группах Галуа и геометрии модулярных многообразий представлены Гончаровым. Интересная унипотентная мотивная фундаментальная группа определена и исследована Делинем и Гончаровым. В данной работе мы кратко упоминаем в рамках (p-адических) L-функций и (p-адических) (кратных) дзета-значений применения подходов Куботы-Леопольдта и Ивасавы, которые основанны на p-адических L-функциях Куботы-Леопольда, и арифметических p-адических L-функциях Ивасавы. Прореферирован ряд недавних работ (и соответствующих результатов): кратные дзета-значения в корнях из единицы, построение семейств мотивных итерированных интегралов с предписанными свойствами по Глэносу (Glanois); явные выражения для круговых p-адических кратных дзета-значений глубины два по Унверу (Unver); связи арифметических степеней циклов Кудлы-Рапопорта на интегральной модели многообразия Шимуры, соответствующей унитарной группе сигнатуры (1,1), с коэффициентами Фурье центральных производных рядов Эйзенштейна рода 2 по Санкарану (Sankaran). Более полно с содержанием статьи можно ознакомиться по приводимому ниже оглавлению: Введение. 1. Сравнения типа Вороного для чисел Бернулли. 2. Римановы дзета-значения. 3. О группах классов колец с теорией дивизоров. Мнимые квадратичные и круговые поля. 4. Ряды Эйзенштейна. 5. Группы классов, поля классов и дзета-функции. 6. Кратные дзета-значения. 7. Элементы неархимедовых локальных полей и неархимедова анализа. 8. Итерированные интегралы и (кратные) дзета-значения. 9. Формальные и p-делимые группы. 10. Мотивы и (p-адические) (кратные) дзета-значения. 11. О рядах Эйзенштейна, ассоциированных с многообразиями Шимуры. Разделы 1-9 и подраздел 11.1 (О некоторых многообразиях Шимуры и модулярных формах Зигеля) можно рассматривать как элементарное введение в результаты раздела 10 и подраздела 11.2 (О несобственном пересечении дивизоров Кудлы-Рапопорта и рядах Эйзенштейна).Я глубоко признателен Н. М. Добровольскому за помощь и поддержку в процессе подготовки статьи к печати.

We use covariants of binary sextics to describe the structure of modules of scalar-valued or vector-valued Siegel modular forms of degree 2 with character, over the ring of scalar-valued Siegel modular forms of even weight. For a modular form defined by a covariant we express the order of vanishing along the locus of products of elliptic curves in terms of the covariant., Comment: 18 pages

This paper is a continuation of Kim et al. (J Inst Math Jussieu, 2018). We supplement four results on a family of holomorphic Siegel cusp forms for $$GSp_4/\mathbb {Q}$$ . First, we improve the result on Hecke fields. Namely, we prove that the degree of Hecke fields is unbounded on the subspace of genuine forms which do not come from functorial lift of smaller subgroups of $$GSp_4$$ . Second, we prove simultaneous vertical Sato–Tate theorem. Namely, we prove simultaneous equidistribution of Hecke eigenvalues at finitely many primes. Third, we compute the n-level density of degree 4 spinor L-functions, and thus we can distinguish the symmetry type depending on the root numbers. This is conditional on certain conjecture on root numbers. Fourth, we consider equidistribution of paramodular forms. In this case, we can prove the conjecture on root numbers. Main tools are the equidistribution theorem in our previous work and Shin–Templier’s (Compos Math 150(12):2003–2053, 2014) work.

Generalizing the method of Faltings–Serre, we rigorously verify that certain abelian surfaces without extra endomorphisms are paramodular. To compute the required Hecke eigenvalues, we develop a method of specialization of Siegel paramodular forms to modular curves. Fil: Brumer, Armand. Fordham University; Estados Unidos Fil: Pacetti, Ariel Martín. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina Fil: Poor, Cris. Fordham University; Estados Unidos Fil: Tornaría, Gonzalo. Universidad de la República; Uruguay Fil: Voight, John. Dartmouth College; Estados Unidos Fil: Yuen, David S.. University of Hawaii; Estados Unidos

We classify the holomorphic Borcherds products of singular weight for all simple lattices of signature $(2,n)$ with $n \geq 3$. In addition to the automorphic products of singular weight for the simple lattices of square free level found by Dittmann, Hagemeier and the second author, we obtain several automorphic products of singular weight $1/2$ for simple lattices of signature $(2,3)$. We interpret them as Siegel modular forms of genus $2$ and explicitly describe them in terms of the ten even theta constants. In order to rule out further holomorphic Borcherds products of singular weight, we derive estimates for the Fourier coefficients of vector valued Eisenstein series, which are of independent interest., 15 pages

We compute the first and second moment of the spinor L-function at the central point of Siegel modular forms of large weight k with power saving error term and give applications to non-vanishing.

In this paper we give a structure theorem for the module of vector valued modular forms in the case of a three dimensional ball with the action of the Picard modular group Γ 3 [ − 3 ] \Gamma _3 [\sqrt {-3}] . The corresponding modular variety of dimension 3 3 is a copy of the Segre cubic.

Let E E be an ordinary elliptic curve over a finite field and g g be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of E g E^g . The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre’s obstruction for principally polarized abelian threefolds isogenous to E 3 E^3 and of the Igusa modular form in dimension 4 4 . We illustrate our algorithms with examples of curves with many rational points over finite fields.

We extend the framework of modular invariant supersymmetric theories to encompass invariance under more general discrete groups $\Gamma$, that allow the presence of several moduli and make connection with the theory of automorphic forms. Moduli span a coset space $G/K$, where $G$ is a Lie group and $K$ is a compact subgroup of $G$, modded out by $\Gamma$. For a general choice of $G$, $K$, $\Gamma$ and a generic matter content, we explicitly construct a minimal K\"ahler potential and a general superpotential, for both rigid and local $N=1$ supersymmetric theories. We also specialize our construction to the case $G=Sp(2g,R)$, $K=U(g)$ and $\Gamma=Sp(2g,Z)$, whose automorphic forms are Siegel modular forms. We show how our general theory can be consistently restricted to multi-dimensional regions of the moduli space enjoying residual symmetries. After choosing $g=2$, we present several examples of models for lepton and quark masses where Yukawa couplings are Siegel modular forms of level 2., Comment: 60 pages, 1 figure

Holomorphic vector valued differential operators acting on Siegel modular forms and preserving automorphy under the restriction to diagonal blocks are important in many respects, including application to critical values of L functions. Such differential operators are associated with vectors of new special polynomials of several variables defined by certain harmonic conditions. They include the classical Gegenbauer polynomial as a prototype, and are interesting as themselves independently of Siegel modular forms. We will give formulas for all such polynomials in two different ways. One is to describe them using polynomials characterized by monomials in off-diagonal block variables. We will give an explicit and practical algorithm to give the vectors of polynomials through these. The other one is rather theoretical but seems much deeper. We construct an explicit generating series of polynomials mutually related under certain mixed Laplacians. Here substituting the variables of the polynomials to partial derivatives, we obtain the generic differential operator from which any other differential operators of this sort are obtained by certain projections. This process exhausts all the differential operators in question. This is also generic in the sense that for any number of variables and block partitions, it is given by a recursive unified expression. As an application, we prove that the Taylor coefficients of Siegel modular forms with respect to off-diagonal block variables, or of corresponding expansion of Jacobi forms, are essentially vector valued Siegel modular forms of lower degrees, which are obtained as images of the differential operators given above. We also show that the original forms are recovered by the images of our operators. This is an ultimate generalization of Eichler–Zagier’s results on Jacobi forms of degree one. Several more explicit results and practical construction are also given.

In this paper, we prove a minimal modularity lifting theorem for Galois representations (conjecturally) associated to Siegel modular forms of genus two which are holomorphic limits of discrete series at infinity., To appear in Duke Math

A second-quantized version of Mathieu moonshine leads to product formulae for functions that are potentially genus-two Siegel Modular Forms analogous to the Igusa Cusp Form. The modularity of these functions do not follow in an obvious manner. For some conjugacy classes, but not all, they match known modular forms. In this paper, we express the product formulae for all conjugacy classes of $M_{24}$ in terms of products of standard modular forms. This provides a new proof of their modularity., Comment: 35 pages; v2: Main proposition has been corrected leading to a single formula; referencing has been improved v3: Improved presentation (final version)

We formulate an explicit refinement of Böcherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of $L$-functions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan–Gross–Prasad conjecture for Bessel periods as proposed by Liu. We note several consequences of our conjecture to arithmetic and analytic properties of $L$-functions and Fourier coefficients of Siegel modular forms.

We extend the usual notion of Petersson inner product on the space of cuspidal Jacobi forms to include non-cuspidal forms as well. This is done by examining carefully the relation between certain "growth-killing" invariant differential operators on $\mathbf H_2$ and those on $\mathbf{H}_1 \times \mathbf{C}$ (here $\mathbf H_n$ denotes the Siegel upper half space of degree $n$). As applications, we can understand better the growth of Petersson norms of Fourier Jacobi coefficients of Klingen Eisenstein series, which in turn has applications to finer issues about representation numbers of quadratic forms, and as a by-product we also show that \textit{any} Siegel modular form of degree $2$ is determined by its `fundamental' Fourier coefficients., 41pp

Let Θ[j] be an analogue of the Ramanujan theta operator for Siegel modular forms. For a given prime p, we give the weights of elements of mod p kernel of Θ[j], where the mod p kernel of Θ[j] is the set of all Siegel modular forms F such that Θ[j](F) is congruent to zero modulo p. In order to construct examples of the mod p kernel of Θ[j] fromany Siegel modular form, we introduce new operators A(j)(M) and show the modularity of F|A(j)(M) when F is a Siegel modular form. Finally, we give some examples of the mod p kernel of Θ[j] and the filtrations of some of them.

We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a s l ( 2 ) ˆ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of s l ( 2 ) ˆ characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients of these vector-valued modular forms are related to multiplicities of roots appearing on the sum side of the Weyl-Kac-Borcherds denominator formulae.

In contrast to the well-known cases of large weights, Sturm’s operator does not realize the holomorphic projection operator for lower weights. We prove its failure for arbitrary Siegel modular forms of genus [Formula: see text] and scalar weight [Formula: see text]. This generalizes a result for genus two in [K. Maurischat and R. Weissauer, Phantom holomorphic projections arising from Sturm’s formula, preprint (2016)].

For positive integers g g and N N , let F N ( g ) \mathcal {F}_N^{(g)} be the field of meromorphic Siegel modular functions of genus g g and level N N whose Fourier coefficients belong to the N N th cyclotomic field. We construct explicit generators of F N ( g ) \mathcal {F}_N^{(g)} over F 1 ( g ) \mathcal {F}_1^{(g)} by making use of a quotient of theta constants, when g ≥ 2 g\geq 2 and N N is even.

It is well known that one can construct Siegel modular forms by theta series with pluriharmonic polynomials which realize the representation of that weight. But it seems we do not have enough refer...

In this paper we establish some algebraic properties of special L-values attached to Siegel modular forms of half-integral weight, often called metaplectic modular forms. These results are motivated by some “exercises” left by Shimura to the reader in his marvellous book “Arithmeticity in the Theory of Automorphic Forms”.

We give a Rankin–Selberg integral representation for the Spin (degree eight) $L$-function on $\operatorname{PGSp}_{6}$ that applies to the cuspidal automorphic representations associated to Siegel modular forms. If $\unicode[STIX]{x1D70B}$ corresponds to a level-one Siegel modular form $f$ of even weight, and if $f$ has a nonvanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin $L$-function $\unicode[STIX]{x1D6EC}(\unicode[STIX]{x1D70B},\text{Spin},s)$ of $\unicode[STIX]{x1D70B}$.

We study the image of the $\ell$-adic Galois representations associated to the four vector valued Siegel modular forms appearing in the work of Chenevier and Lannes. These representations are symplectic of dimension $4$. Following a method of Dieulefait, we determine the primes $\ell$ for which these representations are absolutely irreducible. In addition, we show that their image is "full" for all primes $\ell$ such that the associated residual representation is absolutely irreducible, except in two cases., Comment: in French, 18 pages, 6 tables

Let p be a prime number and \(f\in {S_{n}^{l}}({\Gamma }_{0}(N), \psi )\) 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 = [4ordp(α0(p))] + 1 explained in the Main Theorem with the following properties

The aim of this paper is to give another short proof of the Saito–Kurokawa lift based on a converse theorem of Imai as was already done by Duke and Imamoglu. In contrast to their proof we avoid spectral analysis but use a real analytic Eisenstein series in a suitable Rankin–Selberg integral involving Siegel’s theta series.