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We develop a theory of modular forms on the groups SO ( 3 , n + 1 ) , n ≥ 3 . This is very similar to, but simpler, than the notion of modular forms on quaternionic exceptional groups, which was initiated by Gross-Wallach and Gan-Gross-Savin. We prove the results analogous to those of earlier papers of the author on modular forms on exceptional groups, except now in the familiar setting of classical groups. Moreover, in the setting of SO ( 3 , n + 1 ) , there is a family of absolutely convergent Eisenstein series, which are modular forms. We prove that these Eisenstein series have algebraic Fourier coefficients, like the classical holomorphic Eisenstein series on SO ( 2 , n ) . As an application, using a local result of Savin, we prove that the so-called “next-to-minimal” modular form on quaternionic E 8 has rational Fourier expansion.

For all Eichler orders with a same squarefree level in a definite quaternion algebra over the field of rational numbers, we prove that a weighted sum of Jacobi theta series associated to these orders is a Jacobi Eisenstein series. Multiply the Fourier coefficients of the latter by a power of 2 to get our modified Hurwitz class numbers. As its corollaries, the modified Hurwitz class numbers can be used to calculate the traces of Brandt matrices and the type number of the Eichler orders.

We show that the projection of a product of two or three Eisenstein series of level one onto the cuspidal subspace is not an eigenform unless the dimension of the cuspidal subspace is one.

In recent years, Rogers and Zudilin developed a method to write $L$-values attached to elliptic curves as periods. In order to apply this method to a broader collection of $L$-values, we study Eisenstein series and determine their Fourier series at cusps. Subsequently, we write the $L$-values of an elliptic curve of conductor 32 as an integral of Eisenstein series and evaluate the value at $k>1$ explicitly as a period. As a side result, we give simple integral expressions for the generating functions of $L(E,k)$ when even (or odd) $k$ runs over positive integers., 25 pages

We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on [Formula: see text] in the case where N is prime and equal to the conductor of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O’Sullivan, Petridis, and Risager. In one case we find less cancelation in this sum than would be predicted by the common phenomenon of “square root cancelation”, while in another case we find more cancelation.

We prove the local Kudla–Rapoport conjecture, which is a precise identity between the arithmetic intersection numbers of special cycles on unitary Rapoport–Zink spaces and the derivatives of local representation densities of hermitian forms. As a first application, we prove the global Kudla–Rapoport conjecture, which relates the arithmetic intersection numbers of special cycles on unitary Shimura varieties and the central derivatives of the Fourier coefficients of incoherent Eisenstein series. Combining previous results of Liu and Garcia–Sankaran, we also prove cases of the arithmetic Siegel–Weil formula in any dimension.

We give a lower bound of the number of zeros of the modular forms E k + E l + − E k + l + for large enough k and l on each of the arc boundary and on the vertical boundary of the fundamental domain for the Fricke group Γ 0 + ( 2 ) , where E k + is the Eisenstein series of weight k for Γ 0 + ( 2 ) .

In this article, we first give a complete description of the algebra of integer weight modular forms on the congruence subgroup $$\Gamma _0(2)$$ modulo a prime $$p\ge 3$$ . This result parallels results of Swinnerton-Dyer in the $$SL_2(\mathbb {Z})$$ case, Katz on the subgroup $$\Gamma (N)$$ for $$N\ge 3$$ , Gross on the subgroup $$\Gamma _1(N)$$ for $$N\ge 4$$ and Tupan on modular forms of half-integral weight on $$\Gamma _1(4)$$ . Next, we use the theory of mod p modular forms on $$\Gamma _0(2)$$ to prove the non-existence of simple congruences for Fourier coefficients of quotients of certain integer weight Eisenstein series on $$\Gamma _0(2)$$ . The non-existence of simple congruences for coefficients of quotients of Eisenstein series on $$SL_2(\mathbb {Z})$$ has been shown by Dewar.

We prove a Voronoi–Oppenheim summation formula for divisor functions associated with Gaussian integers. This formula is a direct generalization of Oppenheim’s summation formula for classical divisor functions. To prove the formula we construct an Eisenstein series and study its properties. Our method of proof is similar to Beineke and Bump’s proof of the classical Oppenheim summation formula.

Based on the values of the Weierstrass elliptic function $$\wp (z|\tau )$$ at $$z=\pi \tau /2$$ , $$(\pi +\pi \tau )/{2}, (\pi +\pi \tau )/{4},(\pi +2\pi \tau )/{4}$$ and the theory of modular forms on the arithmetic group $$\Gamma _0(2)$$ , we decompose $$\theta _2^{2n}(\tau )$$ as sum of Eisenstein series and cusp forms. Using the recurrence relation of $$\wp ^{(2n)}(z|\tau )$$ , we provide an algorithm to determine the exact form of these cusp forms.

In this note, we prove a conjecture of Reitzes, Vulakh and Young on the location of the zeros of certain cuspforms of level one.

Based on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp $$\tau = 0$$ τ = 0 . As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane.

We prove that for a graded algebra $$\mathcal {A}$$ with a derivation $$D_{\mathcal {A}}$$ satisfying certain conditions, and a bi-graded algebra $$\mathcal {A}[q]$$ with an extended derivation D of $$D_{\mathcal {A}}$$ , there are only finitely many $$D_{\mathcal {A}}$$ - and D-invariant (or differential with respect to $$D_{\mathcal {A}}$$ and D) principal prime ideals of $$\mathcal {A}$$ and of $$\mathcal {A}[q]$$ , respectively. As its application, we prove the algebraic independence of the values of certain Eisenstein series for the arithmetic Hecke groups H(m) for $$m=4,6, \infty $$ , which is an extension of Nesterenko’s result for the Eisenstein series for SL $$_2(\mathbb {Z})$$ .

We construct ( n + 1 ) (n+1) -variable Hida families of Klingen Eisenstein series on Sp ⁡ ( 2 n + 2 ) \operatorname {Sp}(2n+2) for n n -variable Hida families on Sp ⁡ ( 2 n ) \operatorname {Sp}(2n) , and relate their images under the Siegel operator to p p -adic L L -functions. We also carry out some preliminary calculations of the non-degenerate Fourier coefficients of the constructed Klingen Eisenstein families.

In this paper, we investigate the non-modular solutions to the Schwarz differential equation $\{f,\tau \}=sE_4(\tau)$ where $E_4(\tau)$ is the weight 4 Eisenstein series and $s$ is a complex parameter. In particular, we provide explicit solutions for each $s=2\pi^2(n/6)^2$ with $n\equiv 1\mod 12$. These solutions are obtained as integrals of meromorphic weight 2 modular forms. As a consequence, we find explicit solutions to the differential equation $\displaystyle y''+\frac{\pi^2n^2}{36}\,E_4\,y=0$ for each $n\equiv 1\mod 12$ generalizing the work of Hurwitz and Klein on the case $n=1$. Our investigation relies on the theory of equivariant functions on the complex upper half-plane. This paper supplements a previous work where we determine all the parameters $s$ for which the above Schwarzian equation has a modular solution., Comment: 20 pages

In 1914, Srinivasa Ramanujan published several hypergeometric series for $$1/\pi $$ . One of these series was used by Bill Gosper in 1985 in a world-record-computation of $$\pi $$ . Shortly after this, the Chudnovskys found a faster series for $$1/\pi $$ based on the largest Heegner number and the Borweins proved Ramanujan’s series. Lately, the Chudnovskys’ series has often been used in practice to calculate digits of $$\pi $$ , it reads: $$\begin{aligned} \frac{\sqrt{640320^3}}{12 \pi } = \sum _{n=0}^\infty \frac{\left( 6n\right) !}{\left( 3n\right) !\left( n!\right) ^3}\,\frac{13591409+ 545140134 n}{\left( -640320^3\right) ^n}. \end{aligned}$$ In this paper, we calculate the coefficients in two of Ramanujan’s series for $$1/\pi $$ and those in the Chudnovskys’ series. For our calculation, we don’t require special software packages, but only the Fourier expansions of the Eisenstein series with a precision of $$\approx 20$$ decimals. We also prove the exactness of our calculations by proving that the values of certain non-holomorphic modular functions are algebraic integers. Our proof uses the division values of the Weierstras $$\wp $$ function.

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.

Let E k ( z ) be the normalized Eisenstein series of weight k for the modular group SL ( 2 , Z ) . We study the zeros of E k to prove that the equation ∏ i = 1 n E k i = ∏ j = 1 m E l j has no solutions, except for those given by known relationships between E 4 , E 6 , E 8 , E 10 , and E 14 . We go on to discuss some implications of this result.

By the theory of Eisenstein series, generating functions of various divisor functions arise as modular forms. It is natural to ask whether further divisor functions arise systematically in the theory of mock modular forms. We establish, using the method of Zagier and Zwegers on holomorphic projection, that this is indeed the case for certain (twisted) "small divisors" summatory functions $\sigma_{\psi}^{\mathrm{sm}}(n)$. More precisely, in terms of the weight 2 quasimodular Eisenstein series $E_2(\tau)$ and a generic Shimura theta function $\theta_{\psi}(\tau)$, we show that there is a constant $\alpha_{\psi}$ for which $$ \mathcal{E}^{+}_{\psi}(\tau):= \alpha_{\psi}\cdot\frac{E_2(\tau)}{\theta_{\psi}(\tau)}+ \frac{1}{\theta_{\psi}(\tau)} \sum_{n=1}^\infty \sigma^{\mathrm{sm}}_\psi(n)q^n $$ is a half integral weight (polar) mock modular form. These include generating functions for combinatorial objects such as the Andrews $spt$-function and the "consecutive parts" partition function. Finally, in analogy with Serre's result that the weight $2$ Eisenstein series is a $p$-adic modular form, we show that these forms possess canonical congruences with modular forms., Comment: 11 pages, v2: corrected small error; v3: minor edits in response to referee's report; v4: accepted version

In recent years, a number of papers have been devoted to the study of roots of period polynomials of modular forms. Here, we study cohomological analogues of the Eichler-Shimura period polynomials corresponding to higher $L$-derivatives. We state general conjectures about the locations of the roots of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative "period polynomials" in the case of cusp forms. We prove a special case of this conjecture in the case of Eisenstein series., 21 pages

In this paper we study congruences between Siegel Eisenstein series and Siegel cusp forms for Sp_4(Z)., Comment: 18 pages, 1 figure. Theorem 8.7 of [3] was replaced with Theorem B of[34]

We analyse Virasoro conformal blocks in the regime of heavy intermediate exchange $(h_p \rightarrow \infty)$. For the 1-point block on the torus and the 4-point block on the sphere, we show that each order in the large-$h_p$ expansion can be written in closed form as polynomials in the Eisenstein series. The appearance of this structure is explained using the fusion kernel and, more markedly, by invoking the modular anomaly equations via the 2d/4d correspondence. We observe that the existence of these constraints allows us to develop a faster algorithm to recursively construct the blocks in this regime. We then apply our results to find corrections to averaged heavy-heavy-light OPE coefficients., Comment: 35 pages, 2 figures

Hardy and Ramanujan introduced the Circle Method to study the Fourier expansion of certain meromorphic modular forms on the upper complex half-plane. These led to asymptotic results for the partition numbers and proven and unproven formulas for the coefficients of the reciprocals of Eisenstein series [Formula: see text], especially of weight 4. Berndt et al. finally proved them all. Recently, Bringmann and Kane generalized Petersson’s approach via Poincaré series, to handle the general case. We introduce a third approach. We attach recursively defined polynomials to reciprocals of Eisenstein series. This provides easy access to the signs of the Fourier coefficients of reciprocals of Eisenstein series, sheds some light on reciprocals of [Formula: see text] of general weight, and provides some upper and lower bounds for their growth.

We compute the diagonal restriction of the first derivative with respect to the weight of a p-adic family of Hilbert modular Eisenstein series attached to a general (odd) character of the narrow class group of a real quadratic field, and express the Fourier coefficients of its ordinary projection in terms of the values of a distinguished rigid analytic cocycle in the sense of Darmon and Vonk (Duke Math J, to appear, 2020) at appropriate real quadratic points of Drinfeld’s p-adic upper half-plane. This can be viewed as the p-adic counterpart of a seminal calculation of Gross and Zagier (J Reine Angew Math 355:191–220, 1985, §7) which arose in their “analytic proof” of the factorisation of differences of singular moduli, and whose inspiration can be traced to Siegel’s proof of the rationality of the values at negative integers of the Dedekind zeta function of a totally real field. Our main identity enriches the dictionary between the classical theory of complex multiplication and its extension to real quadratic fields based on RM values of rigid meromorphic cocycles, and leads to an expression for the p-adic logarithms of Gross–Stark units and Stark–Heegner points in terms of the first derivatives of certain twisted Rankin triple product p-adic L-functions.

This paper concerns the study of the Schwartz differential equation { h , τ } = s ⁢ E 4 ⁡ ( τ ) {\{h,\tau\}=s\operatorname{E}_{4}(\tau)} , where E 4 {\operatorname{E}_{4}} is the weight 4 Eisenstein series and s is a complex parameter. In particular, we determine all the values of s for which the solutions h are modular functions for a finite index subgroup of SL 2 ⁡ ( ℤ ) {\operatorname{SL}_{2}({\mathbb{Z}})} . We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of SL 2 ⁡ ( ℤ ) {\operatorname{SL}_{2}({\mathbb{Z}})} . This also leads to the solutions to the Fuchsian differential equation y ′′ + s ⁢ E 4 ⁡ y = 0 {y^{\prime\prime}+s\operatorname{E}_{4}y=0} .

We prove a functional equation for a vector valued real analytic Eisenstein series transforming with the Weil representation of $\operatorname{Sp}(n,\mathbb{Z})$ on $\mathbb{C}[(L^{\prime }/L)^{n}]$. By relating such an Eisenstein series with a real analytic Jacobi Eisenstein series of degree $n$, a functional equation for such an Eisenstein series is proved. Employing a doubling method for Jacobi forms of higher degree established by Arakawa, we transfer the aforementioned functional equation to a zeta function defined by the eigenvalues of a Jacobi eigenform. Finally, we obtain the analytic continuation and a functional equation of the standard $L$-function attached to a Jacobi eigenform, which was already proved by Murase, however in a different way.

The paper gives an identity for a sum of theta-series related to an imaginary quadratic field. This sum is expressed in terms of a certain Eisenstein series. The identity obtained is used in a new proof of the formula for the number of integral points in a system of ellipses. Such formulas are of interest because of their relations to the arithmetic Riemann–Roch theorem.

The purpose of this paper is to consider the computation of the Fourier coefficients of real-analytic Eisenstein series associated to general cusps of Hecke congruence subgroups of arbitrary level. Explicit formulas for some coefficients in the Fourier expansion of the Eisenstein series of weight zero for general cusps are given.

We combine Zagier's theory of renormalizable automorphic functions on the hyperbolic plane with the analytic continuation of representations of SL ( 2 , R ) due to Bernstein and Reznikov to study triple products of Eisenstein series of arbitrary (in particular, non-arithmetic) non-compact finite-volume hyperbolic surfaces.

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.

Boundary behavior of important functions has been an object of intensive research since the time of Riemann. Kurokawa, Kurokawa-Koyama, and Chapman studied the boundary behavior of generalized Eisenstein series which falls into this category. The underlying principle is the use of the Lipschitz summation formula. Our purpose is to show that it is a form of the functional equation for the Lipschitz–Lerch transcendent (and in the long run, it is equivalent to that for the Riemann zeta-function) and that this being indeed a boundary function of the Hurwitz–Lerch zeta-function, one can extract essential information. We also elucidate the relation between Ramanujan’s formula and automorphy of Eisenstein series.

In this paper, we introduce modular polynomials for the congruence subgroup $\Gamma_0(M)$ when $ X_0(M) $ has genus zero and therefore the polynomials are defined by a Hauptmodul of $ X_0(M) $. We show that the intersection number of two curves defined by two modular polynomials can be expressed as the sum of the numbers of $\mathrm{SL}_2(\mathbb{Z})$-equivalence classes of positive definite binary quadratic forms over $\mathbb{Z}$. We also show that the intersection numbers can be also combinatorially written by Fourier coefficients of the Siegel Eisenstein series of degree 2, weight 2 with respect to $\mathrm{Sp}_2(\mathbb{Z})$., Comment: 27 pages, 2 tables

In this paper, we obtain some new Eisenstein series identities containing the Borweins’ cubic theta functions using (p, k)-parameterization. Further, as an application of this, we evaluate the convolution sum i+4 j=n σ (i)σ (j

We prove two results on arithmetic quantum chaos for dihedral Maass forms, both of which are manifestations of Berry's random wave conjecture: Planck scale mass equidistribution and an asymptotic formula for the fourth moment. For level $1$ forms, these results were previously known for Eisenstein series and conditionally on the generalised Lindelof hypothesis for Hecke-Maass eigenforms. A key aspect of the proofs is bounds for certain mixed moments of $L$-functions that imply hybrid subconvexity., Comment: 67 pages

We establish transformation laws for generalized Dedekind sums associated to the Kronecker limit function of non-holomorphic Eisenstein series and their higher-order variants. These results apply to general Fuchsian groups of the first kind, and examples are provided in the cases of the Hecke triangle groups, the Hecke congruence groups $\Gamma_0(N)$, and the non-congruence arithmetic groups $\Gamma_0(N)^+$., Comment: 18 pages

We prove a conjecture of Matsusaka on the analytic continuationof hyperbolic Eisenstein series in weight $2$ on the full modular group $\mathrm{SL}_2(\mathbb{Z})$., Comment: 14 pages, no figures

The classical Kronecker limit formula describes the constant term in the Laurent expansion at the first order pole of the non-holomorphic Eisenstein series associated to the cusp at infinity of the modular group. Recently, the meromorphic continuation and Kronecker limit type formulas were investigated for non-holomorphic Eisenstein series associated to hyperbolic and elliptic elements of a Fuchsian group of the first kind by Jorgenson, Kramer and the first named author. In the present work, we realize averaged versions of all three types of Eisenstein series for Γ0(N) as regularized theta lifts of a single type of Poincaré series, due to Selberg. Using this realization and properties of the Poincaré series we derive the meromorphic continuation and Kronecker limit formulas for the above Eisenstein series. The corresponding Kronecker limit functions are then given by the logarithm of the absolute value of the Borcherds product associated to a special value of the underlying Poincaré series., Journal of Number Theory, 225, ISSN:0022-314X, ISSN:1096-1658

This is a draft version of an invited article for a forthcoming book `The genesis of Langlands Program', eds. Julia Mueller and Freydoon Shahidi, which will be published by Cambridge University Press. This is a survey of Langlands spectral decomposition theorem based on the meromorphic continuation of Eisenstein series and intertwining operators that show up in their functional equations.

There are many families of functions on partitions, such as the shifted symmetric functions, for which the corresponding q-brackets—certain normalized generating series—are quasimodular forms. This provides a tool for enumerative geometers to show that certain generating series of Gromov–Witten invariants or Hurwitz numbers are quasimodular forms. In this thesis, our aim is to study graded algebras of functions on partitions such that all homogeneous elements of the algebra have quasimodular forms as q-brackets. That is, we give explicit constructions answering the following three main questions in the affirmative: (I) Are there other graded algebras than the algebra of shifted symmetric functions such that the q-brackets of its elements are quasimodular forms? (II) Given a congruence subgroup, is there an (even larger) algebra of functions for which the q-bracket is a quasimodular form for this subgroup? (III) What is the class of functions for which the q-bracket is not only a quasimodular form, but even a modular form? The answer to the second and third question follows by studying the following question of independent interest: (IV) What is the modular or quasimodular behavior of the Taylor coefficients of meromorphic quasi-Jacobi forms? This question brings us back to the origin of the results on the q-bracket in enumerative geometry. Namely, we show that the solutions to a differential equation originating from the study of K3 surfaces are quasi-Jacobi forms and describe their transformation.

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.

The purpose of this paper is to establish two new theta function identities and as a consequence we shall offer beautiful concise elementary proofs to Huber, Schultz, and Ye’s Eisenstein series identities of Level 20, where the proofs were presented using the theory of modular forms.

Poincare and Eisenstein series are building blocks for every type of automorphic forms. We define Poincare series for Jacobi forms of lattice index and show that they reproduce Fourier coefficients of cusp forms under the Petersson scalar product. We compute the Fourier expansions of Poincare and Eisenstein series and give an explicit formula for the Fourier coefficients of the trivial Eisenstein series in terms of values of Dirichlet L-functions at negative integers. For even weight and fixed index, finite linear combinations of Fourier coefficients of non-trivial Eisenstein series are equal to finite linear combinations of Fourier coefficients of the trivial one. This result is used to obtain formulas for the Fourier coefficients of Eisenstein series associated with isotropic elements of small order.

We define a generalized hyperbolic Eisenstein series for a pair of a hyperbolic manifold of finite volume and its submanifold. We prove the convergence, the differential equation and the precise spectral expansion associated to the Laplace–Beltrami operator. We also derive the analytic continuation with the location of the possible poles and their residues from the spectral expansion.

Eisenstein derived addition formulas for the Weierstrass zeta function from the addition formula for the cotangent function and the fact that the Weierstrass zeta function can be represented as an infinite sum of the cotangent functions. In this paper, we apply this idea of Eisenstein's to the addition type formula for the double cotangent function, established by the author. We show that the elliptic digamma function, defined by the logarithmic derivative of the elliptic gamma function, satisfies an addition type formula. This formula includes the addition formula for the Weierstrass zeta function, evaluation formulas for the double Eisenstein series introduced by Tsumura and the double shuffle relations for the double Eisenstein series, proved by Gangl-Kaneko-Zagier.