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A partition is said to be ℓ-regular if none of its parts is a multiple of ℓ. Let b5′(n) denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of n. This function has also close connections to representation theory and combinatorics. In this paper, we study arithmetic properties of b5′(n). We provide full characterization of the parity of b5′(2n + 1), present several congruences modulo 4, and prove that the generating function of the sequence (b5′(5n + 1)) is lacunary modulo any arbitrary positive powers of 5. [ABSTRACT FROM AUTHOR]

We count the number of critical points of a modular form with real Fourier coefficients in a γ-translate of the standard fundamental domain ℱ (with γ ∈SL2(ℤ)). Whereas by the valence formula the (weighted) number of zeros of this modular form in γℱ is a constant only depending on its weight, we give a closed formula for this number of critical points in terms of those zeros of the modular form lying on the boundary of ℱ, the value of γ−1(∞) and the weight. More generally, we indicate what can be said about the number of zeros of a quasimodular form. [ABSTRACT FROM AUTHOR]

In this paper, we study the analytic ranks of the Hasse–Weil L-function. We consider the products for the completed Hasse–Weil L-function in the framework of the theory of modular forms. We propose the new conjectures for the deformed Fourier sine and cosine integral formulas related to the completed Hasse–Weil L-functions and Mittag-Leffler function. We show the entire-function solutions for the fractional diffusion equations in mathematical physics. We also discuss the trivial zeros of the Hasse–Weil L-function. [ABSTRACT FROM AUTHOR]

In this paper, we construct generalized L-functions associated to meromorphic modular forms of weight 1 2 for the theta group with a single simple pole in the fundamental domain. We then consider their behavior toward i∞ and relate this to the Riemann zeta function. [ABSTRACT FROM AUTHOR]

Let 풮k(Γ0(N),χ) denote the space of holomorphic cuspforms with Dirichlet character χ and modular subgroup Γ0(N). We will characterize the space of newforms 풮knew(Γ 0(N),χ) as the intersection of eigenspaces of a particular family of Hecke operators, generalizing previous work of Baruch–Purkait to forms with nontrivial character. We achieve this by obtaining representation theoretic results in the p-adic case which we then de-adelize into relations of classical Hecke operators. [ABSTRACT FROM AUTHOR]

We investigate the subspace of the homology of a congruence subgroup Γ of SL 3 (ℤ) with coefficients in the Steinberg module St (ℚ 3) which is spanned by certain modular symbols formed using the units of a totally real cubic field E. By Borel–Serre duality, H 0 (Γ , St (ℚ 3)) is isomorphic to H 3 (Γ , ℚ). The Borel–Serre duals of the modular symbols in question necessarily lie in the cuspidal cohomology H cusp 3 (Γ , ℚ). Their span is a naturally defined subspace C (Γ , E) of H cusp 3 (Γ , ℚ). Using a computer, we study where C (Γ , E) sits between 0 and H cusp 3 (Γ , ℚ). On the basis of our computations, we conjecture that ∑ E C (Γ , E) = H cusp 3 (Γ , ℚ) , and we raise the question as to whether for each E individually it might always be true that C (Γ , E) = H cusp 3 (Γ , ℚ). [ABSTRACT FROM AUTHOR]

Serre showed that for any integer m , a (n) ≡ 0 (mod m) for almost all n , where a (n) is the n th Fourier coefficient of any modular form with rational coefficients. In this paper, we consider a certain class of cuspforms and study # { a (n) (mod m) } n ≤ x over the set of integers with O (1) many prime factors. Moreover, we show that any residue class a ∈ ℤ / m ℤ can be written as the sum of at most 13 Fourier coefficients, which are polynomially bounded as a function of m. [ABSTRACT FROM AUTHOR]

We prove the best possible upper bounds of the gaps between non-vanishing Fourier coefficients of half-integral weight cuspforms. This improves the works of Balog–Ono and Thorner. We also show an asymptotic formula of central modular L -values for short intervals. [ABSTRACT FROM AUTHOR]

Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad. Sci. Paris 287 (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus g > 1 must be integral or half integral. Actually he proved that for a system v (M) of complex numbers of absolute value 1 v (M) det (C Z + D) r (r ∈ ℝ) (0. 1) can be an automorphy factor only if 2 r is integral. We give a different proof for this. It uses Mennicke's result [Zur Theorie der Siegelschen Modulgruppe, Math. Ann. 159 (1965) 115–129] that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from [Solution of the congruence subgroup problem for SL n (n ≥ 3) and Sp 2 n (n ≥ 2) , Publ. Math. Inst. Hautes Études Sci. 33 (1967) 59–137] of Bass–Milnor–Serre. [ABSTRACT FROM AUTHOR]

In this paper, we use the theory of modular forms and give a general method to obtain the convolution sums W d 1 , d 2 , d 3 r 1 , r 2 , r 3 (n) = ∑ l 1 , l 2 , l 3 ∈ ℕ d 1 l 1 + d 2 l 2 + d 3 l 3 = n σ r 1 (l 1) σ r 2 (l 2) σ r 3 (l 3) , for odd integers r 1 , r 2 , r 3 ≥ 1 , and d 1 , d 2 , d 3 , n ∈ ℕ , where σ r (n) is the sum of the r th powers of the positive divisors of n. We consider four cases, namely (i) r 1 = r 2 = r 3 = 1 , (ii) r 1 = r 2 = 1 ; r 3 ≥ 3 (iii) r 1 = 1 ; r 2 , r 3 ≥ 3 and (iv) r 1 , r 2 , r 3 ≥ 3 , and give explicit expressions for the respective convolution sums. We provide several examples of these convolution sums in each case and further use these formulas to obtain explicit formulas for the number of representations of a positive integer n by certain positive definite quadratic forms. The existing formulas for W 1 , 1 , 1 (n) (in [20]), W 1 , 1 , 2 (n) , W 1 , 2 , 2 (n) , W 1 , 2 , 4 (n) (in [7]), W 1 , 1 , 1 1 , 3 , 3 (n) , W 1 , 1 , 3 1 , 3 , 3 (n) , W 1 , 3 , 3 1 , 3 , 3 (n) , W 3 , 1 , 1 1 , 3 , 3 (n) , W 3 , 3 , 1 1 , 3 , 3 (n) (in [35]), W d 1 , d 2 , d 3 (n) , lcm (d 1 , d 2 , d 3) ≤ 6 (in [30]) and lcm (d 1 , d 2 , d 3) = 7 , 8 , 9 (in [31]), which were all obtained by using the theory of quasimodular forms, follow from our method. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to determine all normalized extremal quasimodular forms of depth 1 whose Fourier coefficients are integers. By changing the local parameter at infinity from q = e 2 π i τ to the reciprocal of the elliptic modular j -function, we prove that all normalized extremal quasimodular forms of depth 1 have a hypergeometric series expression and that integrality is not affected by this change of parameters. Furthermore, by transforming these hypergeometric series expressions into a certain manageable form related to the Atkin(-like) polynomials and using the lemmas that appeared in the study of p -adic hypergeometric series by Dwork and Zudilin, the integrality problem can be reduced to the fact that a polynomial vanishes modulo a prime power, which we prove. We also prove that all extremal quasimodular forms of depth 1 with appropriate weight-dependent leading coefficients have integral Fourier coefficients by focusing on the hypergeometric expression of them. [ABSTRACT FROM AUTHOR]

In this paper, we construct a holomorphic Siegel modular form of weight 2 and level 2, and compute its Fourier coefficients explicitly. Moreover, we prove that this modular form equals the generating function of the representative number ρ (n , m , r) associated with the maximal order in the quaternion algebra (− 1 , − 1) ℚ . As a corollary, we can give a new proof of the famous formula for the sums of three squares. As applications, we give an explicit formula for the numbers of solutions of two systems of Diophantine equations related with Sun's "1-3-5 conjecture". Furthermore, we show that "a perfect square" in the integral condition version of Sun's conjecture can be replaced by "a power of 4". [ABSTRACT FROM AUTHOR]

Getz presented a family of level one modular forms f k for which all zeros lie on the unit circle in the fundamental domain. Expanding on work from Nozaki, Griffin et al., and Saha and Saradha, we show that the non-elliptic zeros of these f k satisfy two interlacing properties: standard interlacing, where the zeros of f k and f k + a alternate if and only if a ∈ { 2 , 4 , 6 , 8 , 1 2 } for sufficiently large k ; and Stieltjes interlacing, where for ℓ > k large enough, between any two zeros of f k , there is a zero of f ℓ . [ABSTRACT FROM AUTHOR]

We prove that a nonzero Jacobi form of level N (an odd integer) and square-free index m 1 m 2 with m 1 | N and (N , m 2) = 1 has a nonzero theta component h μ with either (μ , 2 m 1 m 2) = 1 or (μ , 2 m 1 m 2) ∤ 2 m 2 . As an application, we prove that a nonzero Siegel cusp form F of degree 2 and an odd level N in the Atkin–Lehner type newspace is determined by fundamental Fourier coefficients up to a divisor of N. [ABSTRACT FROM AUTHOR]

The anomaly cancellation in superstring theory is known to hold at leading order in the curvature for the gauge groups S O (3 2) and E 8 × E 8 . The coefficients of the next-to-leading order terms may be evaluated, and a mechanism for cancellation is described, which would remain valid at higher genus when there exists a global splitting of the coordinates of supermoduli space. Since the spin- 1 2 fields transform under the adjoint representation in these models, compactification of the heterotic string over G 2 / S U (3) or a fundamental 12-dimensional theory, from which superstrings are produced through an elliptic fibration, over G 2 × S U (2) × U (1) S U (3) × U (1) ′ × U (1) ′ ′ , provides another phenomenologically viable theory at lower energies compatible with the standard description of the elementary particle interactions. The 96 spin- 1 2 fields now would transform under the fundamental representation of G 2 × S U (2) × U (1) and the spin-one gauge fields would belong to the adjoint representation. The sum of the anomaly polynomials for the particle content of the G 2 × S U (2) × U (1) model vanishes at n 1 2 ≃ 1 0 3. 3 4 2 7 1 9 2 4. The contributions to the gravitational anomaly from the particles and antiparticles cancel by the CPT theorem and the duality transformations of polynomials of degree 6 in the curvature and the field strength. The existence of the interaction of a spin-2 charge, which is conserved only over a finite time interval, can be traced to nonlocal terms in the reduction of the string field theory to the gravitational sector. The source of the global gravitational anomaly cancellation in the modular form equations derived from an elliptic fibration of a 12-dimensional theory would restrict the compactifications and provide a method for preserving the absence of anomalies in four dimensions. [ABSTRACT FROM AUTHOR]

Well-known results of Lagrange and Jacobi prove that every m ∈ ℕ can be expressed as a sum of four integer squares, and the number r (m) of such representations can be given by an explicit formula in m. In this paper, we prove that the only real quadratic number field for which the sum of four squares is universal is ℚ (5). We provide explicit formulas for r (m) for K = ℚ (2) and K = ℚ (5). We then consider the theta series of the sum of four squares over any real quadratic number field, providing explicit upper and lower bounds for the Eisenstein coefficients. Last, we include examples of the complete theta series decomposition of the sum of four squares over ℚ (3) , ℚ (1 3) and ℚ (1 7). [ABSTRACT FROM AUTHOR]

On pp. 243 and 244 of his second notebook, Ramanujan recorded seven modular equations of degree 11 followed by two more which he had crossed out. The first modular equation in the list was proved by Berndt using theta function identities. The same was also proved by Venkatachaliengar in a different way. To the best of our knowledge, the only available proofs to the other six modular equations in the list are by Berndt. These proofs employ the theory of modular forms. Interestingly these are the first set of modular equations, the proofs to which by Berndt take a shift from the elementary algebraic methods to the theory of modular forms. In this paper, we reprove all these modular equations of degree 11 (except the first one in the list) by elementary algebraic techniques. First, we use two series identities due to Ye and Liu to prove one of the modular equations and its reciprocal. The remaining modular equations are proved by the method of parametrization. [ABSTRACT FROM AUTHOR]

We prove Rogers–Ramanujan-type identities for the Nahm sums associated with the tadpole Cartan matrix of rank 3. These identities reveal the modularity of these sums, and thereby we confirm a conjecture of Calinescu, Penn and the first author in this case. We show that these Nahm sums together with some shifted sums can be combined into a vector-valued modular function on the full modular group. We also present some conjectures for a general rank. [ABSTRACT FROM AUTHOR]

For a paramodular group of any degree and square free level we study the Hecke algebra and the boundary components. We define paramodular theta series and show that for square free level and large enough weight they generate the space of cusp forms (basis problem), using the doubling or pullback of Eisenstein series method. For this we give a new geometric proof of Garrett's double coset decomposition which works in our more general situation. [ABSTRACT FROM AUTHOR]

Let f be a Drinfeld modular form of level Γ 0 () which is an eigenform for the Hecke operator T ( a prime of q [ T ]). We study the relations between the Fourier coefficients of f and the -adic valuation of its eigenvalue (slope). We use formulas for some of the Fourier coefficients of T f to provide bounds and estimates on the slopes and, in particular, to find necessary conditions for "large" slopes, whose existence is closely connected with conjectures on oldforms and newforms. [ABSTRACT FROM AUTHOR]

We prove two unconditional results on Fourier coefficients of modular forms: Let f be a Hecke eigen cusp form of weight k and level N (N square free), and λ f (n) be the normalized Hecke eigenvalues. Let n f be the least prime p such that λ f (p) < 0. Then we show that in a family of modular forms of weight k and level N , except for a density zero set, n f ≪ (log k 2 N) α for some α > 0. Second, we show that outside a density zero set, a modular form is determined by Fourier coefficients up to O ((log k 2 N) α) for some α > 0. [ABSTRACT FROM AUTHOR]

Following recent studies of vanishing coefficients in arithmetic progressions in infinite q -series products, we further prove that such phenomenon also exists in a family of quotients of theta functions with odd moduli. Moreover, we find that some arithmetic progressions with vanishing coefficients could be shared by an infinite family of quotients of theta functions. Finally, we present several related conjectures that merit further investigation. [ABSTRACT FROM AUTHOR]

We describe the construction and properties of a singular theta lift for the orthogonal group SO (2 , 1). We obtain locally harmonic Maass forms in the sense of Bringmann, Kane and Kohnen with singular sets along geodesics in the upper half plane. We consider these forms as currents and derive properties of the Shimura–Shintani correspondence. This work provides extensions of the theta lifts considered by Hövel and Bringmann, Kane and Viazovska. [ABSTRACT FROM AUTHOR]

In this paper, we have realized the left–right symmetric model (LRSM) with modular symmetry. Most of the previous works on LRSM have been done considering discrete flavor symmetry, but this work has been carried out using Γ (3) modular group which is isomorphic to nonabelian discrete symmetry group A 4 . The advantage of using modular symmetry is the nonrequirement for the use of extra particles called "flavons". In this model, the Yukawa couplings are expressed in terms of modular forms (Y 1 , Y 2 , Y 3). In this work, we have studied minimal LRSM for both type-I and type-II dominances. Here, we have calculated the values for the Yukawa couplings and then plotted it against the sum of the neutrino masses. The results obtained are well within the experimental limits for the desired values of sum of neutrino masses. We have also briefly analyzed the effects of the implications of modular symmetry on neutrinoless double beta decay with the new physics contributions and the correlation of lepton flavor violation and lightest neutrino mass within the framework of modular symmetric LRSM. [ABSTRACT FROM AUTHOR]

We obtain an explicit zero-free region for a weighted sum of L -functions over the orthogonal basis of Hecke eigen cusp forms of a large integral weight for the full modular group. We also estimate the number of such forms whose L value does not vanish at a given point inside this zero-free region. [ABSTRACT FROM AUTHOR]

The goal of this note is to provide a general lower bound on the number of even values of the Fourier coefficients of an arbitrary eta-quotient F , over any arithmetic progression. Namely, if g a , b (x) denotes the number of even coefficients of F in degrees n ≡ b (mod a) such that n ≤ x , then we show that g a , b (x) / x is unbounded for x large. Note that our result is very close to the best bound currently known even in the special case of the partition function p (n) (namely, x log log x , proven by Bellaïche and Nicolas in 2016). Our argument substantially relies upon, and generalizes, Serre's classical theorem on the number of even values of p (n) , combined with a recent modular-form result by Cotron et al. on the lacunarity modulo 2 of certain eta-quotients. Interestingly, even in the case of p (n) first shown by Serre, no elementary proof is known of this bound. At the end, we propose an elegant problem on quadratic representations, whose solution would finally yield a modular form-free proof of Serre's theorem. [ABSTRACT FROM AUTHOR]

We provide comparison results for anticyclotomic p -adic L -functions attached to Hida families of modular forms. The main result is a comparison between the anticyclotomic restriction of the three variable p -adic L -function introduced by Skinner and Urban, and the anticyclotomic L -function constructed by means of p -adic families of Gross points in the setting of definite quaternion algebras. Several auxiliary results involving anticyclotomic p -adic L -functions introduced by Chida-Hsieh and Büyükboduk–Lei are also proved. [ABSTRACT FROM AUTHOR]

Given two eigenforms, it is a natural question to ask if the product of the eigenforms is again an eigenform. In the case of elliptic modular forms, this was answered in the full level case by Duke and Ghate and in the general level case by Johnson. In this paper, we consider the case of genus two Siegel modular forms with general congruence level. [ABSTRACT FROM AUTHOR]

False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular completions of indefinite theta functions of any signature and thereby develop a structure parallel to the recently developed theory of higher depth mock modular forms. We then demonstrate this theoretical base on a number of examples up to depth three coming from characters of modules for the vertex algebra W 0 (p) A n , 1 ≤ n ≤ 3 , and from Ẑ -invariants of three-manifolds associated with gauge group SU(3). [ABSTRACT FROM AUTHOR]

For | q | < 1 , define f i = ∏ n = 1 ∞ (1 − q i n) , and let (A (q) , B (q)) be any of the pairs (f 1 4 , f 1 8 f 2 2 ) , (f 1 4 , f 1 1 0 f 3 2 ) , (f 1 6 , f 2 4 f 1 2 ) , (f 1 6 , f 1 1 4 f 2 4 ) , (f 1 1 0 , f 2 6 f 1 2 ) , (f 1 1 4 , f 3 5 f 1 ) , (f 1 1 4 , f 2 8 f 1 2 ). For any such pair (A (q) , B (q)) , define the sequences { a (n) } and { b (n) } to be the coefficients of q n of A (q) and B (q) , respectively. Then for each pair it is shown that a (n) vanishes if and only if b (n) vanishes. In each case, a criterion is given which states precisely when a (n) = b (n) = 0. Moreover, for the pairs (f 1 2 6 , f 3 9 f 1 ) , (f 1 2 6 , f 2 1 6 f 1 6 ) it is shown that a (n) = b (n) = 0 if 1 2 n + 1 3 satisfies a criteria of Serre for a (n) = 0. [ABSTRACT FROM AUTHOR]

We prove that Rankin–Cohen brackets of Eisenstein series generate the whole space of cuspforms in several cases. This generalizes the classical result that products of two Eisenstein series generate the whole space of modular forms. [ABSTRACT FROM AUTHOR]

We express recent double-sums studied by Wang, Yee, and Liu in terms of two types of Hecke-type double-sum building blocks. When possible we determine the (mock) modularity. We also express a recent q -hypergeometric function of Andrews as a mixed mock modular form. [ABSTRACT FROM AUTHOR]

The study of Andrews–Beck type congruences for partitions has its origin in the work by Andrews, who proved two congruences on the total number of parts in the partitions of n with the Dyson rank, conjectured by George Beck. Recently, Lin, Peng and Toh proved many Andrews–Beck type congruences for k -colored partitions. Moreover, they posed eight conjectural congruences. In this paper, we confirm two congruences modulo 1 1 by utilizing some q -series techniques and the theory of modular forms. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show how the modular method together with different techniques can be used to prove non-existence of primitive non-trivial solutions of the equation x 2 + d y 6 = z p for square-free values 1 ≤ d ≤ 2 0. The key ingredients are: the approach presented in [A. Pacetti and L. V. Torcomian, ℚ -curves, Hecke characters and some Diophantine equations, Math. Comp. 91(338) (2022) 2817–2865] (in particular its recipe for the space of modular forms to be computed) together with the use of the symplectic method (as developed in [E. Halberstadt and A. Kraus, Courbes de Fermat: Résultats et problèmes, J. Reine Angew. Math. 548 (2002) 167–234], although we give a variant over ramified extensions needed in our applications) to discard solutions and the use of a second Frey curve, aiming to prove large image of residual Galois representations. [ABSTRACT FROM AUTHOR]

Andrews and Newman introduced the minimal excludant or " mex " function for an integer partition π of a positive integer n , mex (π) , as the smallest positive integer that is not a part of π. They defined σ mex (n) to be the sum of mex (π) taken over all partitions π of n. We prove infinite families of congruence and multiplicative formulas for σ mex (n). By restricting to the part of π , Andrews and Newman also introduced moex (π) to be the smallest odd integer that is not a part of π and σ moex (n) to be the sum of moex (π) taken over all partitions π of n. In this paper, we show that for any sufficiently large X , the number of all positive integer n ≤ X such that σ moex (n) is an even (or odd) number is at least (log log X). [ABSTRACT FROM AUTHOR]

This paper aims to initiate a systematic investigation of the distribution of L ′ L (1 , f , χ) , where L (s , f , χ) is the L -function attached to a normalized Hecke eigenform f of weight k for Γ 0 (N) and χ mod M is a primitive character. Assuming the Riemann hypothesis for L (s , f , χ) and ζ (s) , we prove an upper bound of L ′ L (1 , f , χ) in terms of N , M and k. This result is analogous to that of Ihara, Kumar Murty and Shimura. Next we show that the upper bound is not far from being optimal by proving an unconditional omega result which is analogous to a result of Mourtada and Kumar Murty. In course of proving the omega result, we prove a zero-density estimate for the L -functions involved. [ABSTRACT FROM AUTHOR]

In this paper, we study holomorphic ℤ -graded vertex superalgebras. We prove that all such vertex superalgebras of central charge 8 and 1 6 are purely even. For the case of central charge 2 4 we prove that the weight-one Lie superalgebra is either zero, of superdimension 2 4 , or else is one of an explicit list of 1332 semisimple Lie superalgebras. [ABSTRACT FROM AUTHOR]

This note is devoted to the study of families of quaternionic modular forms arising from orders defined by Pizer. In this situation, the Hecke eigenspaces are 2-dimensional contrary to the classical case of Eichler orders. The main result is a Control Theorem in the spirit of Hida, interpolating these 2-dimensional Hecke eigenspaces. We restrict our attention to a definite rational quaternion algebra ramified at a single odd prime ℓ. [ABSTRACT FROM AUTHOR]

We study the kernel of the Shimura lifting from modular forms of half-integral weight to integral weight. [ABSTRACT FROM AUTHOR]

In this paper, we construct a model with the help of modular symmetry in the framework of minimal inverse seesaw [ISS(2,3)]. We have used Γ (3) modular group which is isomorphic to non-Abelian discrete symmetry group A 4 . In this group, there are three Yukawa modular forms of weight 2. In this model, we study neutrino masses and mixings for both normal and inverted hierarchies. Use of modular symmetry reduces the need for more number of extra flavons and their specific VEV alignments, as such, minimality of the model is maintained to a great extent. Along with A 4 symmetry group, we have used Z 3 to restrict certain interaction terms in the Lagrangian. Further, we calculate the effective mass to address the phenomena of neutrinoless double-beta decay (0 ν β β). The values of effective mass are found to lie within the bound ( m eff < 0. 1 6 5 eV) as predicted by different 0 ν β β experiments. [ABSTRACT FROM AUTHOR]

In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form Δ should have an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function. This conjecture was proven by Hafner and Stopple. In 2017 and 2019, Chakraborty et al. established an asymptotic relation between Lambert series associated to any primitive cusp form (for full modular group, congruence subgroup and in Maass form case) and the nontrivial zeros of the Riemann zeta function. In this paper, we study Lambert series associated with primitive Hilbert modular form and establish similar kind of asymptotic expansion. [ABSTRACT FROM AUTHOR]

Let f ∈ k (S L 2 (ℤ)) be a normalized cuspidal Hecke eigenform. We give explicit formulas for weighted averages of the rightmost critical values of triple product L -functions L (s , f ⊗ g ⊗ h) , where g and h run over an orthogonal basis of k (S L 2 (ℤ)) consisting of normalized cuspidal Hecke eigenforms. Those explicit formulas provide us an arithmetic expression of the rightmost critical value of the individual triple product L -functions. [ABSTRACT FROM AUTHOR]

In this paper, we give an upper bound of the orders of Jacobi forms. For each fixed weight k , the growth rate of our upper bound divided by the index m as m goes to infinity is zero. Applying this result to Siegel paramodular forms, we know that the space of all formal series of Jacobi forms is finite dimensional. [ABSTRACT FROM AUTHOR]

Let ρ ¯ f , λ be the residual Galois representation attached to a newform f and a prime ideal λ in the integer ring of its coefficient field. In this paper, we prove explicit bounds for the residue characteristic of the prime ideals λ such that ρ ¯ f , λ is exceptional, that is reducible, of projective dihedral image, or of projective image isomorphic to 4 , 4 or 5 . We also develop explicit criteria to check the reducibility of ρ ¯ f , λ , leading to an algorithm that computes the exact set of such λ 's. We have implemented this algorithm in PARI/GP. Along the way, we construct lifts of Katz' operator in characteristic zero, and we prove a new Sturm bound theorem. [ABSTRACT FROM AUTHOR]

Let ψ be a function such that ψ (x) → ∞ as x → ∞. Let λ f (n) be the n th Hecke eigenvalue of a fixed holomorphic cusp form f for S L (2 , ℤ). We show that for any real-valued function h (x) such that (log X) 2 − 2 α ≪ h (X) = o (X) , mean values of | λ f (n) | over intervals [ x , x + h (X) ] are bounded by ψ (X) (log X) α − 1 for all but O f (X ψ (X) − 2) many integers x ∈ [ X , 2 X − h (X) ] , in which α is the average value of | λ f (p) | over primes. We generalize this for | λ f (n) | 2 k for k ∈ ℤ + . [ABSTRACT FROM AUTHOR]

In this work, we define a new type of Eisenstein-like series by using Pell-Lucas numbers and call them the Pell-Lucas–Eisenstein Series. First, we show that the Pell-Lucas–Eisenstein series are convergent on their domain. Afterwards we prove that they satisfy some certain functional equations. Proofs follows from some on calculations on Pell-Lucas numbers. [ABSTRACT FROM AUTHOR]

Let f be a newform of even weight 2 κ for D × , where D is a possibly split indefinite quaternion algebra over ℚ. Let K be a quadratic imaginary field splitting D and p an odd prime split in K. We extend our theory of p -adic measures attached to the power series expansions of f around the Galois orbit of the CM point corresponding to an embedding K ↪ D to forms with any nebentypus and to p dividing the level of f. For the latter we restrict our considerations to CM points corresponding to test objects endowed with an arithmetic p -level structure. Also, we restrict these p -adic measures to ℤ p × and compute the corresponding Euler factor in the formula for the p -adic interpolation of the "square roots" of the Rankin–Selberg special values L (π K ⊗ ξ r , 1 2) , where π K is the base change to K of the automorphic representation of GL 2 associated, up to Jacquet-Langland correspondence, to f and ξ r is a compatible family of grössencharacters of K with infinite type ξ r , ∞ (z) = (z / z ̄) κ + r . [ABSTRACT FROM AUTHOR]

In this paper, we give linear relations between the Fourier coefficients of a special Hilbert modular form of half integral weight and some arithmetic functions. As a result, we have linear relations for the special L -values over certain totally real number fields. [ABSTRACT FROM AUTHOR]

Let F and G be Siegel cusp forms for the group Sp 4 (ℤ) with weights k 1 , k 2 , respectively. Suppose that neither F nor G is a Saito–Kurokawa lift. Further suppose that F and G are Hecke eigenforms lying in distinct eigenspaces. In this paper, we investigate simultaneous arithmetic behavior and related problems of Hecke eigenvalues of these Hecke eigenforms, some of which improve upon results of Gun et al. [ABSTRACT FROM AUTHOR]

Mock theta functions were introduced by Ramanujan in 1920 but a proper understanding of mock modularity has emerged only recently with the work of Zwegers in 2002. In these lectures, we describe three manifestations of this apparently exotic mathematics in three important physical contexts of holography, topology and duality where mock modularity has come to play an important role. [ABSTRACT FROM AUTHOR]