Search

We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of "moduli-friendly" Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic points of modular curves and dispenses us of the need for equations of modular curves and for q-expansion computations in the construction of models of modular Jacobians. The resulting algorithm compares very favourably to our complex-analytic method. [ABSTRACT FROM AUTHOR]

For each d ∈ N , we establish an infinite family of weight 1/2 quantum modular forms from the overpartition M d -rank generating function. Infinite quantum families from both the Dyson rank overpartition generating function and the overpartition M 2 -rank generating function appear as special cases of our work. As a corollary, we obtain explicit closed expressions which may be used to evaluate Eichler integrals of certain weight 3/2 theta functions. [ABSTRACT FROM AUTHOR]

In this study, the development of a multi-layer marking toolkit was investigated to improve construction quality and mitigate the problem of irregular designs in the layout-printing work performed at construction sites. The quality of conventional layout-printing work is dependent on the skill of the worker, and construction quality can suffer owing to inconsistencies in drawings resulting from human error. In this study, these problems were analyzed, and a construction-site-layout-marking toolkit apparatus and mechanical unit, with a structure that allowed for multi-layer installation for automated implementation at construction sites, were developed. The marking toolkit and mechanical unit with the multi-layer structure were developed in a modular form so that each module can operate independently. Furthermore, each module was developed in manual mode to improve the system by acquiring information on the movement of the marking toolkit and multi-layer structure. Additionally, data on the layout-printing method was developed by connecting the system via Ethernet and operating a wireless joystick. Finally, experiments were performed on a road surface covered with B4 paper and concrete panels to confirm the operational feasibility of the system, which was developed to operate manually. [ABSTRACT FROM AUTHOR]

We consider a natural basis for the space of weakly holomorphic modular forms for Γ0+(3). We prove that for some of the basis elements, if z0 in the fundamental domain for Γ0+(3) is one of zeroes of the elements, then either z0 is transcendental or is in .... [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]

We show that the results we had previously obtained on diagonals of 9- and 10-parameter families of rational functions in three variables x , y , and z , using creative telescoping, yielding modular forms expressed as pullbacked 2 F 1 hypergeometric functions, can be obtained much more efficiently by calculating the j -invariant of an elliptic curve canonically associated with the denominator of the rational functions. These results can be drastically generalized by changing the parameters into arbitrary rational functions of the product p = x y z . In other cases where creative telescoping yields pullbacked 2 F 1 hypergeometric functions, we extend this algebraic geometry approach to other families of rational functions in three or more variables. In particular, we generalize this approach to rational functions in more than three variables when the denominator can be associated to an algebraic variety corresponding to products of elliptic curves, or foliations in elliptic curves. We also extend this approach to rational functions in three variables when the denominator is associated with a genus-two curve such that its Jacobian is a split Jacobian, corresponding to the product of two elliptic curves. We sketch the situation where the denominator of the rational function is associated with algebraic varieties that are not of the general type, having an infinite set of birational automorphisms. We finally provide some examples of rational functions in more than three variables, where the telescopers have pullbacked 2 F 1 hypergeometric solutions, because the denominator corresponds to an algebraic variety that has a selected elliptic curve. [ABSTRACT FROM AUTHOR]

In this paper, we address the case of a particular class of function referred to as the rational equivariant functions. We investigate which elliptic zeta functions arising from integrals of power of ℘, where ℘ is the Weierstrass ℘-function attached to a rank two lattice of C , yield rational equivariant functions. Our concern in this survey is to provide certain examples of rational equivariant functions. In this sense, we establish a criterion in order to determine the rationality of equivariant functions derived from ratios of modular functions of low weight. Modular forms play an important role in number theory and many areas of mathematics and physics. [ABSTRACT FROM AUTHOR]

Let π$\pi$ be a cuspidal automorphic representation of GSp4(AQ)$\operatorname{GSp}_4(\mathbf {A}_\mathbf {Q})$, whose archimedean component is a holomorphic discrete series or limit of discrete series representation. If π$\pi$ is not CAP or endoscopic, then we show that its associated ℓ$\ell$‐adic Galois representations are irreducible and crystalline for 100%$100\%$ of primes ℓ$\ell$. If, moreover, π$\pi$ is neither an automorphic induction nor a symmetric cube lift, then we show that, for 100%$100\%$ of primes ℓ$\ell$, the image of its mod ℓ$\ell$ Galois representation contains Sp4(Fℓ)$\operatorname{Sp}_4(\mathbf {F}_\ell)$. [ABSTRACT FROM AUTHOR]

This paper proposes a modelling technique for Synchronous Reluctance Motors (SynRMs) based on a generalized Magnetic Equivalent Circuit (MEC). The proposed model can be used in the design of any number of stator teeth, rotor poles, and rotor barrier combinations. This technique allows elimination of infeasible machine solutions during the initial machine sizing stage, resulting in a lower cohort of feasible machine solutions that can be further optimized using finite element methods. Therefore, saturation effects, however, are not considered in the modelling. This paper focuses on modelling a generic structure of the SynRM in modular form and is then extended to a full SynRM model. The proposed model can be iteratively used for any symmetrical rotor pole and stator teeth combination. The developed technique is applied to model a 4-pole, 36 slot SynRM as an example, and the implemented model is executed following a time stepping strategy. The motor characteristics such as flux distribution and torque of the developed SynRM model is compared with finite elemental analysis (FEA) simulation results. [ABSTRACT FROM AUTHOR]

In this note we realize seven small simple groups as Galois groups over Q. The technique that we employ is the determination of the images of Galois representations attached to modular and automorphic forms, relying in two cases on recent results of Scholze on the existence of Galois representations attached to non-selfdual automorphic forms. [ABSTRACT FROM AUTHOR]

Let f be an elliptic modular form and p an odd prime that is coprime to the level of f. We study the link between divisors of the characteristic ideal of the p-primary fine Selmer group of f over the cyclotomic $\mathbb {Z}_p$ extension of $\mathbb {Q}$ and the greatest common divisor of signed Selmer groups attached to f defined using the theory of Wach modules. One of the key ingredients of our proof is a generalisation of a result of Wingberg on the structure of fine Selmer groups of abelian varieties with supersingular reduction at p to the context of modular forms. [ABSTRACT FROM AUTHOR]

We answer some questions in a paper by Kaneko and Koike ['On modular forms arising from a differential equation of hypergeometric type', Ramanujan J.7(1–3) (2003), 145–164] about the modularity of the solutions of a certain differential equation. In particular, we provide a number-theoretic explanation of why the modularity of the solutions occurs in some cases and does not occur in others. This also proves their conjecture on the completeness of the list of modular solutions after adding some missing cases. [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]

We investigate the parity of the coefficients of certain eta-quotients, extensively examining the case of m -regular partitions. Our theorems concern the density of their odd values, in particular establishing lacunarity modulo 2 for specified coefficients; self-similarities modulo 2; and infinite families of congruences in arithmetic progressions. For all m ≤ 28 , we either establish new results of these types where none were known, extend previous ones, or conjecture that such results are impossible. All of our work is consistent with a new, overarching conjecture that we present for arbitrary eta-quotients, greatly extending Parkin-Shanks' classical conjecture for the partition function. We pose several other open questions throughout the paper, and conclude by suggesting a list of specific research directions for future investigations in this area. [ABSTRACT FROM AUTHOR]

Borcherds lift for an even lattice of signature (p, q) is a lifting from weakly holomorphic modular forms of weight (p-q)/2 for the Weil representation. We introduce a new product operation on the space of such modular forms and develop a basic theory. The product makes this space a finitely generated filtered associative algebra, without unit element and noncommutative in general. This is functorial with respect to embedding of lattices by the quasi-pullback. Moreover, the rational space of modular forms with rational principal part is closed under this product. In some examples with p=2, the multiplicative group of Borcherds products of integral weight forms a subring. [ABSTRACT FROM AUTHOR]

This article uses the example of the Mogadishu International Airport zone and takes a spatio-temporal lens to explore how (sovereign) power unfolds in international interventions that aim at building a sovereign state. I show that the Mogadishu International Airport zone emerges as an elastic frontier zone that contradicts the sovereign imaginary intervenors aim to project and undermines many of the taken-for-granted boundaries that states tend to produce. The Mogadishu International Airport and similar zones emphasize the centrality of logistics and circulation in interventions, but also point towards their temporal and liminal character. Modularity became the material answer to the demand to secure circulation while adapting to the rapid rhythm and short timeframes of statebuilding. Modular designs enable the constant adaptation of the intervention terrain, allow intervenors to deny their power and imprint and facilitate the commercialization of supply chains and intervention materials. Sovereign power that operates through such zones becomes modular itself. It is exercised as an adaptable, in parts exchangeable, and highly mobile form of power that operates through crises and emergencies. The spaces and materials created by modular forms of sovereign power remain elusive, but nonetheless stratify experiences of power and security. [ABSTRACT FROM AUTHOR]

In Tian (Acta Arith. 164:399–412, 2014), the author improved the algorithm proposed by Edixhoven and Couveignes for computing mod ℓ Galois representations associated to eigenforms f for the cases that ℓ ≥ k - 1 and f has level one, where k is the weight of f. In this paper, we generalize the results of Tian (Acta Arith. 164:399–412, 2014) and present a method to find the Jacobians of modular curves of minimal dimensions to realize the modular Galois representations. Our method works for the cases that ℓ ≥ 5 may be any prime without the assumption ℓ ≥ k - 1 and the eigenforms f have arbitrary levels prime to ℓ . Moreover, if k > 2 , we give criteria for realizing the mod ℓ Galois representations in the Jacobians J 0 of X 0 . [ABSTRACT FROM AUTHOR]

In this paper, we present an algorithm to compute the deviation of the cranks from the average by using the theory of modular forms and Jacobi forms. Then applying the Ramanujan-type algorithm developed by Chen, Du and Zhao to each term in the expression of the deviation, we can derive the corresponding dissection formulas. As applications, we revisit the deviation of the cranks modulo 5 and 7, which were given by Garvan, and Mortenson, and also obtain the deviation of the cranks modulo 9 and 14. [ABSTRACT FROM AUTHOR]

The cohomology theory TMF of topological modular forms is a derived algebro-geometric interpretation of the ring of complex modular forms from number theory. In this article, we refine the classical Adams operations, Hecke operators, and Atkin–Lehner involutions from endomorphisms of modular forms to stable operators on TMF. Our algebro-geometric formulation of these operators leads to simple proofs of their many remarkable properties and computations. From these properties, we use techniques from homotopy theory to make simple number-theoretic deductions, including a rederivation of some classical congruences of Ramanujan and providing new infinite families of Hecke operators which satisfy Maeda's conjecture. [ABSTRACT FROM AUTHOR]

The ghost conjecture is first provided by Bergdall and Pollack in [1,2] to study the U p -slopes of spaces of modular forms, which, so far, has already brought plenty of important results. The local version of this conjecture under genericity condition has been solved by Liu-Truong-Xiao-Zhao in [10,11]. In the current paper, we prove a necessary and sufficient condition for a sequence of local ghost series to satisfy that their product has the same Newton polygon to the ghost series build from the direct sum of their associated modules (see Theorem 1.2). That answers a common question asked in both [2,10]. [ABSTRACT FROM AUTHOR]

This paper studies the folding of square, kirigami-inspired space structures consisting of concentrically arranged modular elements formed by thin shells. Localized elastic folds are introduced in the thin shells and different folding strategies can be obtained by varying the location of the folds and the sequence of imposed rotations. Modeling each modular element with rigid rods connected by revolute joints, numerical simulations of the kinematics of folding are obtained, including constraints that represent folding aids and a gravity offload system. These simulations are used to study two specific packaging schemes, and the folding envelopes of a specific structure are analyzed to identify the scheme that is easier to implement in practice. This particular scheme is demonstrated by means of a physical prototype. • We study the folding of space structures consisting of concentric modular elements. • Different packaging schemes are obtained by varying the locations of localized elastic folds. • A numerical model, based on rigid rods with revolute joints, is used to simulate the folding kinematics. • Specific packaging schemes that can be achieved with simple folding aids and gravity offloads are studied. • The scheme that is easier to implement is selected for an experimental demonstration. [ABSTRACT FROM AUTHOR]

Let r ≥ 1 be a positive integer, A a real positive definite symmetric r × r rational matrix, B a rational vector of length r , and C a rational scalar. Nahm's problem is to find all triples (A , B , C) such that the r -fold q -hypergeometric series f A , B , C (q) : = ∑ n = (n 1 , ... , n r) T ∈ (Z ≥ 0) r q 1 2 n T A n + n T B + C (q ; q) n 1 ⋯ (q ; q) n r becomes a modular form, and we call such (A , B , C) a modular triple. When the rank r = 3 , after extensive computer searches, Zagier provided twelve sets of conjectural modular triples and proved three of them. We prove a number of Rogers–Ramanujan type identities involving triple sums. These identities give modular form representations for and thereby verify all of Zagier's rank three examples. In particular, we prove a conjectural identity of Zagier. [ABSTRACT FROM AUTHOR]

The Shimura correspondence is a fundamental tool in the study of half-integral weight modular forms. In this paper, we prove a Shimura-type correspondence for spaces of half-integral weight cusp forms which transform with a power of the Dedekind eta multiplier twisted by a Dirichlet character. We prove that the lift of a cusp form of weight λ + 1 / 2 and level N has weight 2 λ and level 6 N , and is new at the primes 2 and 3 with specified Atkin-Lehner eigenvalues. This precise information leads to arithmetic applications. For a wide family of spaces of half-integral weight modular forms we prove the existence of infinitely many primes ℓ which give rise to quadratic congruences modulo arbitrary powers of ℓ. [ABSTRACT FROM AUTHOR]

We introduce an infinite family of Kronecker series twisted by characters. As an application, we give a closed formula for the sum of all Hecke eigenforms on Γ 0 (N) multiplied by their twisted period polynomials in terms of the product of those twisted Kronecker series, when N is square free. This extends an identity of Zagier among period polynomials, Hecke eigenforms and a quotient of Jacobi theta series. [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]

We study the complexity of Boolean constraint satisfaction problems (CSPs) when the assignment must have Hamming weight in some congruence class modulo M, for various choices of the modulus M. Due to the known classification of tractable Boolean CSPs, this mainly reduces to the study of three cases: \sanstwo -\sansS \sansA \sansT, \sansH \sansO \sansR \sansN -\sansS \sansA \sansT, and \sansL \sansI \sansN -\sanstwo (linear equations mod 2). We classify the moduli M for which these respective problems are polynomial time solvable, and when they are not (assuming the exponential time hypothesis). Our study reveals that this modular constraint lends a surprising richness to these classic, well-studied problems, with interesting broader connections to complexity theory and coding theory. The \sansH \sansO \sansR \sansN -\sansS \sansA \sansT case is connected to the covering complexity of polynomials representing the \sansN \sansA \sansN \sansD function mod M. The \sansL \sansI \sansN -\sanstwo case is tied to the sparsity of polynomials representing the \sansO \sansR function mod M, which in turn has connections to modular weight distribution properties of linear codes and locally decodable codes. In both cases, the analysis of our algorithm as well as the hardness reduction rely on these polynomial representations, highlighting an interesting algebraic common ground between hard cases for our algorithms and the gadgets which show hardness. These new complexity measures of polynomial representations merit further study. The inspiration for our study comes from a recent work by N\"agele, Sudakov, and Zenklusen on submodular minimization with a global congruence constraint. Our algorithm for \sansH \sansO \sansR \sansN -\sansS \sansA \sansT has strong similarities to their algorithm, and in particular identical kinds of set systems arise in both cases. Our connection to polynomial representations leads to a simpler analysis of such set systems and also sheds light on (but does not resolve) the complexity of submodular minimization with a congruency requirement modulo a composite M. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In 1999, Frank Schmidt noted that the number of partitions of integers with distinct parts in which the first, third, fifth, etc., summands add to n is equal to p (n) , the number of partitions of n. The object of this paper is to provide a context for this result which leads directly to many other theorems of this nature and which can be viewed as a continuation of our work on elongated partition diamonds. Again generating functions are infinite products built by the Dedekind eta function which, in turn, lead to interesting arithmetic theorems and conjectures for the related partition functions. [ABSTRACT FROM AUTHOR]

The Birch and Swinnerton-Dyer conjecture – which is one of the seven million-dollar Clay Mathematics Institute Millennium Prize Problems – and its generalizations are a significant focus of number theory research. A 2017 article of Jetchev, Skinner and Wan proved a Birch and Swinnerton-Dyer formula at a prime p for certain rational elliptic curves of rank 1. We generalize and adapt that article's arguments to prove an analogous formula for certain modular forms. For newforms f of even weight higher than 2 with Galois representation V containing a Galois-stable lattice T , let W = V / T and let K be an imaginary quadratic field in which the prime p splits. Our main result is that under some conditions, the p -index of the size of the Shafarevich-Tate group of W with respect to the Galois group of K is twice the p -index of a logarithm of the Abel-Jacobi map of a Heegner cycle defined by Bertolini, Darmon and Prasanna. Significant original adaptations we make to the 2017 arguments are (1) a generalized version of a previous calculation of the size of the cokernel of a localization-modulo-torsion map, and (2) a comparison of different Heegner cycles. [ABSTRACT FROM AUTHOR]

Ahlgren and Samart relate three cusp forms with complex multiplication to certain weakly holomorphic modular forms using p -adic bounds related to their Fourier coefficients. In these three examples, their result strengthens a theorem of Guerzhoy, Kent, and Ono which pairs certain CM forms with weakly holomorphic modular forms via p -adic limits. Ahlgren and Samart use only the theory of modular forms and Hecke operators, whereas Guerzhoy, Kent, and Ono use the theory of harmonic Maass forms. Here we extend Ahlgren and Samart's work to all cases where the cusp form space is one-dimensional and has trivial Nebentypus. Along the way, we obtain a duality result relating two families of weakly holomorphic modular forms that arise naturally in each case. [ABSTRACT FROM AUTHOR]

We consider the Diophantine equation 7 x 2 + y 2 n = 4 z 3. We determine all solutions to this equation for n = 2 , 3 , 4 and 5. We formulate a Kraus type criterion for showing that the Diophantine equation 7 x 2 + y 2 p = 4 z 3 has no non-trivial proper integer solutions for specific primes p > 7. We computationally verify the criterion for all primes 7 < p < 10 9 , p ≠ 13. We use the symplectic method and quadratic reciprocity to show that the Diophantine equation 7 x 2 + y 2 p = 4 z 3 has no non-trivial proper solutions for a positive proportion of primes p. In the paper [10] we consider the Diophantine equation x 2 + 7 y 2 n = 4 z 3 , determining all families of solutions for n = 2 and 3, as well as giving a (mostly) conjectural description of the solutions for n = 4 and primes n ≥ 5. [ABSTRACT FROM AUTHOR]

One of the main open problems in the theory of automorphic products is to classify reflective modular forms. Scheithauer [Invent. Math. 164 (2006), pp. 641-678] classified strongly reflective modular forms of singular weight on lattices of prime level. In this paper we classify symmetric reflective modular forms on lattices of prime level. This yields a full classification of lattices of prime level which have reflective modular forms. We also present some applications. [ABSTRACT FROM AUTHOR]

We prove Deligne's conjecture for symmetric fifth L-functions of elliptic newforms of weight greater than 5. As a consequence, we establish period relations between motivic periods associated to an elliptic newform and the Betti–Whittaker periods of its symmetric cube functorial lift to GL 4 {\operatorname{GL}_{4}}. [ABSTRACT FROM AUTHOR]

By old results with Millson, the generating series for the cohomology classes of special cycles on orthogonal Shimura varieties over a totally real field are Hilbert–Siegel modular forms. These forms arise via theta series. Using this result and the Siegel–Weil formula, we show that the products in the subring of cohomology generated by the special cycles are controlled by the Fourier coefficients of triple pullbacks of certain Siegel–Eisenstein series. As a consequence, there are comparison isomorphisms between special subrings for different Shimura varieties. In the case in which the signature of the quadratic space V is (m , 2) (m,2) at an even number d + d_{+} of archimedean places, the comparison gives a "combinatorial model" for the special cycle ring in terms of the associated totally positive definite space. [ABSTRACT FROM AUTHOR]

Following Ryan and Tornaría, we prove that moduli of congruences of Hecke eigenvalues, between Saito–Kurokawa lifts and non-lifts (certain Siegel modular forms of genus 2), occur (squared) in denominators of central spinor L-values (divided by twists) for the non-lifts. This is conditional on Böcherer's conjecture and its analogues and is viewed in the context of recent work of Furusawa, Morimoto and others. It requires a congruence of Fourier coefficients, which follows from a uniqueness assumption or can be proved in examples. We explain these factors in denominators via a close examination of the Bloch–Kato conjecture. [ABSTRACT FROM AUTHOR]

The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$ , for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5). [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]

We describe the ring of modular forms of degree |$2$| in characteristic |$2$| using its relation with curves of genus |$2$|⁠. [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]

We revisit the low-energy effective U(1) action of topologically twisted N = 2 SYM theory with gauge group of rank one on a generic oriented smooth four-manifold X with nontrivial fundamental group. After including a specific new set of Q -exact operators to the known action, we express the integrand of the path integral of the low-energy U(1) theory as an anti-holomorphic derivative. This allows us to use the theory of mock modular forms and indefinite theta functions for the explicit evaluation of correlation functions of the theory, thus facilitating the computations compared to previously used methods. As an explicit check of our results, we compute the path integral for the product ruled surfaces X = Σ g × CP 1 for the reduction on either factor and compare the results with existing literature. In the case of reduction on the Riemann surface Σ g , via an equivalent topological A-model on CP 1 , we will be able to express the generating function of genus zero Gromov–Witten invariants of the moduli space of flat rank one connections over Σ g in terms of an indefinite theta function, whence we would be able to make concrete numerical predictions of these enumerative invariants in terms of modular data, thereby allowing us to derive results in enumerative geometry from number theory. [ABSTRACT FROM AUTHOR]

There is a lifting from a non-CM elliptic curve E / Q to a paramodular form f of degree 2 and weight 3 given by the symmetric cube map. We find the level of f in terms of the coefficients of the Weierstrass equation of E. In order to compute the paramodular level, we use the available description of the local representations of GL (2 , Q p) attached to E for p ≥ 5 and determine the local representation of GL (2 , Q 3) attached to E. [ABSTRACT FROM AUTHOR]

We study p -adic families of cohomological automorphic forms for GL (2) over imaginary quadratic fields and prove that families interpolating a Zariski-dense set of classical cuspidal automorphic forms only occur under very restrictive conditions. We show how to computationally determine when this is not the case and establish concrete examples of families interpolating only finitely many Bianchi modular forms. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a special function on the Drinfeld period domain \Omega ^{r} for r\geq 2 which gives the false Eisenstein series of Gekeler when r=2. We also study its functional equation and relation with quasi-periodic functions of a Drinfeld module as well as transcendence of its values at CM points. [ABSTRACT FROM AUTHOR]