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In this paper, we solve three Diophantine equations: 2 x ± (2 k p) y = z 2 and − 2 x + (2 k 3) y = z 2 with k ≥ 0 and prime p ≡ ± 3 (mod 8) . We obtain all the non-negative integer solutions by using elementary methods and the database of elliptic curves in "The L-functions and modular forms database" (LMFDB). [ABSTRACT FROM AUTHOR]

We study Hilbert Poincaré series associated to general seed functions and construct Cohen’s kernels and double Eisenstein series as series of Hilbert Poincaré series. Then we calculate the Rankin–Cohen brackets of Hilbert Poincaré series and Hilbert modular forms and extend Zagier’s kernel formula to totally real number fields. Finally, we show that the Rankin–Cohen brackets of two different types of Eisenstein series are special values of double Eisenstein series up to a constant. [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]

In this paper, we explore the adelic Shintani lifting of a weight 2 cusp form f. For a finite Schwartz function related to an auxiliary quadratic character χ , we express certain Fourier coefficients of this lifting in terms of the twisted L-value L (f , χ , 1). [ABSTRACT FROM AUTHOR]

We study the range of validity of the density hypothesis for the zeros of L-functions associated with cusp Hecke eigenforms f of even integral weight, and prove that Nf​(σ,T)≪T2(1−σ)+ε holds for σ≥1407/1601. This improves upon a result of Ivić, who had previously shown the zero-density estimate in the narrower range σ≥53/60. Our result relies on an improvement of the large value estimates for Dirichlet polynomials based on mixed moment estimates for the Riemann zeta function. The main ingredients in our proof are the Halász–Montgomery inequality, Ivić’s mixed moment bounds for the zeta function, Huxley’s subdivision argument, Bourgain’s dichotomy approach, and Heath-Brown’s bound for double zeta sums. [ABSTRACT FROM AUTHOR]

In this paper, we study the zeros of polynomials obtained from the L-functions and their derivatives associated to non-cuspidal modular forms in Eisenstein spaces of prime levels as a generalization of work by Diamantis and Rolen. [ABSTRACT FROM AUTHOR]

In [9] , Gun, Murty and Rath studied non-vanishing and transcendental nature of special values of a varying class of L -functions and their derivatives. This led to a number of works by several authors in different set-ups including studying higher derivatives. However, all these works were focused around the central point of the critical strip. In this article, we extend the study to arbitrary points in the critical strip. [ABSTRACT FROM AUTHOR]

Let p be an odd prime. Let f 1 and f 2 be weight 2 cuspidal Hecke eigenforms with isomorphic residual Galois representations at p. Greenberg–Vatsal and Emerton–Pollack–Weston showed that if p is a good ordinary prime for the two forms, the Iwasawa invariants of their p-primary Selmer groups and p-adic L-functions over the cyclotomic ℤ p -extension of ℚ are closely related. The goal of this article is to generalize these results to the anticyclotomic setting. More precisely, let K be an imaginary quadratic field where p splits. Suppose that the generalized Heegner hypothesis holds with respect to both (f 1 , K) and (f 2 , K) . We study relations between the Iwasawa invariants of the BDP Selmer groups and the BDP p-adic L-functions of f 1 and f 2 . [ABSTRACT FROM AUTHOR]

For f a cuspidal modular form for the group Γ 0 (N) of integral or half-integral weight, N a multiple of 4 in case the weight is half-integral, we study the zeros of the L -function attached to f twisted by an additive character e 2 π i n p q with p q ∈ Q. We prove that for certain f and p q ∈ Q , the additively twisted L -function has infinitely many zeros on the critical line. We develop a variant of the Hardy-Littlewood method which uses automorphic distributions to prove the result. [ABSTRACT FROM AUTHOR]

Harris and Venkatesh made a conjecture relating the derived Hecke operators and the adjoint motivic cohomology in the setting of weight one modular forms. This conjecture was proved under some conditions in the dihedral case by Darmon–Harris–Rotger–Venkatesh. We use a new approach to prove more general cases of the conjecture (up to sign). Our approach relies on Waldspurger's formula for the central value of Rankin L-series and Ichino's formula for the triple product L-function. [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]

For a positive integer N, let X0(N)$X_0(N)$ be the modular curve over Q$\mathbf {Q}$ and J0(N)$J_0(N)$ its Jacobian variety. We prove that the rational cuspidal subgroup of J0(N)$J_0(N)$ is equal to the rational cuspidal divisor class group of X0(N)$X_0(N)$ when N=p2M$N=p^2M$ for any prime p and any squarefree integer M. To achieve this, we show that all modular units on X0(N)$X_0(N)$ can be written as products of certain functions Fm,h$F_{m, h}$, which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on X0(N)$X_0(N)$ under a mild assumption. [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]

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]

By the modularity theorem, every rigid Calabi–Yau threefold X has associated modular form f such that the equality of L‐functions L(X,s)=L(f,s)$L(X,s)=L(f,s)$ holds. In this case, period integrals of X are expected to be expressible in terms of the special values L(f,1)$L(f,1)$ and L(f,2)$L(f,2)$. We propose a similar interpretation of period integrals of a nodal model of X. It is given in terms of certain variants of a Mellin transform of f. We provide numerical evidence toward this interpretation based on a case of double octics. [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]

In the present paper, we prove that the first homology group of Bianchi 3-fold is generated by the special Bianchi modular symbols. Also, we construct the integral valued p -adic L -function of p -ordinary Bianchi Hecke eigenforms by taking the Lefschetz-Poincaré Pairing of the cohomology class attached to Bianchi modular forms and Bianchi modular symbols on Bianchi 3-folds. By using the homology generation result, we prove that the μ -invariant of some isotopic components of the p -adic L -function of certain Bianchi Hecke eigenforms vanishes for a positive proportion of ordinary prime ideals. The main ingredient of the proof of the generation result is analyzing the fast convergent series expression of the integration of the Bianchi modular forms along the special modular symbols. [ABSTRACT FROM AUTHOR]

Let f and g be two different holomorphic cusp froms or Maass cusp forms for the full modular group S L (2 , Z) . We are interested in coefficients of Rankin–Selberg L-functions, and establish some bounds for ∑ n ≤ x λ sym i f × sym j g (n) , ∑ n ≤ x λ f (n i) λ g (n j) , ∑ n ≤ x | λ sym i f × sym j g (n) | , ∑ n ≤ x | λ f (n i) λ g (n j) | , and ∑ n ≤ x max { | λ sym i f × sym j g (n) | 2 φ , | λ sym i f × sym j g (n + h) | 2 φ } , where φ > 0 and h is a fixed positive integer. [ABSTRACT FROM AUTHOR]

We show that the Fourier coefficients of the symmetric power L-functions of modular forms are O((log n)-12 +-) except for a density zero set. [ABSTRACT FROM AUTHOR]

The main purpose of this note is to provide an algorithm for approximating the value of the balanced p -adic L -function, as constructed in [Hsi21] , at the point (2 , 1 , 1) , which is lying outside of the interpolation region. The algorithmic procedure is obtained building on the work of [FM14] and considering finite-length geodesics on the Bruhat–Tits tree for G L 2 (Q p). We are interested in the case where at least one of the Hida families is associated with an elliptic curve over the rationals and we further restrict ourselves to the case where only one finite local sign of the functional equation is −1. [ABSTRACT FROM AUTHOR]

Recently, Chen and Kiming studied the theta operator on modular forms modulo prime powers p m , where p ≥ 5 and m ≥ 2 . In this paper, we study mod p m filtrations and mod p m theta cycles. We give a bound on some elements in the mod p m theta cycle ( m ≥ 2 ), and we exactly compute those values in the case that m = 2 . [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]

Let f (z) be a Maass cusp form for the full modular group S L 2 (Z). And let λ sym 2 f (n) be the n -th coefficient of symmetric square L -function associated with f (z). For A ≥ 2 fixed we define M (A) as the infimum of all numbers M such that ∫ 1 T | L (1 / 2 + i t , sym 2 f) | A d t ≪ f T M + ϵ , where L (s , sym 2 f) is the automorphic L -function attached to f. In this paper, we will establish the upper bounds of M (A) and get the zero density estimates for L (s , sym 2 f). [ABSTRACT FROM AUTHOR]

We investigate the sixth moment of the family of L -functions associated to holomorphic modular forms on G L 2 with respect to a congruence subgroup Γ 1 (q). We improve on previous work and obtain an unconditional upper bound of the correct order of magnitude. [ABSTRACT FROM AUTHOR]

Let K be an imaginary quadratic field where p splits. We study signed Selmer groups for nonordinary modular forms over the anticyclotomic Zp-extension of K, showing that one inclusion of an Iwasawa main conjecture involving the p-adic L-function of Bertolini--Darmon--Prasanna implies that their p-invariants vanish. This gives an alternative method to reprove a recent result of Matar on the vanishing of the μ-invariants of plus andminus signed Selmer groups for elliptic curves. [ABSTRACT FROM AUTHOR]

Let K be an imaginary quadratic field with class number 1, in this paper we obtain the functional equation of the p -adic L -function of small slope p -stabilised Bianchi modular forms. Then, using p -adic families of Bianchi modular forms, we extend our result to Σ-smooth base-change Bianchi modular forms. [ABSTRACT FROM AUTHOR]

We investigate meromorphic quasi-modular forms and their L -functions. We study the space of meromorphic quasi-modular forms and Rankin–Cohen brackets of meromorphic modular forms. Then we define their L -functions by using multiple techniques of regularized integral. Explicit formulas for the L -functions are given. [ABSTRACT FROM AUTHOR]

Let F be a newform of weight 2k on Γ 0 (N) with an odd integer N and a positive integer k, and ℓ be a prime larger than or equal to 5 with (ℓ , N) = 1 . For each fundamental discriminant D, let χ D be a quadratic character associated with quadratic field Q (D) . Assume that for each D, the ℓ -adic valuation of the algebraic part of L (F ⊗ χ D , k) is non-negative. Let W ℓ + (resp. W ℓ - ) be the set of positive (resp. negative) fundamental discriminants D with (D , N) = 1 such that the ℓ -adic valuation of the algebraic part of L (F ⊗ χ D , k) is zero. We prove that for each sign ϵ , if W ℓ ϵ is a non-empty finite set, then W ℓ ϵ ⊂ 1 , (- 1) ℓ - 1 2 ℓ. By this result, we prove that if ϵ is the sign of (- 1) k , then k ≥ ℓ - 1 or k = ℓ - 1 2. These are applied to obtain a lower bound for # { D ∈ W ℓ ϵ : | D | ≤ X } and the indivisibility of the order of the Shafarevich–Tate group of an elliptic curve over Q . To prove these results, first we refine Waldspurger's formula on the Shimura correspondence for general odd levels N. Next we study mod ℓ modular forms of half-integral weight with few non-vanishing coefficients. To do this, we use the filtration of mod ℓ modular forms and mod ℓ Galois representations. [ABSTRACT FROM AUTHOR]

Let G be the group (GL 2 × GU (1)) / GL 1 over a totally real field F, and let X be a Hida family for G . Revisiting a construction of Howard and Fouquet, we construct an explicit section P of a sheaf of Selmer groups over X . We show, answering a question of Howard, that P is a universal Heegner class, in the sense that it interpolates geometrically defined Heegner classes at all the relevant classical points of X . We also propose a 'Bertolini–Darmon' conjecture for the leading term of P at classical points. We then prove that the p-adic height of P is given by the cyclotomic derivative of a p-adic L-function. This formula over X (which is an identity of functionals on some universal ordinary automorphic representations) specialises at classical points to all the Gross–Zagier formulas for G that may be expected from representation-theoretic considerations. Combined with a result of Fouquet, the formula implies the p-adic analogue of the Beilinson–Bloch–Kato conjecture in analytic rank one, for the selfdual motives attached to Hilbert modular forms and their twists by CM Hecke characters. It also implies one half of the first example of a non-abelian Iwasawa main conjecture for derivatives, in 2 [ F : Q ] variables. Other applications include two different generic non-vanishing results for Heegner classes and p-adic heights. [ABSTRACT FROM AUTHOR]

Let D be a definite quaternion algebra over Q and O an Eichler order in D of square-free level. We study distribution of the toric periods of algebraic modular forms of level O . We focus on two problems: non-vanishing and sign changes. Firstly, under certain conditions on O , we prove the non-vanishing of the toric periods for positive proportion of imaginary quadratic fields. This improves the known lower bounds toward Goldfeld's conjecture in some cases and provides evidence for similar non-vanishing conjectures for central values of twisted automorphic L-functions. Secondly, we show that the sequence of toric periods has infinitely many sign changes. This proves the sign changes of the Fourier coefficients { a (n) } n of weight 3 2 modular forms, where n ranges over fundamental discriminants. In the final section, we present numerical experiments in some cases and formulate several conjectures based on them. [ABSTRACT FROM AUTHOR]

We prove hybrid subconvexity bounds for twisted L-functions L(s,f\times \chi) at the central point using a fourth moment estimate, including a new instance of the Burgess subconvexity bound. [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]

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]

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]

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]

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 describe an algorithm for computing p-adic L-functions of characters of totally real fields, using the Fourier expansions of diagonal restrictions of Hilbert modular forms. [ABSTRACT FROM AUTHOR]

Inspired by Lehmer's conjecture on the non-vanishing of the Ramanujan τ -function, one may ask whether an odd integer α can be equal to τ (n) or any coefficient of a newform f(z). Balakrishnan, Craig, Ono and Tsai used the theory of Lucas sequences and Diophantine analysis to characterize non-admissible values of newforms of even weight k ≥ 4 . We use these methods for weight 2 and 3 newforms and apply our results to L-functions of modular elliptic curves and certain K3 surfaces with Picard number ≥ 19 . In particular, for the complete list of weight 3 newforms f λ (z) = ∑ a λ (n) q n that are η -products, and for N λ the conductor of some elliptic curve E λ , we show that if | a λ (n) | < 100 is odd with n > 1 and (n , 2 N λ) = 1 , then a λ (n) ∈ { - 5 , 9 , ± 11 , 25 , ± 41 , ± 43 , - 45 , ± 47 , 49 , ± 53 , 55 , ± 59 , ± 61 , ± 67 , - 69 , ± 71 , ± 73 , 75 , ± 79 , ± 81 , ± 83 , ± 89 , ± 93 ± 97 , 99 }. Assuming the Generalized Riemann Hypothesis, we can rule out a few more possibilities leaving a λ (n) ∈ { - 5 , 9 , ± 11 , 25 , - 45 , 49 , 55 , - 69 , 75 , ± 81 , ± 93 , 99 }. [ABSTRACT FROM AUTHOR]

In earlier work, the first named author generalized the construction of Darmon-style $\mathcal {L}$ -invariants to cuspidal automorphic representations of semisimple groups of higher rank, which are cohomological with respect to the trivial coefficient system and Steinberg at a fixed prime. In this paper, assuming that the Archimedean component of the group has discrete series we show that these automorphic $\mathcal {L}$ -invariants can be computed in terms of derivatives of Hecke eigenvalues in $p$ -adic families. Our proof is novel even in the case of modular forms, which was established by Bertolini, Darmon and Iovita. The main new technical ingredient is the Koszul resolution of locally analytic principal series representations by Kohlhaase and Schraen. As an application of our results we settle a conjecture of Spieß: we show that automorphic $\mathcal {L}$ -invariants of Hilbert modular forms of parallel weight $2$ are independent of the sign character used to define them. Moreover, we show that they are invariant under Jacquet–Langlands transfer and, in fact, equal to the Fontaine–Mazur $\mathcal {L}$ -invariant of the associated Galois representation. Under mild assumptions, we also prove the equality of automorphic and Fontaine–Mazur $\mathcal {L}$ -invariants for representations of definite unitary groups of arbitrary rank. Finally, we study the case of Bianchi modular forms to show how our methods, given precise results on eigenvarieties, can also work in the absence of discrete series representations. [ABSTRACT FROM AUTHOR]

Let Π be a cohomological irreducible cuspidal automorphic representation of GL 2 (A F) with central character ω Π over a totally real number field F. In this paper, we prove the algebraicity of the near central non-critical value of the symmetric fourth L -function of Π twisted by ω Π − 2. The algebraicity is expressed in terms of the Petersson norm of the normalized newform of Π and the top degree Whittaker period of the Gelbart–Jacquet lift Sym 2 Π of Π. [ABSTRACT FROM AUTHOR]

We prove that the cuspidal eigencurve Ccusp is étale over the weight space at any classical weight 1 Eisenstein point ƒ and meets two Eisenstein components of the eigencurve C transversally at f. Further, we prove that the local ring of C at ƒ is Cohen-Macaulay but not Gorenstein and compute the Fourier coefficients of a basis of overconvergent weight 1 modular forms lying in the same generalised eigenspace as ƒ. In addition, we prove an R = T theorem for the local ring at ƒ of the closed subspace of C given by the union of Ccusp and one Eisenstein component and prove unconditionally, via a geometric construction of the residue map, that the corresponding congruence ideal is generated by the Kubota-Leopoldt p-adic L-function. Finally we obtain a new proof of the Ferrero-Greenberg Theorem and Gross' formula for the derivative of the p-adic L-function at the trivial zero. [ABSTRACT FROM AUTHOR]

In this paper, we consider general divisor problems involving the coefficients of triple product L-function L(f x g x h; s) attached to holomorphic cusp forms f(z); g(z) and h(z) of even integral weight k for the full modular group SL(2; Z). [ABSTRACT FROM AUTHOR]

Let τ (n) be Ramanujan's tau function, defined by the discriminant modular form Δ (z) = q ∏ j = 1 ∞ (1 − q j) 2 4 = ∑ n = 1 ∞ τ (n) q n , q = e 2 π i z (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer's conjecture asserts that τ (n) ≠ 0 for all n ≥ 1 ; since τ (n) is multiplicative, it suffices to study primes p for which τ (p) might possibly be zero. Assuming standard conjectures for the twisted symmetric power L -functions associated to τ (including GRH), we prove that if x ≥ 1 0 5 0 , then # { x < p ≤ 2 x : τ (p) = 0 } ≤ 1. 2 2 × 1 0 − 5 x 3 / 4 log x , a substantial improvement on the implied constant in previous work. To achieve this, under the same hypotheses, we prove an explicit version of the Sato–Tate conjecture for primes in arithmetic progressions. [ABSTRACT FROM AUTHOR]

We express the Mahler measures of 23 families of Laurent polynomials in terms of Eisenstein–Kronecker series. These Laurent polynomials arise as Landau–Ginzburg potentials on Fano 3-folds, sixteen of which define K3 hypersurfaces of generic Picard rank 19, and the rest are of generic Picard rank less than 19. We relate the Mahler measure at each rational singular moduli to the value at 3 of the L-function of some weight-3 newform. Moreover, we find ten exotic relations among the Mahler measures of these families. [ABSTRACT FROM AUTHOR]

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F, and let p be a prime of F at which f is new. Let K be a quadratic extension of F, and L(f/K,s) the L‐function of the base‐change of f to K. Under certain hypotheses on f and K, the functional equation of L(f/K,s) ensures that it vanishes at the central point. The Bloch–Kato conjecture predicts that this should force the existence of non‐trivial classes in an appropriate global Selmer group attached to f and K. In this paper, we use the theory of double integrals developed by Salazar and the second author to construct certain p‐adic Abel–Jacobi maps, which we use to propose a construction of such classes via Stark–Heegner cycles. This builds on ideas of Darmon and in particular generalises an approach of Rotger and Seveso in the setting of classical modular forms. [ABSTRACT FROM AUTHOR]

In this work, we use the Rankin–Selberg method to obtain results on the analytic properties of the standard L -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 L -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. [ABSTRACT FROM AUTHOR]

The article discusses the supersingular Rankin-Selberg p-adic L-functions by constructing a 1-variable anticyclotomic p-adic L-function linked to a given normalized eigenform like a newform or normalized Eisenstein series. Also cited are the use of an imaginary quadratic field and the fundamental discriminant of K to create a Heegner hypothesis for N, and the equation for a complex-valued Hecke character.