Weight 2k Research Articles

Nonvanishing of second coefficients of Hecke polynomials

Let Tm(N,2k) be the mth Hecke operator on the space S(N,2k) of cuspforms of weight 2k and level N. This paper shows that in all but finitely many cases, which we list, the second coefficient of the characteristic polynomial of T2(N,2k) does not vanish when 2 and N are coprime.

Jacobi-Eisenstein series over number fields

For any given totally real number field K, we compute the Fourier developments of the Jacobi Eisenstein series over K at the cusp at infinity. As main application we prove, for any K with class number 1, that the L-series of the Jacobi Eisenstein series of weight k≥3 for indices with rank and modified level 1 coincide with the L-series of the Eisenstein series of weight 2k−2 on the full Hilbert modular group of K. Moreover, under this correspondence the Fourier coefficients of the Jacobi Eisenstein series are related to the twisted L-series of the Hilbert Eisenstein series at the critical point by a Waldspurger type identity. This is a first step in the proof that Skoruppa's and Zagier's lifting from Jacobi forms over Q to elliptic modular forms holds true over arbitrary totally real number fields too.

Variants of Lehmer's speculation for newforms

In the spirit of Lehmer's unresolved speculation on the nonvanishing of Ramanujan's tau-function, it is natural to ask whether a fixed integer α is a value of τ(n) or is a Fourier coefficient af(n) of any given newform f(z). We offer a method, which applies to newforms with integer coefficients and trivial residual mod 2 Galois representation, that answers this question for odd α. We determine infinitely many spaces for which the primes 3≤ℓ≤37 are not absolute values of coefficients of newforms with integer coefficients, and we obtain many explicit examples for τ(n). We also obtain sharp lower bounds for the number of prime factors of such newform coefficients. In the weight aspect, for powers of odd primes ℓ, we prove that ±ℓm is not a coefficient of any such newform f with weight 2k>M±(ℓ,m) and even level coprime to ℓ, where M±(ℓ,m) are effectively computable constants that are Oℓ(m).

Hyperelliptic curves and newform coefficients

We study which integers are admissible as Fourier coefficients of even integer weight newforms. In the specific case of the tau-function, we show that for all odd primes ℓ<100 and all integers m≥1, we haveτ(n)≠±ℓ,±5m. For general newforms f with even integer weight 2k and integer coefficients, we prove for most integers j dividing 2k−1 and all ordinary primes p that af(p2) is never a j-th power. We prove a similar result for af(p4), conditional on the Frey-Mazur Conjecture. Our primary method involves relating questions about values of newforms to the existence of perfect powers in certain binary recurrence sequences, and makes use of bounds from the theory of linear forms in logarithms. The method extends without difficulty to a large family of Lebesgue-Nagell equations with fixed exponent. To prove results about general newforms, we also make use of the modular method and Ribet's level-lowering theorem.

Newforms of half-integral weight: the minus space of Sk+1/2(Γ0(8M))

We compute generators and relations for a certain 2-adic Hecke algebra of level 8 associated with the double cover of SL2 and a 2-adic Hecke algebra of level 4 associated with PGL2. We show that these two Hecke algebras are isomorphic as expected from the Shimura correspondence. We use the 2-adic generators to define classical Hecke operators on the space of holomorphic modular forms of weight k + 1/2 and level 8M where M is odd and square-free. Using these operators and our previous results on half-integral weight forms of level 4M we define a subspace of the space of half-integral weight forms as a common -1 eigenspace of certain Hecke operators. Using the relations and a result of Ueda we show that this subspace, which we call the minus space, is isomorphic as a Hecke module under the Ueda correspondence to the space of new forms of weight 2k and level 4M. We observe that the forms in the minus space satisfy a Fourier coefficient condition that gives the complement of the plus space but does not define the minus space.

Effects of Macromonomeric Length of Ureidopyrimidinone-Induced Supramolecular Polymers on Their Crystalline Structure and Mechanical/Rheological Properties

To investigate the effect of the macromonomeric length of ureidopyrimidinone-based supramolecular polymers on their crystalline structure and mechanical/rheological properties, we synthesized ureidopyrimidinone-end functional poly(tetramethylene glycol), U(PTMG), with varied molecular weight (1k, 2k, and 3k g·mol−1). From 1H NMR, FT-IR, and specific viscosity analysis, we confirmed chain extension of U(PTMG)s by the quadruple hydrogen bonding of ureidopyrimidinone, indicating successful formation of U(PTMG) supramolecular polymers. We found that the U(PTMG) supramolecular polymers had dual-crystalline structures composed of PTMG and UPy crystals. The rigid UPy crystals, which had relatively high melting temperatures, were dominant with decreasing macromonomeric lengths of the U(PTMG) supramolecular polymers, whereas the chain-folded PTMG crystals were drastically reduced. As a result, the mechanical and rheological properties of the U(PTMG) supramolecular polymers became rigid and elastic, with decreasing macromonomer lengths of the U(PTMG) supramolecular polymers, due to their increased portions of UPy crystals.

Zeros of certain combinations of Eisenstein series of weight 2k,3k, and k + l

We locate the zeros of the modular forms Ek2(τ)+E2k(τ),Ek3(τ)+E3k(τ), and Ek(τ)El(τ)+Ek+l(τ), where Ek(τ) is the Eisenstein series for the full modular group SL2(Z). By utilizing work of F.K.C. Rankin and Swinnerton-Dyer, we prove that for sufficiently large k,l, all zeros in the standard fundamental domain are located on the lower boundary A={eiθ:π/2≤θ≤2π/3}.

On a theta lift related to the Shintani lift

We study the Millson theta lift which maps weight −2k to weight 1/2−k harmonic weak Maass forms for k∈Z,k≥0, and which is closely related to the classical Shintani lift from weight 2k+2 to weight k+3/2 cusp forms. We compute the Fourier expansion of the theta lift and show that it involves twisted traces of CM values and geodesic cycle integrals of the input function. As an application, we recover Zagier's generating series of twisted traces of singular moduli of weight 1/2 as Millson lifts.

On central L-derivative values of automorphic forms

We study the average and nonvanishing of the central L-derivative values of L(s, f) and $$L(s,f_{K_{\scriptscriptstyle D}})$$ for f in an orthogonal Hecke eigenbasis $$\mathcal {H}_{2k}$$ of weight 2k cusp forms of level 1 for large odd k. Here $$f_{K_{\scriptscriptstyle D}}$$ is the base change of f to an imaginary quadratic field $$K_{\scriptscriptstyle D}=\mathbb {Q}(\sqrt{D})$$ with fundamental discriminant D. We prove asymptotic formulas for the first and second moments of $$L'(\frac{1}{2},f)$$ , as well as the first moment of $$L'(\frac{1}{2},f_{K_{\scriptscriptstyle D}})$$ , over $$\mathcal {H}_{2k}$$ as odd $$k\rightarrow \infty $$ . Further, we employ mollifiers to establish that for sufficiently large k there are positive proportion of Hecke eigenforms f in $$\mathcal {H}_{2k}$$ with $$L'(\frac{1}{2},f)\ne 0$$ . We also give applications of our results to Heegner cycles of high weights of the modular curve $$X_0(1)$$ .

Synthesis and in vitro evaluation of pH-sensitive PEG-I-dC16 block polymer micelles for anticancer drug delivery.

To develop an acid trigger release of antitumour drug delivery carriers, pH-sensitive amphiphilic poly (ethyleneglycol)-imine-benzoic-dipalmitate (PEG-I-dC16 ) polymers were designed and synthesized and the drug-loaded micelles were evaluated in vitro. PEG-I-dC16 synthesized by Schiff base synthetic method and characterized by (1) H-NMR. To determine the drug-loading capacity, doxorubicin (DOX) was encapsulated in the micelles using membrane dialysis method. Zeta potential, particle size, drug-loading capacity, in vitro drug release in different pH conditions and cytotoxicity evaluation of micelles were carried out comparing with non-acid liable PEG-amide-benzoic-dipalmitate (PEG-A-dC16) polymers micelles. The cellular uptake and intracellular distribution of DOX were detected by flow cytometry and confocal laser scanning microscope. Drug-loading capacity and encapsulation efficiency of micelle (PEG molecular weight 2k) were 12.7 ± 1.1% and 49.8 ± 2.2%, respectively. The average particle size was 72.3 ± 2.5 nm. The DOX release rate of PEG-I-dC16 micelles is much higher at pH 6.5 than at pH 7.4. DOX cellular uptake and nuclear accumulation of PEG-I-dC16 micelles were more efficiency than that of PEG-A-dC16 micelles. The pH-sensitive PEG-I-dC16 micelles could be a promising drug delivery system for anticancer drugs.

A formula for generating weakly modular forms with weight 12

In this short paper, generally, we define a family of functions fk depends on the Eisenstein series with weight 2k, for k ( N. More detail, by considering the function fk, we define a derivative formula for generating weakly modular forms with weight 12. As a result for this, we claim that this formula gives an advantage to find the special solutions of some differential equations.

Ikeda's conjecture on the period of the Duke–Imamoḡlu–Ikeda lift

Let k and n be positive even integers. For a cus- pidal Hecke eigenform h in the Kohnen plus subspace of weight k − n/2 + 1/2 for 0(4), let In(h) be the Duke-Imamoglu-Ikeda lift of h in the space of cusp forms of weight k for Spn(Z), and f the primitive form of weight 2k − n for SL2(Z) corresponding to h under the Shimura correspondence. We then express the ratio hIn(h),In(h)i/hh,hi of the period of In(h) to that of h in terms of special values of certain L-functions of f. This proves the conjecture proposed by Ikeda concerning the period of the Duke- Imamoglu-Ikeda lift. One of the fascinating problems in the theory of modular forms is to find the relation between the periods (or the Petersson products) of cuspidal Hecke eigenforms which are related with each other through their L-functions. In particular, there are several important results concerning the relation between the period of a cuspidal Hecke eigen- form g for an elliptic modular group Γ � SL2(Z) and that of its lift bHere, by a lift of g we mean a cuspidal Hecke eigenform for another modular group Γ 0 (e.g. the symplectic group, the orthogonal group, the unitary group, etc.) whose certain L-function can be expressed in terms of certain L-functions related with g:Thus we propose the following problem: Problem A. Express the ratio hbbi=hg;gi e in terms of arithmetic invariants of g;for example, the special values of certain L-functions related with g for some integer e: For instance, Zagier (Zag77) solved Problem A for the Doi-Naganuma

Dirichlet series with functional equations and arithmetical identities

Exploiting the functional equation of Hecke-type associated with a function satisfying a modular relation with a residual function as developed in Bochner (J Indian Math Soc 16:99–102, 1952), Chandrasekharan and Narasimhan (Ann Math 74:1–23, 1961) derived the equivalence of the functional equation to two arithmetical identities. Hawkins and Knopp (Contemp Math 143:451–475, 1993) showed the equivalence of the functional equation to modular integrals with rational period functions of weight 2k, \(k \in \mathbb {Z}^+\) on the theta group \(\Gamma _\vartheta \). The aim of the current work is to show that results analogous to those of Chandrasekharan and Narasimhan can be developed in the Hawkins and Knopp context, but with respect to the full modular group \(\Gamma (1)\), rather than the theta group \(\Gamma _\vartheta \).

On the second moment of L-series of holomorphic cusp forms on the critical line

New uniform estimates are obtained for the second moment of L-series of cusp forms of even weight 2k ≥ 4 with respect to the congruence subgroup Γ0(N).

Rational period functions in higher level cases

Extending Knopp's results [9,10] we investigate examples and properties of rational period functions in higher level cases, including location of poles and behavior under the action of Hecke operators. More precisely, we prove that a rational period function may have poles only at 0 or at real quadratic irrationalities. Moreover by applying the action of Hecke operators we prove that for positive odd integer k and p∈{2,3}, the space of all rational period functions of weight 2k for Γ0+(p) is infinite dimensional.

Koecher–Maass series of the Ikeda lift for U ( m , m )

Let K be an imaginary quadratic field with discriminant -D, and x the Dirichlet character corresponding to the extension K/Q. Let m=2n or 2n+1 with n a positive integer. Let f be a primitive form of weight 2k+1 and level D with Neben character x, or a primitive form of weight 2k for SL(2,Z) according as m=2n, or m=2n+1. For such an f let I(m,f) be the lift of f to the space of Hermitian modular forms of degree m constructed by Ikeda. We then give an explicit formula of the Koecher-Maass series L(s,I(m,f)) of I(m,f). This is a generalization of [Y. Mizuno, Manuscripta Math. 119(2006), 175-216].

Modular forms of half-integral weights on SL(2, ℤ)

AbstractIn this paper, we prove that, for an integer r with (r, 6) = 1 and 0 &lt; r &lt; 24 and a nonnegative even integer s, the setis isomorphic toas Hecke modules under the Shimura correspondence. Here Ms(1) denotes the space of modular forms of weight is the space of newforms of weight 2k on Γ0 (6) that are eigenfunctions with eigenvalues €2 and €3 for Atkin-Lehner involutions W2 and W3, respectively, and the notation ⊕(12/.) means the twist by the quadratic character (12/-). There is also an analogous result for the cases (r, 6) = 3.

On the Cuspidality of Pullbacks of Siegel Eisenstein Series and Applications to the Bloch-Kato Conjecture

Let k > 9 be an even integer and p a prime with p > 2k −2. Let f be a newform of weight 2k −2 and level so that f is ordinary at p and is irreducible. Under some additional hypotheses, we prove that , where S is the Pontryagin dual of the Selmer group associated to with ϵ the p-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito–Kurokawa lift of f and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain the lower bound on the Selmer group.

A canonical subspace of modular forms of half-integral weight

We characterise the space of newforms of weight k + 1/2 on Γ0(4N), N odd and square-free (studied by the second and third authors with Vasudevan) under the Atkin-Lehner W(4) operator. As an application, we show that the (±1)-eigensubspaces of the W(4) operator on the space of modular forms of weight k + 1/2 on Γ0(4N) is mapped to modular forms of weight 2k on Γ0(N), under a class of Shimura maps. The existence of such subspaces having this mapping property was conjectured by Zagier in a private communication. One of the special features of the (±1)-eigensubspaces is that the (2k + 1)-th power of the classical theta series of weight 1/2 belongs to the + eigensubspace and hence this gives interesting congruences for r2k+1(p2).

Stark–Heegner points and the Shimura correspondence

AbstractLet $g = \sum c(D)q^D$ and $f=\sum a_n q^n$ be modular forms of half-integral weight k+1/2 and integral weight 2k respectively that are associated to each other under the Shimura–Kohnen correspondence. For suitable fundamental discriminants D, a theorem of Waldspurger relates the coefficient c(D) to the central critical value L(f,D,k) of the Hecke L-series of f twisted by the quadratic Dirichlet character of conductor D. This paper establishes a similar kind of relationship for central critical derivatives in the special case k=1, where f is of weight 2. The role of c(D) in our main theorem is played by the first derivative in the weight direction of the Dth Fourier coefficient of a p-adic family of half-integral weight modular forms. This family arises naturally, and is related under the Shimura correspondence to the Hida family interpolating f in weight 2. The proof of our main theorem rests on a variant of the Gross–Kohnen–Zagier formula for Stark–Heegner points attached to real quadratic fields, which may be of some independent interest. We also formulate a more general conjectural formula of Gross–Kohnen–Zagier type for Stark–Heegner points, and present numerical evidence for it in settings that seem inaccessible to our methods of proof based on p-adic deformations of modular forms.