International Journal of Number Theory

Given a Hecke eigenform f of weight 2 and square-free level N, by the work of Kohnen, there is a unique weight 3/2 modular form of level 4N mapping to f under the Shimura correspondence. Furthermore, by the work of Waldspurger the Fourier coefficients of such a form are related to the quadratic twists of the form f. Gross gave a construction of the half integral weight form when N is prime, and such construction was later generalized to square-free levels. However, in the non-square free case, the situation is more complicated since the natural construction is vacuous. The problem being that there are too many special points so that there is cancellation while trying to encode the information as a linear combination of theta series. In this paper, we concentrate in the case of level p², for p > 2 a prime number, and show how the set of special points can be split into subsets (indexed by bilateral ideals for an order of reduced discriminant p²) which gives two weight 3/2 modular forms mapping to f under the Shimura correspondence. Moreover, the splitting has a geometric interpretation which allows to prove that the forms are indeed a linear combination of theta series associated to ternary quadratic forms. Once such interpretation is given, we extend the method of Gross–Zagier to the case where the level and the discriminant are not prime to each other to prove a Gross-type formula in this situation.

Let G be a finite abelian group, and let m > 0 with exp(G) | m. Let s_m(G) be the generalized Erdős–Ginzburg–Ziv invariant which denotes the smallest positive integer d such that any sequence of elements in G of length d contains a subsequence of length m with sum zero in G. For any integer r > 0, let be the collection of all r-uniform intersecting families of size m. Let be the smallest positive integer d such that any G-coloring of the edges of the complete r-uniform hypergraph yields a zero-sum copy of some intersecting family in . Among other results, we mainly prove that , where Ω(s_m(G)) denotes the least positive integer n such that , and we show that if r | Ω(s_m(G)) – 1 then .

We prove a functional equation for the characteristic ideal of the "big" Selmer group 𝒳(𝒯_ℱ/F_cyc) associated to an ordinary Hida family of elliptic modular forms over the cyclotomic ℤ_p extension of a general number field F, under the assumption that there is at least one arithmetic specialization whose Selmer group is torsion over its Iwasawa algebra. For a general number field, the two-variable cyclotomic Iwasawa main conjecture for ordinary Hida family is not proved and this can be thought of as an evidence to the validity of the Iwasawa main conjecture. The central idea of the proof is to prove a variant of the result of Perrin-Riou [Groupes de Selmer et accouplements; cas particulier des courbes elliptiques, Doc. Math.2003 (2003) 725–760, Extra Volume: Kazuya Kato's fiftieth birthday] by constructing a generalized pairing on the individual Selmer groups corresponding to the arithmetic points and make use of the appropriate specialization techniques of Ochiai [Euler system for Galois deformations, Ann. Inst. Fourier (Grenoble)55(1) (2005) 113–146].

In this paper, we construct several infinite families of diagonal quartic surfaces ax⁴ + by⁴ + cz⁴ + dw⁴ = 0 (where a, b, c, d are non-zero integers) with infinitely many rational points and satisfying the condition abcd is not a square. In particular, we present an infinite family of diagonal quartic surfaces defined over ℚ with Picard number equal to one and possessing infinitely many rational points. Further, we present some sextic surfaces of type ax⁶ + by⁶ + cz⁶ + dwⁱ = 0, i = 2, 3, or 6, with infinitely many rational points.

Let K be a local field with finite residue field, we define a normal form for Eisenstein polynomials depending on the choice of a uniformizer π_K and of residue representatives. The isomorphism classes of extensions generated by the polynomials in the family exhaust all totally ramified extensions, and the multiplicity with which each isomorphism class L/K appears is always smaller than the number of conjugates of L over K. An algorithm to recover the set of all special polynomials generating the extension determined by a general Eisenstein polynomial is described. We also give a criterion to quickly establish that a polynomial generates a different extension from that generated by a set of special polynomials, such criterion does not only depend on the usual distance on the set of Eisenstein polynomials considered by Krasner and others. We conclude with an algorithm for the construction of the unique special equation determining a totally ramified class field in general degree, given a suitable representation of a group of norms.

In this paper, we study a balancing problem of ordinary binomial coefficients. The diop-hantine equation

It was shown by Granville that the ABC conjecture allows one to prove asymptotic estimates on the number of squarefree values of polynomials. However, his proof gives no information on the error term of the asymptotic formula. On the ABC conjecture, we prove an asymptotic formula with error term using a different technique. From the ABC conjecture we also deduce an asymptotic formula with error term for the number of squarefree values of polynomials on certain sets of integers that are residually well distributed in a suitable sense.

The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a+b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an expression as an infinite product. This provides linear and quadratic recurrences for the hypergeometric zeta function. A family of associated polynomials is characterized as Appell polynomials and the underlying distribution is given explicitly in terms of the zeros of the associated hypergeometric function. These properties are also given a probabilistic interpretation in the framework of beta distributions.

We recall the well-known notion of the set of uniqueness for arithmetical functions, introduced by Kátai and several other mathematicians like Indlekofer, Elliot and Hoffman, independently. We define its analogue for completely additive complex-valued functions over the set of non-zero Gaussian integers with some examples. We show that the set of "Gaussian prime plus one's" along with finitely many Gaussian primes of norm up to some constant K is a set of uniqueness with respect to Gaussian integers. This is analogous to Kátai's result in the case of positive integers [I. Kátai, On sets characterizing number theoretical functions, II, Acta Arith.16 (1968) 1–14].

This paper generalizes work of Buzzard and Kilford to the case p = 3, giving an explicit bound for the overconvergence of the quotient E_κ/V(E_κ) and using this bound to prove that the eigencurve is a union of countably many annuli over the boundary of weight space.

Fix an integer d ≥ 1. In 2008, David and Weston showed that, on average, an elliptic curve over Q picks up a nontrivial p-torsion point defined over a finite extension K of the p-adics of degree at most d for only finitely many primes p. This paper proves an analogous averaging result for principally polarized abelian surfaces A over Q with real multiplication by and a level- structure. Furthermore, we indicate how the result on abelian surfaces with real multiplication relates to the deformation theory of modular Galois representations.

We construct a class of multiple Legendre polynomials and prove that they satisfy an Apéry-like recurrence. We give new upper bounds of the approximation measures of logarithms of rational numbers by algebraic numbers of bounded degree. We prove, e.g., that the nonquadraticity exponent of log 2 is bounded from above by 12.841618…, thus improving upon a recent result of the author. Our construction also yields some other known results.

We generalize the correspondence between Dirichlet series with finitely many poles that satisfy a functional equation and automorphic integrals with log-polynomial sum period functions. In particular, we extend the correspondence to hold for Dirichlet series with finitely many essential singularities. We also study Dirichlet series with infinitely many poles in a vertical strip. For Hecke groups with λ ≥ 2 and some weights, we prove a similar correspondence for these Dirichlet series. For this case, we provide a way to estimate automorphic integrals with infinite log-polynomial periods by automorphic integrals with finite log-polynomial periods.

We extend the method of Cassels for computing the Cassels–Tate pairing on the 2-Selmer group of an elliptic curve, to the case of 3-Selmer groups. This requires significant modifications to both the local and global parts of the calculation. Our method is practical in sufficiently small examples, and can be used to improve the upper bound for the rank of an elliptic curve obtained by 3-descent.

Recent works of F. G. Garvan ([Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank, Int. J. Number Theory6(12) (2010) 281–309; MR2646759 (2011j:05032)]) and Y. Yang ([Congruences of the partition function, Int. Math. Res. Not.2011(14) (2011) 3261–3288; MR2817679 (2012e:11177)] and [Modular forms for half-integral weights on SL₂(ℤ), to appear in Nagoya Math. J.]) concern a certain family of half-integral weight Hecke-invariant subspaces which arise as multiples of fixed odd powers of the Dedekind eta-function multiplied by SL₂(ℤ)-forms of fixed weight. In this paper, we study the image of Hecke operators on subspaces which arise as multiples of fixed even powers of eta multiplied by SL₂(ℤ)-forms of fixed weight.