Using elementary means, we derive an explicit formula for a3(n), the number of 3-core partitions of n, in terms of the prime factorization of 3n+1. Based on this result, we are able to prove several infinite families of arithmetic results involving a3(n), one of which specializes to the recent result of Baruah and Berndt which states that, for all n≥0, a3(4n+1)=a3(n).

Qualified difference sets (QDS) composed of unions of cyclotomic classes are discussed. An exhaustive computer search for such QDS and modified QDS that also possess the zero residue has been conducted for all powers n=4,6,8 and 10. Two new families were discovered in the case n=8 and some new isolated systems were discovered for n=6 and n=10.

Given an integer n≥2, let λ(n):=(log n)/(log γ(n)), where γ(n)=∏ p∣np, denote the index of composition of n, with λ(1)=1. Letting ϕ and σ stand for the Euler function and the sum of divisors function, we show that both λ(ϕ(n)) and λ(σ(n)) have normal order 1 and mean value 1. Given an arbitrary integer k≥2, we then study the size of min {λ(ϕ(n)),λ(ϕ(n+1)),…,λ(ϕ(n+k−1))} and of min {λ(σ(n)),λ(σ(n+1)),…,λ(σ(n+k−1))} as n becomes large.

We continue our study of the reduction of PEL Shimura varieties with parahoric level structure at primes p at which the group defining the Shimura variety ramifies. We describe ‘good’ p-adic integral models of these Shimura varieties and study their étale local structure. In the present paper we mainly concentrate on the case of unitary groups for a ramified quadratic extension. Some of our results are applications of the theory of twisted affine flag varieties that we developed in a previous paper.

In this paper, we consider certain classes of Eisenstein-type series involving hyperbolic functions, and prove some formulas for them which can be regarded as relevant analogues of our previous results. We can also regard these formulas as certain generalizations of the famous formulas for the ordinary Eisenstein series given by Hurwitz.

We study the Gowers norm for periodic binary sequences and relate it to correlation measures for such sequences. The case of periodic binary sequences derived from inversive pseudorandom numbers is considered in detail.

We compute the spherical functions on the symmetric space Sp2n/Spn×Spn and derive a Plancherel formula for functions on the symmetric space. As an application of the Plancherel formula, we prove an identity which amounts to the fundamental lemma of a relative trace identity between Sp2n and .

The main goal of this paper is to provide asymptotic expansions for the numbers #{p≤x:p prime,sq(p)=k} for k close to ((q−1)/2)log qx, where sq(n) denotes the q-ary sum-of-digits function. The proof is based on a thorough analysis of exponential sums of the form (where the sum is restricted to p prime), for which we have to extend a recent result by the second two authors.

An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

Kudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura varieties to Fourier coefficients of Eisenstein series. The lowest dimensional case, in which one intersects two codimension one cycles on the integral model of a Shimura curve, has been completed by Kudla, Rapoport and Yang. In the present paper we prove results in a higher dimensional setting. On the integral model of a Shimura surface we consider the intersection of a Shimura curve with a codimension two cycle of complex multiplication points, and relate the intersection to certain cycle classes constructed by Kudla, Rapoport and Yang. As a corollary we deduce that our intersection multiplicities appear as Fourier coefficients of a Hilbert modular form of half-integral weight.

Let K be a real quadratic number field and let p be a prime number which is inert in K. We denote the completion of K at the place p by Kp. We propose a p-adic construction of special elements in Kp× and formulate the conjecture that they should be p-units lying in narrow ray class fields of K. The truth of this conjecture would entail an explicit class field theory for real quadratic number fields. This construction can be viewed as a natural generalization of a construction obtained by Darmon and Dasgupta who proposed a p-adic construction of p-units lying in narrow ring class fields of K.

Let q≥2 and N≥1 be integers. W. Zhang recently proved that for any fixed ε>0 and qε≤N≤q1/2−ε, where the sum is taken over all nonprincipal characters χ modulo q, L(1,χ) denotes the L-functions corresponding to χ, and αq=qo(1) is some explicit function of q. Here we improve this result and show that the same asymptotic formula holds in the essentially full range qε≤N≤q1−ε.

Let E/ℚ be an elliptic curve and p a prime of supersingular reduction for E. Denote by the anticyclotomic ℤp-extension of an imaginary quadratic field K which satisfies the Heegner hypothesis. Assuming that p splits in K/ℚ, we prove that has trivial Λ-corank and, in the process, also show that and both have Λ-corank two.

Suppose that G is an abelian group and that A ⊂ G is finite and contains no non-trivial three-term arithmetic progressions. We show that |A+A| »ε|A|(log|A|)⅓−ε.

A problem posed in the early eighteenth century asks for right-angled triangles, each of whose sides exceeds double the area by a perfect square. We summarize known results and find such triangles with the smallest possible standard generators.

Let $p$ be a prime, and let $f:\mathbb{Z}/p\mathbb{Z}\to \mathbb{R}$ be a function with $\mathbb{E}f=0$ and $||\hat{f}|{{|}_{1}}\le 1$. Then ${{\min }_{x\in \mathbb{Z}/p\mathbb{Z}}}|f\left( x \right)|=O{{\left( \log p \right)}^{-1/3+\in }}$. One should think of $f$ as being “approximately continuous”; our result is then an “approximate intermediate value theorem”.

As an immediate consequence we show that if $A\subseteq \mathbb{Z}/p\mathbb{Z}$ is a set of cardinality $\left\lfloor {p}/{2}\; \right\rfloor $, then ${{\sum }_{r}}\widehat{|\,{{1}_{A}}}\left( r \right)|\gg {{\left( \log p \right)}^{1/3-\in }}$. This gives a result on a “$\,\bmod \,p$” analogue of Littlewood's well-known problem concerning the smallest possible ${{L}^{1}}$-norm of the Fourier transform of a set of $n$ integers.

Another application is to answer a question of Gowers. If $A\,\subseteq \,{\mathbb{Z}}/{p\mathbb{Z}}\;$ is a set of size $\left\lfloor {p}/{2}\; \right\rfloor $, then there is some $x\,\in \,\mathbb{Z}/p\mathbb{Z}$ such that

$$||A\cap \left( A+x \right)\,-\,p/4|\,=o\left( p \right).$$

We prove quantitative versions of the Balog–Szemerédi–Gowers and Freiman theorems in the model case of a finite field geometry 𝔽2n, improving the previously known bounds in such theorems. For instance, if is such that ∣A+A∣≤K∣A∣ (thus A has small additive doubling), we show that there exists an affine subspace H of 𝔽2n of cardinality such that . Under the assumption that A contains at least ∣A∣3/K quadruples with a1+a2+a3+a4=0, we obtain a similar result, albeit with the slightly weaker condition ∣H∣≫K−O(K)∣A∣.

We prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL2(AF) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R=T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.

In this article, we explore a beautiful idea of Skinner and Wiles in the context of GSp(4) over a totally real field. The main result provides congruences between automorphic forms which are Iwahori-spherical at a certain place ω, and forms with a tamely ramified principal series at ω, Thus, after base change to a finite solvable totally real extension, one can often lower the level at ω. For the proof, we first establish an analogue of the Jacquet–Langlands correspondence, using the stable trace formula. The congruences are then obtained on inner forms, which are compact at infinity modulo the centre, and split at all the finite places. The crucial ingredient allowing us to do so, is an important result of Roche on types for principal series representations of split reductive groups.

We associate two almost Cp-representations to a (ϕ,Γ)-module, and we compute their dimensions and heights. As a corollary, we get a full faithfulness result for Be-representations.