We present an explicit construction of the Macdonald polynomials and of the Heckman–Opdam generalized Jacobi polynomials associated with root systems.

In this paper, we develop machinery to solve ternary Diophantine equations of the shape Axn + Byn = C z3 for various choices of coefficients (A, B, C). As a byproduct of this, we show, if p is prime, that the equation xn + yn = pz3 has no solutions in coprime integers x and y with |xy| > 1 and prime n > p4p2. The techniques employed enable us to classify all elliptic curves over $\mathbb{Q}$ with a rational 3-torsion point and good reduction outside the set {3, p}, for a fixed prime p.

In this paper we study a kind of spherical function, which we call a Siegel–Whittaker function, related to Fourier expansions of automorphic forms on Hermitian symmetric domains of type IV. We obtain a multiplicity-one theorem and an integral representation of this spherical function.

We give a natural construction of framed mixed Tate motives unramified over $\mathbb{Z}$ whose periods are the multiple $\zeta$-values. Namely, for each convergent multiple $\zeta$-value we define two boundary divisors A and B in the moduli space $\overline{\mathcal{M}}_{0,n+3}$ of stable curves of genus zero. The corresponding multiple zeta-motive is the nth cohomology of the pair $(\overline{\mathcal{M}}_{0,n+3}-A,B)$.

We apply Zagier's result for the traces of singular moduli to construct Borcherds products in higher level cases.

One of the main results announced in Part I (Compositio Math. 139 (2003), 197–237) is proved. The main technique consists in developing a coordinate free version of some of the theory of Serre operators on differential modular forms introduced by Barcau.

In this paper we suppose G is a finite group acting tamely on a regular projective curve $\mathcal{X}$ over $\mathbb{Z}$ and V is an orthogonal representation of G of dimension 0 and trivial determinant. Our main result determines the sign of the $\epsilon$-constant $\epsilon(\mathcal{X}/G,V)$ in terms of data associated to the archimedean place and to the crossing points of irreducible components of finite fibers of $\mathcal{X}$, subject to certain standard hypotheses about these fibers.

In this paper, we resolve a conjecture of Schäffer on the solvability of Diophantine equations of the shape $1^k + 2^k + \dotsb + x^k = y^n$, for $1 \leq k \leq 11$. Our method, which may, with a modicum of effort, be extended to higher values of k, combines a wide variety of techniques, classical and modern, in Diophantine analysis.

We give estimates for exponential sums of the form $\sum_{n \leq N}\Lambda(n)\exp(2 \pi i a g^n/m)$, where m is a positive integer, a and g are integers relatively prime to m, and $\Lambda$ is the von Mangoldt function. In particular, our results yield bounds for exponential sums of the form $\sum_{p \leq N}\exp(2 \pi i a M_p/m)$, where Mp is the Mersenne number; $M_p=2^p-1$ for any prime p. We also estimate some closely related sums, including $\sum_{n \leq N}\mu(n)\exp(2 \pi i a g^n/m)$ and $\sum_{n \leq N}\mu^2(n)\exp(2 \pi i a g^n/m)$, where $\mu$ is the Möbius function.

In this paper we show how to predict relative trace identities from the computation of Jacquet modules of the Weil representations. Many previously considered special cases of relative trace identities fit the principle we develop here, including those with important applications on L-functions. We also show how to prove these identities using the Weil representation. We give a proof of the relative trace identities between the distributions on SO(n + 1, n) and $\widetilde{\textit{Sp}}(m)$$(n\geq m)$. The proof should serve as a model to the other cases conjectured in the paper.

Let $\pi$ be a generic irreducible representation of either a symplectic group or a split special even orthogonal group over a local field of characteristic zero. We prove that $\varepsilon(\frac12,\pi,\psi)=\pi(-1)$.

We investigate the arithmetic and combinatorial significance of the values of the polynomials jn(x) defined by the q-expansion \[\sum_{n=0}^{\infty}j_n(x)q^n:=\frac{E_4(z)^2E_6(z)}{\Delta(z)}\cdot\frac{1}{j(z)-x}.\] They allow us to provide an explicit description of the action of the Ramanujan Theta-operator on modular forms. There are a substantial number of consequences for this result. We obtain recursive formulas for coefficients of modular forms, formulas for the infinite product exponents of modular forms, and new p-adic class number formulas.

We generalize a result of Anderson by showing that torsion points of certain orders cannot lie on a theta divisor in the Jacobians of hyperelliptic images of Fermat curves. The proofs utilize the explicit geometry of hyperelliptic Jacobians and their formal groups at the origin.

We determine exactly when a quadratic form is represented by a spinor genus of another quadratic form of three or four variables. We apply this to extend the embedding theorem for quaternion and also answer a question by Borovoi.

We prove various characterizations of the period torsor of abelian varieties.

We prove a relation between a generating series for the heights of Heegner cycles on the arithmetic surface associated with a Shimura curve and the second term in the Laurent expansion at s = ½ of an Eisenstein series of weight $\frac32$ for SL(2). On the geometric side, a typical coefficient of the generating series involves the Faltings heights of abelian surfaces isogenous to a product of CM elliptic curves, an archimedean contribution, and contributions from vertical components in the fibers of bad reduction. On the analytic side, these terms arise via the derivatives of local Whittaker functions. It should be noted that s = ½ is not the central point for the functional equation of the Eisenstein series in question. Moreover, the first term of the Laurent expansion at s = ½ coincides with the generating function for the degrees of the Heegner cycles on the generic fiber and, in particular, does not vanish.

In a recent paper, Rosen and Silverman showed that Tate's conjecture on algebraic cycles implies a formula of Nagao, which gives the rank of an elliptic surface in terms of a weighted average of fibral Frobenius trace values. In this article, we extend their result to the case of elliptic threefolds. The main ingredients of our argument are a Shioda–Tate-like formula for elliptic threefolds, and a relation between the ‘average’ number of rational points on singular fibers and the Galois action on those fibers.

We prove a portion of a conjecture of Conrad, Diamond, and Taylor, yielding some new cases of the Fontaine–Mazur conjectures, specifically, the modularity of certain potentially Barsotti–Tate Galois representations. The proof follows the template of Wiles, Taylor–Wiles, and Breuil–Conrad–Diamond–Taylor, and relies on a detailed study of the descent, across tamely ramified extensions, of finite flat group schemes over the ring of integers of a local field. This makes crucial use of the filtered $\phi_1$-modules of Breuil.

We give an elementary proof of the function field (or geometric) case of the Mordell–Lang conjecture in characteristic zero.

We construct multiple zeta functions as absolute tensor products of usual zeta functions. The Euler product expression is established for the most basic case $\zeta(s,\mathbf{F}_p)\otimes\zeta(s,\mathbf{F}_q)$ by using the signed double Poisson summation formula and the theory of the double sine function.