We develop an explicit descent theory in the context of Whitehead groups of non-commutative Iwasawa algebras. We apply this theory to describe the precise connection between main conjectures of non-commutative Iwasawa theory (in the spirit of Coates, Fukaya, Kato, Sujatha and Venjakob) and the equivariant Tamagawa number conjecture. The latter result is both a converse to a theorem of Fukaya and Kato and also provides an important means of deriving explicit consequences of the main conjecture and proving special cases of the equivariant Tamagawa number conjecture.

For a closed d-dimensional subvariety X of an abelian variety A and a canonically metrized line bundle L on A, Chambert-Loir has introduced measures c1(L∣X)∧d on the Berkovich analytic space associated to A with respect to the discrete valuation of the ground field. In this paper, we give an explicit description of these canonical measures in terms of convex geometry. We use a generalization of the tropicalization related to the Raynaud extension of A and Mumford’s construction. The results have applications to the equidistribution of small points.

We determine the real counting function N(q) (q∈[1,∞)) for the hypothetical ‘curve’ over 𝔽1, whose corresponding zeta function is the complete Riemann zeta function. We show that such a counting function exists as a distribution, is positive on (1,∞) and takes the value −∞ at q=1 as expected from the infinite genus of C. Then, we develop a theory of functorial 𝔽1-schemes which reconciles the previous attempts by Soulé and Deitmar. Our construction fits with the geometry of monoids of Kato, is no longer limited to toric varieties and it covers the case of schemes associated with Chevalley groups. Finally we show, using the monoid of adèle classes over an arbitrary global field, how to apply our functorial theory of -schemes to interpret conceptually the spectral realization of zeros of L-functions.

We prove a mean-value result for derivatives of L-functions at the center of the critical strip for a family of forms obtained by twisting a fixed form by quadratic characters with modulus which can be represented as sum of two squares. Such a family of forms is related to elliptic fibrations given by the equation q(t)y2=f(x) where q(t)=t2+1 and f(x) is a cubic polynomial. The aim of the paper is to establish a prototype result for such quadratic families. Though our method can be generalized to prove similar results for any positive definite quadratic form in place of sum of two squares, we refrain from doing so to keep the presentation as clear as possible.

Let A be an n×m matrix with real entries. Consider the set BadA of x∈[0,1)n for which there exists a constant c(x)>0 such that for any q∈ℤm the distance between x and the point {Aq} is at least c(x)|q|−m/n. It is shown that the intersection of BadA with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.

Let {En} be the Euler numbers. We give a general congruence modulo 2(m+2)n for E2mk+b, where k,m,n are positive integers and b∈{0,2,4,…}.

Given a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

We study the automorphism groups of the reduction of a modular curve X0(N) over primes p∤N.

When p is a prime number, and k1,…,kt are natural numbers with 1≤k1<k2<⋯<kt<p, we show that the simultaneous congruences ∑ t1xkji≡∑ t1ykjimod p (1≤j≤t) possess at most k1⋯ktpt solutions with 1≤xi,yi≤p (1≤i≤t). Analogous conclusions are provided when one or more of the exponents ki is negative.

We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V𝔭(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G𝔭(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G𝔭(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V𝔭(M). The second states that the connected component of G𝔭(M) is reductive if M is semisimple and has a separable endomorphism algebra.

Let ℓ be a prime number. It is not known whether every finite ℓ-group of rank n≥1 can be realized as a Galois group over with no more than n ramified primes. We prove that this can be done for the (minimal) family of finite ℓ-groups which contains all the cyclic groups of ℓ-power order and is closed under direct products, (regular) wreath products and rank-preserving homomorphic images. This family contains the Sylow ℓ-subgroups of the symmetric groups and of the classical groups over finite fields of characteristic not ℓ. On the other hand, it does not contain all finite ℓ-groups.

In this paper, we introduce cell-forms on 𝔐0,n, which are top-dimensional differential forms diverging along the boundary of exactly one cell (connected component) of the real moduli space 𝔐0,n(ℝ). We show that the cell-forms generate the top-dimensional cohomology group of 𝔐0,n, so that there is a natural duality between cells and cell-forms. In the heart of the paper, we determine an explicit basis for the subspace of differential forms which converge along a given cell X. The elements of this basis are called insertion forms; their integrals over X are real numbers, called cell-zeta values, which generate a ℚ-algebra called the cell-zeta algebra. By a result of F. Brown, the cell-zeta algebra is equal to the algebra of multizeta values. The cell-zeta values satisfy a family of simple quadratic relations coming from the geometry of moduli spaces, which leads to a natural definition of a formal version of the cell-zeta algebra, conjecturally isomorphic to the formal multizeta algebra defined by the much-studied double shuffle relations.

In a previous paper, the potential automorphy of certain Galois representations to GLn for n even was established, following the work of Harris, Shepherd–Barron and Taylor and using the lifting theorems of Clozel, Harris and Taylor. In this paper, we extend those results to n=3 and n=5, and conditionally to all other odd n. The key additional tools necessary are results which give the automorphy or potential automorphy of symmetric powers of elliptic curves, most notably those of Gelbert, Jacquet, Kim, Shahidi and Harris.

We discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We obtain such elements by evaluating rational functions on elliptic curves, at points whose order is small with respect to their degree. We discuss several special cases, including an old construction of Wiedemann, giving the first nontrivial estimate for the order of the elements in this construction.

Let X be a projective cubic hypersurface of dimension 11 or more, which is defined over ℚ. We show that X(ℚ) is non-empty provided that the cubic form defining X can be written as the sum of two forms that share no common variables.

We generalise results of Buzzard, Taylor and Kassaei on analytic continuation of p-adic overconvergent eigenforms over ℚ to the case of p-adic overconvergent Hilbert eigenforms over totally real fields F, under the assumption that p splits completely in F. This includes weight-one forms and has applications to generalisations of Buzzard and Taylor’s main theorem. Next, we follow an idea of Kassaei’s to generalise Coleman’s well-known result that ‘an overconvergent Up-eigenform of small slope is classical’ to the case of p-adic overconvergent Hilbert eigenforms of Iwahori level.

Abstract. Let $H$ be the Hilbert class field of a $\text{CM}$ number field $K$ with maximal totally real subfield $F$ of degree $n$ over $\mathbb{Q}$ . We evaluate the second term in the Taylor expansion at $s\,=\,0$ of the Galois-equivariant $L$ -function ${{\Theta }_{{{S}_{\infty }}\,}}\left( s \right)$ associated to the unramified abelian characters of $\text{Gal}\left( H/K \right)$ . This is an identity in the group ring $\mathbb{C}\left[ \text{Gal}\left( H/K \right) \right]$ expressing $\Theta _{{{S}_{\infty }}}^{(n)}\,\left( 0 \right)$ as essentially a linear combination of logarithms of special values $\left\{ \Psi ({{z}_{\sigma }}) \right\}$ , where $\Psi :\,{{\mathbb{H}}^{n}}\,\to \,\mathbb{R}$ is a Hilbert modular function for a congruence subgroup of $S{{L}_{2}}\left( {{\mathcal{O}}_{F}} \right)$ and $\left\{ {{z}_{\sigma }}\,:\,\sigma \,\in \,\text{Gal}\left( H/K \right) \right\}$ are $\text{CM}$ points on a universal Hilbert modular variety. We apply this result to express the relative class number ${{h}_{H}}/{{h}_{K}}$ as a rational multiple of the determinant of an $\left( {{h}_{K}}\,-\,1 \right)\,\times \,\left( {{h}_{K}}\,-\,1 \right)$ matrix of logarithms of ratios of special values $\Psi ({{z}_{\sigma }})$ , thus giving rise to candidates for higher analogs of elliptic units. Finally, we obtain a product formula for $\Psi ({{z}_{\sigma }})$ in terms of exponentials of special values of $L$ -functions.

We introduce a differential operator on quasimodular polynomials that corresponds to the derivative operator on quasimodular forms. We then prove that such a differential operator is compatible with a heat operator on Jacobi-like forms in certain cases. These results show in those cases that the derivative operator on quasimodular forms corresponds to a heat operator on Jacobi-like forms.

Following R. A. Rankin’s method, D. Zagier computed the nth Rankin–Cohen bracket of a modular form g of weight k1 with the Eisenstein series of weight k2, computed the inner product of this Rankin–Cohen bracket with a cusp form f of weight k=k1+k2+2n and showed that this inner product gives, up to a constant, the special value of the Rankin–Selberg convolution of f and g. This result was generalized to Jacobi forms of degree 1 by Y. Choie and W. Kohnen. In this paper, we generalize this result to Jacobi forms defined over ℋ×ℂ(g,1).

In Boyer [Monodromy of perverse sheaves on vanishing cycles on some Shimura varieties, Invent. Math. 177 (2009), 239–280 (in French)], a sheaf version of the monodromy-weight conjecture for some unitary Shimura varieties was proved by giving explicitly the monodromy filtration of the complex of vanishing cycles in terms of local systems introduced in Harris and Taylor [The geometry and cohomology of some simple Shimura varieties (Princeton University Press, Princeton, NJ, 2001)]. The main result of this paper is the cohomological version of the monodromy-weight conjecture for these Shimura varieties, which we prove by means of an explicit description of the groups of cohomology in terms of automorphic representations and the local Langlands correspondence.