We generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of the plus and minus Coleman maps for normalised new forms of arbitrary weights and relate Pollack’s p-adic L-functions to the plus and minus Selmer groups. In addition, by generalising works of Pollack and Rubin on CM elliptic curves, we prove the ‘main conjecture’ for CM modular forms.

In this paper, we prove some one level density results for the low-lying zeros of families of L-functions. More specifically, the families under consideration are that of L-functions of holomorphic Hecke eigenforms of level 1 and weight k twisted with quadratic Dirichlet characters and that of cubic and quartic Dirichlet L-functions.

We use the theory of Kolyvagin systems to prove (most of) a refined class number formula conjectured by Darmon. We show that, for every odd prime p, each side of Darmon’s conjectured formula (indexed by positive integers n) is ‘almost’ a p-adic Kolyvagin system as n varies. Using the fact that the space of Kolyvagin systems is free of rank one over ℤp, we show that Darmon’s formula for arbitrary n follows from the case n=1, which in turn follows from classical formulas.

Let L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring ℤ(p)[G] that annihilates the p-part of the class group of L.

This work is the geometric part of our proof of the weighted fundamental lemma, which is an extension of Ngô Bao Châu’s proof of the Langlands–Shelstad fundamental lemma. Ngô’s approach is based on a study of the elliptic part of the Hichin fibration. The total space of this fibration is the algebraic stack of Hitchin bundles and its base space is the affine space of ‘characteristic polynomials’. Over the elliptic set, the Hitchin fibration is proper and the number of points of its fibers over a finite field can be expressed in terms of orbital integrals. In this paper, we study the Hitchin fibration over an open set larger than the elliptic set, namely the ‘generically regular semi-simple set’. The fibers are in general neither of finite type nor separated. By analogy with Arthur’s truncation, we introduce the substack of ξ-stable Hitchin bundles. We show that it is a Deligne–Mumford stack, smooth over the base field and proper over the base space of ‘characteristic polynomials’. Moreover, the number of points of the ξ-stable fibers over a finite field can be expressed as a sum of weighted orbital integrals, which appear in the Arthur–Selberg traceformula.

Let β∈(1,2) be a Pisot number and let Hβ denote Garsia’s entropy for the Bernoulli convolution associated with β. Garsia, in 1963, showed that Hβ<1 for any Pisot β. For the Pisot numbers which satisfy xm=xm−1+xm−2+⋯+x+1 (with m≥2), Garsia’s entropy has been evaluated with high precision by Alexander and Zagier for m=2 and later by Grabner, Kirschenhofer and Tichy for m≥3, and it proves to be close to 1. No other numerical values for Hβ are known. In the present paper we show that Hβ>0.81 for all Pisot β, and improve this lower bound for certain ranges of β. Our method is computational in nature.

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.

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 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.

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.

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.

For a number field K and a finite abelian group G, we determine the probabilities of various local completions of a random G-extension of K when extensions are ordered by conductor. In particular, for a fixed prime ℘ of K, we determine the probability that ℘ splits into r primes in a random G-extension of K that is unramified at ℘. We find that these probabilities are nicely behaved and mostly independent. This is in analogy to Chebotarev’s density theorem, which gives the probability that in a fixed extension a random prime of K splits into r primes in the extension. We also give the asymptotics for the number of G-extensions with bounded conductor. In fact, we give a class of extension invariants, including conductor, for which we obtain the same counting and probabilistic results. In contrast, we prove that neither the analogy with the Chebotarev probabilities nor the independence of probabilities holds when extensions are ordered by discriminant.

Let u(n)=f(gn), where g > 1 is integer and f(X) ∈ ℤ[X] is non-constant and has no multiple roots. We use the theory of -unit equations as well as bounds for character sums to obtain a lower bound on the number of distinct fields among for n ∈ . Fields of this type include the Shanks fields and their generalizations.

We define a topological space over the p-adic numbers, in which Euler products and Dirichlet series converge. We then show how the classical Riemann zeta function has a (p-adic) Euler product structure at the negative integers. Finally, as a corollary of these results, we derive a new formula for the non-Archimedean Euler–Mascheroni constant.

Let ρ be a two-dimensional modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that ρ has a large image and admits a low-weight crystalline modular deformation we show that any low-weight crystalline deformation of ρ unramified outside a finite set of primes will be modular. We follow the approach of Wiles as generalized by Fujiwara. The main new ingredient is an Ihara-type lemma for the local component at ρ of the middle degree cohomology of a Hilbert modular variety. As an application we relate the algebraic p-part of the value at one of the adjoint L-function associated with a Hilbert modular newform to the cardinality of the corresponding Selmer group.

The main theorem of the author’s thesis suggests that it should be possible to lift the Kolyvagin systems of Stark units, constructed by the author in an earlier paper, to a Kolyvagin system over the cyclotomic Iwasawa algebra. In this paper, we verify that this is indeed the case. This construction of Kolyvagin systems over the cyclotomic Iwasawa algebra from Stark units provides the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank (in the sense of Mazur and Rubin) is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement in the context of local Iwasawa theory, assuming the truth of the Rubin–Stark conjecture and Leopoldt’s conjecture. This statement in the local Iwasawa theory context turns out to be interesting in its own right, as it suggests a relation between the solutions to p-adic and complex Stark conjectures.

We prove that the relative class number of a nonabelian normal CM-field of degree 2pq (where p and q are two distinct odd primes) is always greater than four. Not only does this result solve the class number one problem for the nonabelian normal CM-fields of degree 42, but it has also been used elsewhere to solve the class number one problem for the nonabelian normal CM-fields of degree 84.

We break the convexity bound in the t-aspect for L-functions attached to cusp forms f for GL2(k) over arbitrary number fields k. The argument uses asymptotics with error term with a power saving, for second integral moments over spectral families of twists L(s,f⊗χ) by Grossencharacters χ, from our previous paper on integral moments.

Let k be an algebraically closed field of characteristic greater than 2, and let F=k((t)) and G=𝕊p2d. In this paper we propose a geometric analog of the Weil representation of the metaplectic group . This is a category of certain perverse sheaves on some stack, on which acts by functors. This construction will be used by Lysenko (in [Geometric theta-lifting for the dual pair S𝕆2m, 𝕊p2n, math.RT/0701170] and subsequent publications) for the proof of the geometric Langlands functoriality for some dual reductive pairs.

We obtain second integral moments of automorphic L-functions on adele groups GL2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL1 and GL2. This requires reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL2(ℚ) or GL2(ℚ(i)) with a single, coherent, global object that makes sense over a number field. This is the first expression of integral moments in adele-group terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers ℚ, we recover the classical results on moments.