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The theoretical background that led to the discovery and excavation of the Late Mesolithic semi-subterranean dwelling at Vuollerim is presented. This is followed by an interpretation of the social function of the house. A comparison with contemporary dwellings is made and the cultural context of the Vuollerim site is explored. The paper ends with some observations and thoughts concerning the impact that the archaeological discovery in Vuollerim had on its present day inhabitants and how it changed their view of themselves, their history and their future.

We study GL_2(F)-invariant periods on representations of GL_2(A), where F is a nonarchimedean local field and A/F a product of field extensions of total degree 3. For irreducible representations, a theorem of Prasad shows that the space of such periods has dimension at most 1, and is non-zero when a certain epsilon-factor condition holds. We give an extension of this result to a certain class of reducible representations (of Whittaker type), extending results of Harris--Scholl when A is the split algebra F x F x F., Comment: Revised version, to appear in Forum Mathematicum

We construct an Euler system associated to regular algebraic, essentially conjugate self-dual cuspidal automorphic representations of GL3 over imaginary quadratic fields, using the cohomology of Shimura varieties for GU(2,1)., Mathematische Annalen, 382 (3-4), ISSN:1432-1807, ISSN:0025-5831

The Asai (or twisted tensor) $L$-function of a Bianchi modular form $\Psi$ is the $L$-function attached to the tensor induction to $\mathbb{Q}$ of its associated Galois representation. In this paper, when $\Psi$ is ordinary at $p$ we construct a $p$-adic analogue of this $L$-function: that is, a $p$-adic measure on $\mathbb{Z}_p^\times$ that interpolates the critical values of the Asai $L$-function twisted by Dirichlet characters of $p$-power conductor. The construction uses techniques analogous to those used by Lei, Zerbes and the first author in order to construct an Euler system attached to the Asai representation of a quadratic Hilbert modular form., Comment: Final version, to appear in Algebra & Number Theory

In this paper we make a systematic study of certain motivic cohomology classes ("Rankin-Eisenstein classes") attached to the Rankin--Selberg convolution of two modular forms of weight $\ge 2$. The main result is the computation of the $p$-adic syntomic regulators of these classes. As a consequence we prove many cases of the Perrin-Riou conjecture for Rankin--Selberg convolutions of cusp forms., Comment: Updated version with minor corrections. To appear in Amer. J. Math

The arithmetic of Hilbert modular forms has been extensively studied under the assumption that the forms concerned are "paritious" -- all the components of the weight are congruent modulo 2. In contrast, non-paritious Hilbert modular forms have been relatively little studied, both from a theoretical and a computational standpoint. In this article, we aim to redress the balance somewhat by studying the arithmetic of non-paritious Hilbert modular eigenforms. On the theoretical side, our starting point is a theorem of Patrikis, which associates projective l-adic Galois representations to these forms. We show that a general conjecture of Buzzard and Gee actually predicts that a strengthening of Patrikis' result should hold, giving Galois representations into certain groups intermediate between GL(2) and PGL(2), and we verify that the predicted Galois representations do indeed exist. On the computational side, we give an algorithm to compute non-paritious Hilbert modular forms using definite quaternion algebras. To our knowledge, this is the first time such a general method has been presented. We end the article with a selection of examples., Final version, to appear in Math Zeitschrift

We use the "higher Hida theory" recently introduced by the second author to p-adically interpolate periods of non-holomorphic automorphic forms for GSp(4), contributing to coherent cohomology of Siegel threefolds in positive degrees. We apply this new method to construct p-adic L-functions associated to the degree 4 (spin) L-function of automorphic representations of GSp(4), and the degree 8 L-function of GSp(4) x GL(2)., Final version, now published in "Duke Math Journal"

We construct two-parameter analytic families of Galois cohomology classes interpolating the etale Abel--Jacobi images of generalised Heegner cycles, with both the modular form and Grossencharacter varying in p-adic families., Comment: 23 pages. Revised version, to appear in Proc. London Math. Soc

Let $f$ and $g$ be two modular forms which are non-ordinary at $p$. The theory of Beilinson-Flach elements gives rise to four rank-one non-integral Euler systems for the Rankin-Selberg convolution $f \otimes g$, one for each choice of $p$-stabilisations of $f$ and $g$. We prove (modulo a hypothesis on non-vanishing of $p$-adic $L$-fuctions) that the $p$-parts of these four objects arise as the images under appropriate projection maps of a single class in the wedge square of Iwasawa cohomology, confirming a conjecture of Lei-Loeffler-Zerbes. Furthermore, we define an explicit logarithmic matrix using the theory of Wach modules, and show that this describes the growth of the Euler systems and $p$-adic $L$-functions associated to $f \otimes g$ in the cyclotomic tower. This allows us to formulate "signed" Iwasawa main conjectures for $f\otimes g$ in the spirit of Kobayashi's $\pm$-Iwasawa theory for supersingular elliptic curves; and we prove one inclusion in these conjectures under our running hypotheses., We have expanded the introduction and corrected a few minor imprecisions

In this paper, we define a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of $\mathbf{Q}_p$, over a Galois extension of $\mathbf{Q}_p$ whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localisation at the primes above p are extensions of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a conjecture on the existence of a "zeta element", whose image under the regulator map is a p-adic L-function. We show that this conjecture implies the known properties of the 2-variable p-adic L-functions constructed by Perrin-Riou and Kim., Final version, to appear in Int. J. Number Theory

Euler systems are certain compatible families of cohomology classes, which play a key role in studying the arithmetic of Galois representations. We briefly survey the known Euler systems, and recall a standard conjecture of Perrin-Riou predicting what kind of Euler system one should expect for a general Galois representation. Surprisingly, several recent constructions of Euler systems do not seem to fit the predictions of this conjecture, and we formulate a more general conjecture which explains these extra objects. The novel aspect of our conjecture is that it predicts that there should often be Euler systems of several different ranks associated to a given Galois representation, and we describe how we expect these objects to be related., Comment: Expository article, 13 pages. To appear in proceedings of the Iwasawa 2017 conference

We show that the image of the adelic Galois representation attached to a non-CM modular form is open in the adelic points of a suitable algebraic group. We also show a similar result for the adelic Galois representation attached to a finite set of modular forms., Updated version, to appear in Glasgow J Math

We construct an Euler system for Galois representations associated to cohomological cuspidal automorphic representations of the group GSp(4), using the pushforwards of Eisenstein classes for GL(2) x GL(2)., Comment: 41 pages. To appear in "Journal of the European Mathematical Society" (see https://www.ems-ph.org/journals/forthcoming.php?jrn=jems )

We show that the Euler system associated to Rankin--Selberg convolutions of modular forms, introduced in our earlier works with Lei and Kings, varies analytically as the modular forms vary in $p$-adic Coleman families. We prove an explicit reciprocity law for these families, and use this to prove cases of the Bloch--Kato conjecture for Rankin--Selberg convolutions., Updated version, to appear in "Research in the Mathematical Sciences" (Robert Coleman memorial volume)

This paper is a sequel to our earlier paper "Wach modules and Iwasawa theory for modular forms" (arXiv: 0912.1263), where we defined a family of Coleman maps for a crystalline representation of the Galois group of Qp with nonnegative Hodge-Tate weights. In this paper, we study these Coleman maps using Perrin-Riou's p-adic regulator L_V. Denote by H(\Gamma) the algebra of Qp-valued distributions on \Gamma = Gal(Qp(\mu (p^\infty) / Qp). Our first result determines the H(\Gamma)-elementary divisors of the quotient of D_{cris}(V) \otimes H(\Gamma) by the H(\Gamma)-submodule generated by (\phi * N(V))^{\psi = 0}, where N(V) is the Wach module of V. By comparing the determinant of this map with that of L_V (which can be computed via Perrin-Riou's explicit reciprocity law), we obtain a precise description of the images of the Coleman maps. In the case when V arises from a modular form, we get some stronger results about the integral Coleman maps, and we can remove many technical assumptions that were required in our previous work in order to reformulate Kato's main conjecture in terms of cotorsion Selmer groups and bounded p-adic L-functions., Comment: 27 pages

For a crystalline p-adic representation of the absolute Galois group of Qp, we define a family of Coleman maps (linear maps from the Iwasawa cohomology of the representation to the Iwasawa algebra), using the theory of Wach modules. Let f = sum(a_n q^n) be a normalized new modular eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f, we define two Coleman maps with values in the Iwasawa algebra of Zp^* (after extending scalars to some extension of Qp). Applying these maps to the Kato zeta elements gives a decomposition of the (generally unbounded) p-adic L-functions of f into linear combinations of two power series of bounded coefficients, generalizing works of Pollack (in the case a_p=0) and Sprung (when f corresponds to a supersingular elliptic curve). Using ideas of Kobayashi for elliptic curves which are supersingular at p, we associate to each of these power series a cotorsion Selmer group. This allows us to formulate a "main conjecture". Under some technical conditions, we prove one inclusion of the "main conjecture" and show that the reverse inclusion is equivalent to Kato's main conjecture., Comment: 44 pages. To appear in the Asian Journal of Mathematics

We study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic Zp-extension of Q under the assumption that f is non-ordinary at p. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using p-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara and Sprung for supersingular elliptic curves., To appear in Canad. J. Math

In this paper, we prove the “local $\varepsilon$ -isomorphism” conjecture of Fukaya and Kato for a particular class of Galois modules, obtained by interpolating the twists of a fixed crystalline representation of $G_{ \mathbf {Q}_{p}}$ by a family of characters of $G_{ \mathbf {Q}_{p}}$ . This can be regarded as a local analogue of the Iwasawa main conjecture for abelian $p$ -adic Lie extensions of $ \mathbf {Q}_{p}$ , extending earlier work of Kato for rank one modules and of Benois and Berger for the cyclotomic extension. We show that such an $\varepsilon$ -isomorphism can be constructed using the 2-variable version of the Perrin-Riou regulator map constructed by the first and third authors.

We construct three-variable $p$-adic families of Galois cohomology classes attached to Rankin convolutions of modular forms, and prove an explicit reciprocity law relating these classes to critical values of L-functions. As a consequence, we prove finiteness results for the Selmer group of an elliptic curve twisted by a 2-dimensional odd irreducible Artin representation when the associated $L$-value does not vanish., Final version, to appear in Cambridge J Math; small correction to acknowlegements

We construct a compatible family of global cohomology classes (an Euler system) for the symmetric square of a modular form, and apply this to bounding Selmer groups of the symmetric square Galois representation and its twists., Comment: Updated version, to appear in J Reine Angew Math

We give a computationally effective criterion for determining whether a finite-index subgroup of $\mathrm{SL}_2(\mathbf{Z})$ is a congruence subgroup, extending earlier work of Hsu for subgroups of $\mathrm{PSL}_2(\mathbf{Z})$.

We study the theory of p-adic finite-order functions and distributions on ray class groups of number fields, and apply this to the construction of (possibly unbounded) p-adic L-functions for automorphic forms on \(\mathop{\text{GL}}\nolimits _{2}\) which may be non-ordinary at the primes above p. As a consequence, we obtain a “plus-minus” decomposition of the p-adic L-functions of automorphic forms for \(\mathop{\text{GL}}\nolimits _{2}\) over an imaginary quadratic field with p split and Hecke eigenvalues 0 at the primes above p, confirming a conjecture of B.D. Kim.

These are the notes of a five-lecture course presented at the Computations with Modular Forms summer school, aimed at graduate students in number theory and related areas. Sections 1–4 give a sketch of the theory of reductive algebraic groups over Q, and of Gross’s purely algebraic definition of automorphic forms in the special case when G(R) is compact. Sections 5–9 describe how these automorphic forms can be explicitly computed, concentrating on the case of definite unitary groups; and Sects. 10 and 11 describe how to relate the results of these computations to Galois representations, and present some examples where the corresponding Galois representations can be identified, giving illustrations of various instances of Langlands’ functoriality conjectures.

We construct an Euler system attached to a weight 2 modular form twisted by a Groessencharacter of an imaginary quadratic field, and apply this to bounding Selmer groups., Comment: Revised version, to appear in Compositio Math

We construct an Euler system in the cohomology of the tensor product of the Galois representations attached to two modular forms, using elements in the higher Chow groups of products of modular curves. We use this Euler system to prove a finiteness theorem for the strict Selmer group of the Galois representation when the associated p-adic Rankin--Selberg L-function is non-vanishing at s = 1., Revised version with many updates and corrections

For ordinary modular forms, there are two constructions of a p-adic L-function attached to the non-unit root of the Hecke polynomial, which are conjectured but not known to coincide. We prove this conjecture for modular forms of CM type, by calculating the the critical-slope L-function arising from Kato's Euler system and comparing this with results of Bellaiche on the critical-slope L-function defined using overconvergent modular symbols., 14 pages

Let G be a reductive algebraic group over a number field k. It is shown how Emerton's methods may be applied to the problem of p-adically interpolating the metaplectic forms on G, i.e. the automorphic forms on metaplectic covers of G, as long as the metaplectic covers involved split at the infinite places of k., Comment: 37 pages

Proposition 4.1 (1) of this article is not correct as stated. For instance, if e is the quadratic Dirichlet character of conductor 15, then there are two newforms in Sk(Γ1(15), e) with coefficients in Q, and these are twists of each other by the quadratic character of conductor 3, whereas the quantity u in the proposition is 0 here. I am grateful to Steve Donnelly for pointing out this example. The correct statement is the following. Let f be a newform of level Np, with p N , and character e = eNep, where eN has conductor prime to p and the conductor of ep is p . Let u = min(b r 2c, r − c).

In this short note, we study the geometry of the eigenvariety parametrising p-adic automorphic forms for GL(1) over a number field K, as constructed by Buzzard. We show that if K is not totally real and contains no CM subfield, points in this space arising from classical automorphic forms (i.e. algebraic Grossencharacters of K) are not Zariski-dense in the eigenvariety (as a rigid space); but the eigenvariety posesses a natural formal scheme model, and the set of classical points is Zariski-dense in the formal scheme. We also sketch the theory for GL(2) over an imaginary quadratic field, following Calegari and Mazur, emphasising the strong formal similarity with the case of GL(1) over a general number field., 6 pages

Osteochondral fracture of lateral femoral condyle can result from support and other twisting injuries of the knee. Arthroscopy is a better diagnostic and therapeutic tool as standard radiographs can mislead regarding the size and location of the fragment. If fragment is large and displaced, arthrotomy may become necessary. We present a case of large osteochondral fracture of lateral femoral condyle involving the entire articular surface in a 12 year old male. This was treated with open reduction and internal fixation using biodegradable screws. MRI at six weeks after surgery showed satisfactory results. After the surgery full weight bearing was allowed at three months.

This paper studies Emerton's Jacquet module functor for locally analytic representations of p-adic reductive groups. When P is a parabolic subgroup whose Levi factor M is not commutative, we show that passing to an isotypical subspace for the derived subgroup of M gives rise to essentially admissible locally analytic representations of the torus Z(M), which have a natural interpretation in terms of rigid geometry. We use this to extend Emerton's representation-theoretic construction of eigenvarieties by constructing eigenvarieties interpolating automorphic representations whose local components at p are not necessarily principal series., Comment: 25 pages

We present an algorithm for computing the $p$-component of the automorphic representation arising from a cuspidal newform $f$ for a prime $p$. This is equivalent to computing the restriction to the decomposition group at $p$ of the $\ell$-adic Galois representations attached to $f$ for any $\ell\neq p$. The situation is most interesting when $p^2$ divides the level of $f$, in which case the $p$-component could be supercuspidal. In the supercuspidal case, the local component is induced from an irreducible character of a compact-mod-center subgroup of $\text{GL}_2(\mathbf{Q}_p)$; our algorithm outputs both the group and the irreducible character. We provide examples which illustrate how the local Galois representation can be completely read off from the local component., Comment: 21 pages

I present a general theory of overconvergent p-adic automorphic forms and eigenvarieties for connected reductive algebraic groups G whose real points are compact modulo centre, extending earlier constructions due to Buzzard, Chenevier and Yamagami for forms of GL(n). This leads to some new phenomena, including the appearance of intermediate spaces of "semi-classical" automorphic forms; this gives a hierarchy of interpolation spaces (eigenvarieties) interpolating classical automorphic forms satisfying different finite slope conditions (corresponding to a choice of parabolic subgroup of G at p). The construction of these spaces relies on methods of locally analytic representation theory, combined with the theory of compact operators on Banach modules., Comment: Updated Dec 2016: includes corrigendum fixing an error in section 2.6 of the published version

The main result of this paper is an instance of the conjecture made by Gouvea and Mazur (Math. Res. Lett., 1995) which asserts that for certain values of r the space of r-overconvergent p-adic modular forms of tame level N and weight k should be spanned by the finite slope Hecke eigenforms. Using methods adapted from the work of Buzzard and Calegari I show that for p=2, k = 0, N = 1, this holds for all r in (5/12, 7/12). The proof relies on a certain factorisation of the U_p operator which is known in this case but I conjecture also holds for p = 3 and 5; this conjecture also implies exact formulae for the set of slopes similar to those proved for p=2 by Buzzard and Calegari. The same methods also provide an efficient approach to explicit computations of q-expansions of small slope overconvergent eigenforms, extending the computations of Gouvea and Mazur., Comment: Updated version fixing some minor typos

Nekovar and Niziol have introduced in [arxiv:1309.7620] a version of syntomic cohomology valid for arbitrary varieties over p-adic fields. This uses a mapping cone construction similar to the rigid syntomic cohomology of the first author in the good-reduction case, but with Hyodo--Kato (log-crystalline) cohomology in place of rigid cohomology. \ud In this short note, we describe a cohomology theory which is a modification of the theory of Nekovar and Niziol, modified by replacing 1 - Phi (where Phi is the Frobenius map) with other polynomials in Phi. This is the analogue for general varieties of the finite-polynomial cohomology defined by the first author for varieties with good reduction. We use this cohomology theory to give formulae for p-adic regulator maps on curves or products of curves, without imposing any good reduction hypotheses.