We study the values of finite multiple harmonic $q$-series at a primitive root of unity and show that these specialize to the finite multiple zeta value (FMZV) and the symmetric multiple zeta value (SMZV) through an algebraic and analytic operation, respectively. Further, we prove the duality formula for these values, as an example of linear relations, which induce those among FMZVs and SMZVs simultaneously. This gives evidence towards a conjecture of Kaneko and Zagier relating FMZVs and SMZVs. Motivated by the above results, we define cyclotomic analogues of FMZVs, which conjecturally generate a vector space of the same dimension as that spanned by the finite multiple harmonic $q$-series at a primitive root of unity of sufficiently large degree.

Let $G\subseteq \widetilde{G}$ be two quasisplit connected reductive groups over a local field of characteristic zero and having the same derived group. Although the existence of L-packets is still conjectural in general, it is believed that the L-packets of $G$ should be the restriction of those of $\widetilde{G}$ . Motivated by this, we hope to construct the L-packets of $\widetilde{G}$ from those of $G$ . The primary example in our mind is when $G=\text{Sp}(2n)$ , whose L-packets have been determined by Arthur [The endoscopic classification of representations: orthogonal and symplectic groups, Colloquium Publications, vol. 61 (American Mathematical Society, Providence, RI, 2013)], and $\widetilde{G}=\text{GSp}(2n)$ . As a first step, we need to consider some well-known conjectural properties of L-packets. In this paper, we show how they can be deduced from the conjectural endoscopy theory. As an application, we obtain some structural information about L-packets of $\widetilde{G}$ from those of  $G$ .

We study the asymptotical behaviour of the moduli space of morphisms of given anticanonical degree from a rational curve to a split toric variety, when the degree goes to infinity. We obtain in this case a geometric analogue of Manin’s conjecture about rational points of bounded height on varieties defined over a global field. The study is led through a generating series whose coefficients lie in a Grothendieck ring of motives, the motivic height zeta function. In order to establish convergence properties of this function, we use a notion of motivic Euler product. It relies on a construction of Denef and Loeser which associates a virtual motive to a first order logic ring formula.

We prove that, given a smooth projective curve C of genus g≥2, the forgetful morphism (respectively ) from the moduli space of orthogonal (respectively symplectic) bundles to the moduli space of all vector bundles over C is an embedding. Our proof relies on an explicit description of a set of generators for the polynomial invariants on the representation space of a quiver under the action of a product of classical groups.

We set up a formalism of endoscopy for metaplectic groups. By defining a suitable transfer factor, we prove an analogue of the Langlands–Shelstad transfer conjecture for orbital integrals over any local field of characteristic zero, as well as the fundamental lemma for units of the Hecke algebra in the unramified case. This generalizes prior work of Adams and Renard in the real case and serves as a first step in studying the Arthur–Selberg trace formula for metaplectic groups.

Let X⊂P be a variety (respectively an open subset of an analytic submanifold) and let x∈X be a point where all integer valued differential invariants are locally constant. We show that if the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Segre P× P, n,m[ges ]2, a Grassmaniann G(2,n+2), n[ges ]4, or the Cayley plane OP$^2$, then X is the corresponding homogeneous variety (resp. an open subset of the corresponding homogeneous variety). The case of the Segre P$^2$×P$^2$had been conjectured by Griffiths and Harris in [GH]. If the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Veronese v$_2$(P) and the Fubini cubic form of X at x is zero, then X=v$_2$(P) (resp. an open subset of v$_2$(P)). All these results are valid in the real or complex analytic categories and locally in the C$^∞$category if one assumes the hypotheses hold in a neighborhood of any point x. As a byproduct, we show that the systems of quadrics I$_2$(P [sqcup ]P)⊂ S$^2$C, I$_2$(P$^1$× P)⊂ S$^2$C and I$_2$(S$_5$)⊂ S$^2$C$^16$are stable in the sense that if A ⊂S$^*$ is an analytic family such that for t≠0,A≃A, then A$_0$≃A.

We also make some observations related to the Fulton–:Hansen connectedness theorem.

We study the Galois tower generated by iterates of a quadratic polynomial $f$ defined over an arbitrary field. One question of interest is to find the proportion $a_n$ of elements at level $n$ that fix at least one root; in the global field case these correspond to unramified primes in the base field that have a divisor at level $n$ of residue class degree one. We thus define a stochastic process associated to the tower that encodes root-fixing information at each level. We develop a uniqueness result for certain permutation groups, and use this to show that for many $f$ each level of the tower contains a certain central involution. It follows that the associated stochastic process is a martingale, and convergence theorems then allow us to establish a criterion for showing that $a_n$ tends to 0. As an application, we study the dynamics of the family $x^2 + c \in\overline{\mathbb{F}}_p[x]$, and this in turn is used to establish a basic property of the $p$-adic Mandelbrot set.

We show that the semisimple part of the trace formula converges for a wide class of test functions.

Lazard showed in his seminal work (Groupes analytiques p-adiques, Publ. Math. Inst. Hautes Études Sci. 26 (1965), 389–603) that for rational coefficients, continuous group cohomology of p-adic Lie groups is isomorphic to Lie algebra cohomology. We refine this result in two directions: first, we extend Lazard’s isomorphism to integral coefficients under certain conditions; and second, we show that for algebraic groups over finite extensions K/ℚp, his isomorphism can be generalized to K-analytic cochains andK-Lie algebra cohomology.

We define a natural discriminant for a hyperelliptic curve X of genus g over a field K as a canonical element of the (8g+4)th tensor power of the maximal exterior product of the vectorspace of global differential forms on X. If v is a discrete valuation on K and X has semistable reduction at v, we compute the order of vanishing of the discriminant at v in terms of the geometry of the reduction of X over v. As an application, we find an upper bound for the Arakelov self-intersection of the relative dualizing sheaf on a semistable hyperelliptic arithmetic surface.

Let A=A$_g, 1, n$ denote the moduli scheme over Z[1/N] of p.p. g-dimensional abelian varieties with a level n structure; its generic fibre can be described as a Shimura variety. We study its ’Shimura subvarieties‘. If x ∈ A is an ordinary moduli point in characteristic p, then we formulate a local ’linearity property‘ in terms of the Serre–Tate group structure on the formal deformation space (= formal completion of A at x). We prove that an irreducible algebraic subvariety of A is a ’Shimura subvariety‘ if, locally at an ordinary point x, it is ’formally linear‘. We show that there is a close connection to a differential-geometrical linearity property in characteristic 0.

We apply our results to the study of Oort‘s conjecture on subvarieties Z [rarrhk] A with a dense collection of CM-points. We give a reformulation of this conjecture, and we prove it in a special case.

Let D denote the fundamental discriminant of a real quadratic field, and let h(D) denote its associated class number. If p is prime, then the ’Cohen and Lenstra Heuristics‘ give a probability that p[nmid]h(D). If p>3 is prime, then subject to a mild condition, we show that $\# \{0<D<X|p\nmid h(D)\}\gg_p \frac{\sqrt{X}}{\log X}.$ This condition holds for each 3<p<5000.

Despite the presence of many famous examples, the precise interplay between basic hypergeometric series and modular forms remains a mystery. We consider this problem for canonical spaces of weight 3/2 harmonic Maass forms. Using recent work of Zwegers, we exhibit forms that have the property that their holomorphic parts arise from Lerch-type series, which in turn may be formulated in terms of the Rogers–Fine basic hypergeometric series.

In this paper we prove two results concerning Vinogradov’s three primes theorem with primes that can be called almost twin primes. First, for any $m$, every sufficiently large odd integer $N$ can be written as a sum of three primes $p_{1},p_{2}$ and $p_{3}$ such that, for each $i\in \{1,2,3\}$, the interval $[p_{i},p_{i}+H]$ contains at least $m$ primes, for some $H=H(m)$. Second, every sufficiently large integer $N\equiv 3~(\text{mod}~6)$ can be written as a sum of three primes $p_{1},p_{2}$ and $p_{3}$ such that, for each $i\in \{1,2,3\}$, $p_{i}+2$ has at most two prime factors.

By use of a natural extension map and a power series method, we obtain a local stability theorem for $p$-Kähler structures with the $(p,p+1)$th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$-lemma under small differentiable deformations.

The number N of rational points on an algebraic curve of genus g over a finite field ${\mathbb F}_q$ satisfies the Hasse–Weil bound $N \leq q + 1 +2g\sqrt {q}$. A curve that attains this bound is called maximal. With $g_0 =\textstyle {1\over 2}(q -\sqrt {q})$ and $g_1= \textstyle {1\over 4}(\sqrt {q}- 1)^2$, it is known that maximalcurves have $g= g_0$ or $g \leq g_1$. Maximal curves with $g=g_0$ or $g_1$ have been characterized up to isomorphism. A natural genus to be studied is $g_2 =\textstyle {1 \over 8}(\sqrt {q}- 1)(\sqrt {q}- 3),$ and for this genus there are two non-isomorphic maximal curves known when $\sqrt {q}\equiv 3\;({\rm mod}\; {4})$. Here, a maximal curve with genus $g_2$ and a non-singular plane model is characterized as a Fermat curve of degree $ {1\over 2}(\sqrt {q} + 1)$.

Let $(S,\mathfrak{m})$ be an $n$-dimensional regular local ring essentially of finite type over a field and let $\mathfrak{a}$ be an ideal of $S$. We prove that if $\text{depth}\,S/\mathfrak{a}\geqslant 3$, then the cohomological dimension $\text{cd}(S,\mathfrak{a})$ of $\mathfrak{a}$ is less than or equal to $n-3$. This settles a conjecture of Varbaro for such an $S$. We also show, under the assumption that $S$ has an algebraically closed residue field of characteristic zero, that if $\text{depth}\,S/\mathfrak{a}\geqslant 4$, then $\text{cd}(S,\mathfrak{a})\leqslant n-4$ if and only if the local Picard group of the completion $\widehat{S/\mathfrak{a}}$ is torsion. We give a number of applications, including a vanishing result on Lyubeznik’s numbers, and sharp bounds on the cohomological dimension of ideals whose quotients satisfy good depth conditions such as Serre’s conditions $(S_{i})$.

To any finite group $\Gamma \subset {SL}_2({\mathbb {C}})$ and each element τ in the center of the group algebra of Γ, we associate a category, ${\mathcal {C}}\! oh({\mathbb {P}}^2_{_{\! \Gamma ^{\! },^{\! }\tau }},{\mathbb {P}}^1).$ It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed preprojective algebra introduced by Crawley-Boevey and Holland. The category ${\mathcal {C}}\! oh({\mathbb {P}}^2_{_{\! \Gamma ^{\! },^{\! }\tau }},{\mathbb {P}}^1)$ should be thought of as the category of coherent sheaves on a ‘noncommutative projective space’, ${\mathbb {P}}^2_{_{\! \Gamma ^{\! },^{\! }\tau }},$ equipped with a framing at ${\mathbb {P}}^1$, the line at infinity. Our first result establishes an isomorphism between the moduli space of torsion free objects of ${\mathcal {C}}\! oh({\mathbb {P}}^2_{_{\! \Gamma ^{\! },^{\! }\tau }},{\mathbb {P}}^1)$ and the Nakajima quiver variety arising from Γ via the McKay correspondence. We apply the above isomorphism to deduce a generalization of the Crawley-Boevey and Holland conjecture, saying that the moduli space of `rank 1' projective modules over the deformed preprojective algebra is isomorphic to a particular quiver variety. This reduces, for $\Gamma =\{ 1\}$, to the recently obtained parametrisation of the isomorphism classes of right ideals in the first Weyl algebra, $\sf {A_1}$, by points of the Calogero–Moser space, due to Cannings and Holland and Berest and Wilson. Our approach is algebraic and is based on a monadic description of torsion free sheaves on ${\mathbb {P}}^2_{_{\! \Gamma ^{\! },^{\! }\tau }}$. It is totally different from the one used by Berest and Wilson, involving $\tau $-functions.

The Jackson integrals associated with the non-reduced root system are defined as multiple sums which are generalization of the Bailey's very-well-poised 6ψ6 sum. They are classified by the number of their parameters when they can be expressed as a product of the Jacobi elliptic theta functions. The sums which appear in the classification list coincide with those investigated individually by Gustafson and van Diejen.

In this paper, we consider the family of hyperelliptic curves over $\mathbb{Q}$ having a fixed genus $n$ and a marked rational non-Weierstrass point. We show that when $n\geqslant 9$, a positive proportion of these curves have exactly two rational points, and that this proportion tends to one as $n$ tends to infinity. We study rational points on these curves by first obtaining results on the 2-Selmer groups of their Jacobians. In this direction, we prove that the average size of the 2-Selmer groups of the Jacobians of curves in our family is bounded above by 6, which implies a bound of $5/2$ on the average rank of these Jacobians. Our results are natural extensions of Poonen and Stoll [Most odd degree hyperelliptic curves have only one rational point, Ann. of Math. (2) 180 (2014), 1137–1166] and Bhargava and Gross [The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, in Automorphic representations and$L$-functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22 (Tata Institute of Fundamental Research, Mumbai, 2013), 23–91], where the analogous results are proved for the family of hyperelliptic curves with a marked rational Weierstrass point.