In the recent paper by Pakovich and Muzychuk [Solution of the polynomial moment problem, Proc. Lond. Math. Soc. (3) 99 (2009), 633–657] it was shown that any solution of ‘the polynomial moment problem’, which asks to describe polynomials $Q$ orthogonal to all powers of a given polynomial $P$ on a segment, may be obtained as a sum of so-called ‘reducible’ solutions related to different decompositions of $P$ into a composition of two polynomials of lower degrees. However, the methods of that paper do not permit us to estimate the number of necessary reducible solutions or to describe them explicitly. In this paper we provide a description of polynomial solutions of the functional equation $P_1\circ W_1=P_2\circ W_2=\cdots =P_r\circ W_r,$and on this base describe solutions of the polynomial moment problem in an explicit form suitable for applications.

We study the distribution of the size of Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with a non-trivial $2$-torsion point over $\mathbb {Q}$. This complements the work [Xiong and Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math.219 (2008), 523–553] which studied the same subject for elliptic curves with full 2-torsions over $\mathbb {Q}$ and generalizes [Feng and Xiong, On Selmer groups and Tate–Shafarevich groups for elliptic curves $y^2=x^3-n^3$. Mathematika 58 (2012), 236–274.] for the special elliptic curves $y^2=x^3-n^3$. It is shown that the 2-ranks of these groups all follow the same distribution and in particular, the mean value is $\sqrt {\frac {1}{2}\log \log X}$ for square-free positive integers $n \le X$ as $X \to \infty $.

Given an elliptic curve E over a field of positive characteristic p, we consider how to efficiently determine whether E is ordinary or supersingular. We analyze the complexity of several existing algorithms and then present a new approach that exploits structural differences between ordinary and supersingular isogeny graphs. This yields a simple algorithm that, given E and a suitable non-residue in 𝔽p2, determines the supersingularity of E in O(n3log 2n) time and O(n) space, where n=O(log p) . Both these complexity bounds are significant improvements over existing methods, as we demonstrate with some practical computations.

We study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic T7, T8 singularities. We define discrete symplectic invariants (the Lagrangian tangency orders) and compare them with the index of isotropy. We use these invariants to distinguish symplectic singularities of classical T7 singularity. We also give the geometric description of symplectic classes of the singularity.

Given a prime q and a negative discriminant D, the CM method constructs an elliptic curve E/Fq by obtaining a root of the Hilbert class polynomial HD(X) modulo q. We consider an approach based on a decomposition of the ring class field defined by HD, which we adapt to a CRT setting. This yields two algorithms, each of which obtains a root of HD mod q without necessarily computing any of its coefficients. Heuristically, our approach uses asymptotically less time and space than the standard CM method for almost all D. Under the GRH, and reasonable assumptions about the size of log q relative to ∣D∣, we achieve a space complexity of O((m+n)log q) bits, where mn=h(D) , which may be as small as O(∣D∣1/4 log q) . The practical efficiency of the algorithms is demonstrated using ∣D∣>1016 and q≈2256, and also ∣D∣>1015 and q≈233220. These examples are both an order of magnitude larger than the best previous results obtained with the CM method.

We consider the Prym map from the space of double coverings of a curve of genus g with r branch points to the moduli space of abelian varieties. We prove that 𝒫:ℛg,r→𝒜δg−1+r/2 is generically injective if We also show that a very general Prym variety of dimension at least 4 is not isogenous to a Jacobian.

Explicit generators are found for the group G2 of automorphisms of the algebra of one-sided inverses of a polynomial algebra in two variables over a field. Moreover, it is proved that

where S2 is the symmetric group, is the 2-dimensional algebraic torus, E∞() is the subgroup of GL∞() generated by the elementary matrices. In the proof, we use and prove several results on the index of an operator. The final argument is the proof of the fact that K1() ≃ K*. The algebras and are noncommutative, non-Noetherian, and not domains.

We study the distribution of the size of Selmer groups and Tate–Shafarevich groups arising from a 2-isogeny and its dual 2-isogeny for elliptic curves En:y2=x3−n3. We show that the 2-ranks of these groups all follow the same distribution. The result also implies that the mean value of the 2-rank of the corresponding Tate–Shafarevich groups for square-free positive integers n≤X is as X→∞. This is quite different from quadratic twists of elliptic curves with full 2-torsion points over ℚ [M. Xiong and A. Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math. 219 (2008), 523–553], where one Tate–Shafarevich group is almost always trivial while the other is much larger.

We compare the cohomology of (parabolic) Hitchin fibers for Langlands dual groups G and G∨. The comparison theorem fits in the framework of the global Springer theory developed by the author. We prove that the stable parts of the parabolic Hitchin complexes for Langlands dual group are naturally isomorphic after passing to the associated graded of the perverse filtration. Moreover, this isomorphism intertwines the global Springer action on one hand and Chern class action on the other. Our result is inspired by the mirror symmetric viewpoint of geometric Langlands duality. Compared to the pioneer work in this subject by T. Hausel and M. Thaddeus, R. Donagi and T. Pantev, and N. Hitchin, our result is valid for more general singular fibers. The proof relies on a variant of Ngô’s support theorem, which is a key point in the proof of the Fundamental Lemma.

Let C be a locally planar curve. Its versal deformation admits a stratification by the genera of the fibres. The strata are singular; we show that their multiplicities at the central point are determined by the Euler numbers of the Hilbert schemes of points on C. These Euler numbers have made two prior appearances. First, in certain simple cases, they control the contribution of C to the Pandharipande–Thomas curve counting invariants of three-folds. In this context, our result identifies the strata multiplicities as the local contributions to the Gopakumar–Vafa BPS invariants. Second, when C is smooth away from a unique singular point, a conjecture of Oblomkov and the present author identifies the Euler numbers of the Hilbert schemes with the ‘U(∞)’ invariant of the link of the singularity. We make contact with combinatorial ideas of Jaeger, and suggest an approach to the conjecture.

We prove the conjectural relations between Mahler measures and L-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions for L-values of elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family (1+X) (1+Y )(X+Y )−αXY, α∈ℝ.

We develop the theory of Abelian functions associated with algebraic curves. The growth in computer power and the advancement of efficient symbolic computation techniques have allowed for recent progress in this area. In this paper we focus on the genus three cases, comparing the two canonical classes of hyperelliptic and trigonal curves. We present new addition formulae, derive bases for the spaces of Abelian functions and discuss the differential equations such functions satisfy.

Supplementary materials are available with this article.

We prove that the moduli spaces of n-pointed m-stable curves introduced in our previous paper have projective coarse moduli. We use the resulting spaces to run an analogue of Hassett’s log minimal model program for .

The set of non-constant holomorphic mappings between two given compact Riemann surfaces of genus greater than 1 is always finite. This classical statement was made by de Franchis. Furthermore, bounds on the cardinality of the set depending only on the genera of the surfaces have been obtained by a number of mathematicians. The analysis is carried over in this paper to the case of Riemann surfaces of finite analytic type (i.e. compact Riemann surfaces minus a finite set of points) so that the finiteness result, together with a crude but explicit bound depending only on the topological data, may be extended for the number of holomorphic mappings between such surfaces.

The moduli space of holomorphic maps from Riemann surfaces to the Grassmannian is known to have two kinds of compactifications: Kontsevich’s stable map compactification and Marian–Oprea–Pandharipande’s stable quotient compactification. Over a non-singular curve, the latter moduli space is Grothendieck’s Quot scheme. In this paper, we give the notion of ‘ ϵ-stable quotients’ for a positive real number ϵ, and show that stable maps and stable quotients are related by wall-crossing phenomena. We will also discuss Gromov–Witten type invariants associated to ϵ-stable quotients, and investigate them under wall crossing.

Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno–Drinfeld Lie algebra . We show that Gaudin subalgebras form a variety isomorphic to the moduli space of stable curves of genus zero with n+1 marked points. In particular, this gives an embedding of in a Grassmannian of (n−1)-planes in an n(n−1)/2-dimensional space. We show that the sheaf of Gaudin subalgebras over is isomorphic to a sheaf of twisted first-order differential operators. For each representation of the Kohno–Drinfeld Lie algebra with fixed central character, we obtain a sheaf of commutative algebras whose spectrum is a coisotropic subscheme of a twisted version of the logarithmic cotangent bundle of .

Let C be a proper smooth geometrically connected hyperbolic curve over a field of characteristic 0 and ℓ a prime number. We prove the injectivity of the homomorphism from the pro-ℓ mapping class group attached to the two dimensional configuration space of C to the one attached to C, induced by the natural projection. We also prove a certain graded Lie algebra version of this injectivity. Consequently, we show that the kernel of the outer Galois representation on the pro-ℓ pure braid group on C with n strings does not depend on n, even if n = 1. This extends a previous result by Ihara–Kaneko. By applying these results to the universal family over the moduli space of curves, we solve completely Oda's problem on the independency of certain towers of (infinite) algebraic number fields, which has been studied by Ihara, Matsumoto, Nakamura, Ueno and the author. Sequentially we obtain certain information of the image of this Galois representation and get obstructions to the surjectivity of the Johnson–Morita homomorphism at each sufficiently large even degree (as Oda predicts), for the first time for a proper curve.

We introduce a sequence of isolated curve singularities, the elliptic m-fold points, and an associated sequence of stability conditions, generalizing the usual definition of Deligne–Mumford stability. For every pair of integers 1≤m<n, we prove that the moduli problem of n-pointed m-stable curves of arithmetic genus one is representable by a proper irreducible Deligne–Mumford stack . We also consider weighted variants of these stability conditions, and construct the corresponding moduli stacks . In forthcoming work, we will prove that these stacks have projective coarse moduli and use the resulting spaces to give a complete description of the log minimal model program for .

We discuss the Mordell–Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the Mordell–Weil sieve algorithm and discuss its efficiency.

Supplementary materials are available with this article.

We exhibit a strong link between the Hall algebra HX of an elliptic curve X defined over a finite field 𝔽l (or, more precisely, its spherical subalgebra U+X) and Cherednik’s double affine Hecke algebras of type GLn, for all n. This allows us to obtain a geometric construction of the Macdonald polynomials Pλ(q,t−1) in terms of certain functions (Eisenstein series) on the moduli space of semistable vector bundles on the elliptic curve X.