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In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves X_0(N) of genus up to 8, and genus up to 10 with N prime, for which they were previously unknown. The values of N we consider are contained in the set \begin{equation*} \mathcal {L}=\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \}. \end{equation*} We obtain that all the non-cuspidal quadratic points on X_0(N) for N\in \mathcal {L} are complex multiplication (CM) points, except for one pair of Galois conjugate points on X_0(103) defined over \mathbb {Q}(\sqrt {2885}). We also compute the j-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings. [ABSTRACT FROM AUTHOR]

In this paper, we construct a compactification of the space of Bridgeland stability conditions on a smooth projective curve, as an analogue of Thurston compactifications in Teichmüller theory. In the case of elliptic curves, we compare our results with the classical one of the torus via homological mirror symmetry and give the Nielsen–Thurston classification of autoequivalences using the compactification. Furthermore, we observe an interesting phenomenon in the case of the projective line. [ABSTRACT FROM AUTHOR]

Building on Mazur's 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over \mathbb {Q}. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields \mathbb {Q}(\sqrt {d}) with |d| < 10^4 we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over 19 quadratic fields, including \mathbb {Q}(\sqrt {213}) and \mathbb {Q}(\sqrt {-2289}). To make this procedure work, we determine all of the finitely many quadratic points on the modular curves X_0(125) and X_0(169), which may be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, we study moments of Hurwitz class numbers associated to imaginary quadratic orders restricted into fixed arithmetic progressions. In particular, we fix t in an arithmetic progression t\equiv m\ \, \left (\operatorname {mod} \, M \right) and consider the ratio of the 2k-th moment to the zeroeth moment for H(4n-t^2) as one varies n. The special case n=p^r yields as a consequence asymptotic formulas for moments of the trace t\equiv m\ \, \left (\operatorname {mod} \, M \right) of Frobenius on elliptic curves over finite fields with p^r elements. [ABSTRACT FROM AUTHOR]

Given an elliptic curve E defined over \mathbb {C}, let E^{\times } be an open subset of E obtained by removing a point. In this paper, we show that the i-th Betti number of the unordered configuration space \mathrm {Conf}^{n}(E^{\times }) of n points on E^{\times } appears as a coefficient of an explicit rational function in two variables. We also compute its Hodge numbers as coefficients of another explicit rational function in four variables. Our result is interesting because these rational functions resemble the generating function of the \mathbb {F}_{q}-point counts of \mathrm {Conf}^{n}(E^{\times }), which can be obtained from the zeta function of E over any fixed finite field \mathbb {F}_{q}. We show that the mixed Hodge structure of the i-th singular cohomology group H^{i}(\mathrm {Conf}^{n}(E^{\times })) with complex coefficients is pure of weight w(i), an explicit integer we provide in this paper. This purity statement implies our main result about the Betti numbers and the Hodge numbers. Our proof uses Totaro's spectral sequence computation that describes the weight filtration of the mixed Hodge structure on H^{i}(\mathrm {Conf}^{n}(E^{\times })). [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve over \mathbb {Q} with discriminant \Delta _E. For primes p of good reduction, let N_p be the number of points modulo p and write N_p=p+1-a_p. In 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies \begin{equation*} \lim _{x\to \infty }\frac {1}{\log x}\sum _{\substack {p\leq x\\ p\nmid \Delta _{E}}}\frac {a_p\log p}{p}=-r+\frac {1}{2}, \end{equation*} where r is the order of the zero of the L-function L_{E}(s) of E at s=1, which is predicted to be the Mordell-Weil rank of E(\mathbb {Q}). We show that if the above limit exits, then the limit equals -r+1/2. We also relate this to Nagao's conjecture. This paper also includes an appendix by Andrew V. Sutherland which gives evidence for the convergence of the above-mentioned limit. [ABSTRACT FROM AUTHOR]

In this paper, we initiate a systematic study of entanglements of division fields from a group theoretic perspective. For a positive integer n and a subgroup G\subseteq GL_2(\mathbb {Z}/n\mathbb {Z}) with surjective determinant, we provide a definition for G to represent an (a,b)-entanglement and give additional criteria for G to represent an explained or unexplained (a,b)-entanglement. Using these new definitions, we determine the tuples ((p,q),T), with p

Let A be a principally polarized abelian variety of dimension g over a number field K. Assume that the image of the adelic Galois representation of A is an open subgroup of GSp_{2g}(\hat {\mathbb {Z}}). Then there exists a positive integer m so that the Galois image of A is the full preimage of its reduction modulo m. The least m with this property, denoted m_A, is called the image conductor of A. Jones [Pacific J. Math. 308 (2020), pp. 307–331] recently established an upper bound for m_A, in terms of standard invariants of A, in the case that A is an elliptic curve without complex multiplication. In this paper, we generalize the aforementioned result to provide an analogous bound in arbitrary dimension. [ABSTRACT FROM AUTHOR]

We provide some minor corrections to our paper "Statistics for Iwasawa invariants of elliptic curves". [ABSTRACT FROM AUTHOR]

The goal of this paper is the study of simple rank 2 parabolic vector bundles over a 2-punctured elliptic curve C. We show that the moduli space of these bundles is a non-separated gluing of two charts isomorphic to P1 × P1. We also showcase a special curve Γ isomorphic to C embedded in this space, and this way we prove a Torelli theorem. This moduli space is related to the moduli space of semistable parabolic bundles over P1 via a modular map which turns out to be the 2:1 cover ramified in Γ. We recover the geometry of del Pezzo surfaces of degree 4 and we reconstruct all their automorphisms via elementary transformations of parabolic vector bundles. [ABSTRACT FROM AUTHOR]

In this paper, we consider maximal and irreducible quartic orders which arise from integral binary quartic forms, via the construction of Birch and Merriman, and whose field of fractions is a quartic D_4-field. By a theorem of Wood, such quartic orders may be regarded as quartic D_4-fields whose ring of integers has a monogenic cubic resolvent. We shall determine the asymptotic number of such objects when ordered by conductor. We shall also give a lower bound, which we suspect has the correct order of magnitude, and a slightly larger upper bound for the number of such objects when ordered by discriminant. A simplified version of the techniques used allows us to give a count for those elliptic curves with a marked rational 2-torsion point when ordered by discriminant. [ABSTRACT FROM AUTHOR]

In this paper, we remove certain hypotheses in the theorem of Bertolini-Darmon on the rationality of Stark-Heegner points over narrow genus class fields of real quadratic fields. Along the way, we establish that certain normalized special values of L-functions are squares of rational numbers, a result that is of independent interest, and can be regarded as instances of the rank zero case of the Birch and Swinnerton-Dyer conjecture modulo squares. [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field of characteristic different from 2, let g be a positive integer, let ƒ(x) \in K[x] be a degree 2g+1 monic polynomial without multiple roots, let Cƒ: y2 = ƒ(x) be the corresponding genus g hyperelliptic curve over K, and let J be the Jacobian of Cƒ. We identify Cƒ with the image of its canonical embedding into J (the infinite point of Cƒ goes to the zero of the group law on J). It is known [Izv. Math. 83 (2019), pp. 501-520] that if g ≥ 2, then Cƒ(K) contains no points of orders lying between 3 and 2g. In this paper we study torsion points of order 2g + 1 on Cƒ(K). Despite the striking difference between the cases of g = 1 and g ≥ 2, some of our results may be viewed as a generalization of well-known results about points of order 3 on elliptic curves. E.g., if p = 2g + 1 is a prime that coincides with char(K), then every odd degree genus g hyperelliptic curve contains at most two points of order p. If g is odd and ƒ(x) has real coefficients, then there are at most two real points of order 2g + 1 on Cƒ. If ƒ(x) has rational coefficients and g ≤ 51, then there are at most two rational points of order 2g+1 on Cƒ. (However, there exist odd degree genus 52 hyperelliptic curves over Q that have at least four rational points of order 105.) [ABSTRACT FROM AUTHOR]

A recent paper of Shekhar compares the ranks of elliptic curves E1 and E2 for which there is an isomorphism E1[p] ∼ E2[p]asGal(.../Q)- modules, where p is a prime of good ordinary reduction for both curves. In this paper we prove an analogous result in the case where p is a prime of good supersingular reduction. [ABSTRACT FROM AUTHOR]

We present a quasi-linear algorithm to compute (separable) isogenies of degree lg, for l an odd prime number, between Jacobians of curves of genus g = 2 and 3 starting from the equation of the curve C and a maximal isotropic subgroup V of the l-torsion, generalizing Vélu's formula from genus 1. Denoting by JC the Jacobian of C, the isogeny is JC → JCV. Thus V is the kernel of the isogeny and we compute only isogenies with such kernels. This work is based on the paper Computing functions on Jacobians and their quotients of Jean-Marc Couveignes and Tony Ezome. We improve their genus 2 algorithm, generalize it to genus 3 hyperelliptic curves, and introduce a way to deal with the genus 3 nonhyperelliptic case, using algebraic theta functions. [ABSTRACT FROM AUTHOR]

Bott proved a strong vanishing theorem for sheaf cohomology on projective space. It holds for toric varieties, but not for most other varieties. We prove Bott vanishing for the quintic del Pezzo surface, also known as the moduli space M0,5 of 5-pointed stable curves of genus zero. This is the first non-toric Fano variety for which Bott vanishing has been shown, answering a question by Achinger, Witaszek, and Zdanowicz. In another direction, we prove sharp results on which K3 surfaces satisfy Bott vanishing. For example, a K3 surface with Picard number 1 satisfies Bott vanishing if and only if the degree is 20 or at least 24. For K3 surfaces of any Picard number, we have complete results when the degree is big enough. We build on Beauville, Mori, and Mukai's work on moduli spaces of K3 surfaces, as well as recent advances by Arbarello-Bruno-Sernesi, Ciliberto-Dedieu-Sernesi, and Feyzbakhsh. The most novel aspect of the paper is our analysis of Bott vanishing for K3 surfaces with an elliptic curve of low degree. (In other terminology, this concerns K3 surfaces that are monogonal, hyperelliptic, trigonal, or tetragonal.) It turns out that the crucial issue is whether an elliptic fibration has a certain special type of singular fiber. [ABSTRACT FROM AUTHOR]

Motivated by the relationship between numerical Grothendieck groups induced by the embedding of a smooth anticanonical elliptic curve into a del Pezzo surface, we define the notion of a quasi-del Pezzo homomorphism between pseudolattices and establish its basic properties. The primary aim of the paper is then to prove a classification theorem for quasi-del Pezzo homomorphisms, using a pseudolattice variant of the minimal model program. Finally, this result is applied to the classification of a certain class of genus 1 Lefschetz fibrations over discs. [ABSTRACT FROM AUTHOR]

The Birch and Swinnerton-Dyer conjecture predicts that the parity of the algebraic rank of an abelian variety over a global field should be controlled by the expected sign of the functional equation of its L-function, known as the global root number. In this paper, we give explicit formulae for the local root numbers as a product of Jacobi symbols. This enables one to compute the global root number, generalising work of Rohrlich, who studies the case of elliptic curves. We provide similar formulae for the root numbers after twisting the abelian variety by a self-dual Artin representation. As an application, we find a rational genus two hyperelliptic curve with a simple Jacobian whose root number is invariant under quadratic twist. [ABSTRACT FROM AUTHOR]

Let p = 4, 7 mod 9 be a rational prime number such that 3 mod p is not a cube. In this paper, we prove the 3-part of |III(Ep)| · |III(E3p²)| is as predicted by the Birch and Swinnerton-Dyer conjecture, where Ep : x3+y3 = p and E3p² : x³ + y³ = 3p² are the elliptic curves related to the Sylvester conjecture and cube sum problems. [ABSTRACT FROM AUTHOR]

In this paper, we study the geometry of the SYZ transform on a semiflat Lagrangian torus fibration. Our starting point is an investigation on the relation between Lagrangian surgery of a pair of straight lines in a symplectic 2-torus and the extension of holomorphic vector bundles over the mirror elliptic curve, via the SYZ transform for immersed Lagrangian multisections defined by Arinkin and Joyce [Fukaya category and Fourier transform, AMS/IP Stud. Adv. Math., Amer. Math. Soc., Providence, RI, 2001] and Leung, Yau, and Zaslow [Adv. Theor. Math. Phys. 4 (2000), no. 6, 1319-1341]. This study leads us to a new notion of equivalence between objects in the immersed Fukaya category of a general compact symplectic manifold (M, ω), under which the immersed Floer cohomology is invariant; in particular, this provides an answer to a question of Akaho and Joyce [J. Differential Geom. 86 (2010), no. 3, 831-500, Question 13.15]. Furthermore, if M admits a Lagrangian torus fibration over an integral affine manifold, we prove, under some additional assumptions, that this new equivalence is mirror to an isomorphism between holomorphic vector bundles over the dual torus fibration via the SYZ transform. [ABSTRACT FROM AUTHOR]

In this paper, we give a resolution of the generalized Fermat equations x5 + y5 = 3zn and x13 + y13 = 3zn, for all integers n ≥ 2 and all integers n ≥ 2 which are not a power of 7, respectively, using the modular method with Frey elliptic curves over totally real fields. The results require a refined application of the multi-Frey technique, which we show to be effective in new ways to reduce the bounds on the exponents n. We also give a number of results for the equations x5 + y5 = dzn, where d = 1, 2, under additional local conditions on the solutions. This includes a result which is reminiscent of the second case of Fermat’s Last Theorem and which uses a new application of level raising at p modulo p. [ABSTRACT FROM AUTHOR]

Under a non-torsion assumption on Heegner points, results of Kolyvagin describe the structure of Shafarevich–Tate groups of elliptic curves. In this paper we prove analogous results for (p-primary) Shafarevich–Tate groups associated with higher weight modular forms over imaginary quadratic fields satisfying a “Heegner hypothesis”. More precisely, we show that the structure of Shafarevich–Tate groups is controlled by cohomology classes built out of Nekovář's Heegner cycles on Kuga–Sato varieties. As an application of our main theorem, we improve on a result of Besser giving a bound on the order of these groups. As a second contribution, we prove a result on the structure of (p-primary) Selmer groups of modular forms in the sense of Bloch–Kato. [ABSTRACT FROM AUTHOR]

The 4-dimensional Sklyanin algebras, over ℂ., A(E, τ), are constructed from an elliptic curve E and a translation automorphism τ of E. The Klein vierergruppe Γ acts as graded algebra automorphisms of A(E, τ). There is also an action of Γ as automorphisms of the matrix algebra M2(ℂ.) making it isomorphic to the regular representation. The main object of study in this paper is the invariant subalgebra à := A(E, τ)⊗M2(ℂ))Γ. Like A(E, τ), à is noetherian, generated by 4 degree-one elements modulo six quadratic relations, Koszul, Artin-Schelter regular of global dimension 4, has the same Hilbert series as the polynomial ring on 4 variables, satisfies the x condition, and so on. These results are special cases of general results proved for a triple (A, T, H) consisting of a Hopf algebra H, an (often graded) H-comodule algebra A, and an H-torsor T. Those general results involve transferring properties between A, A ⊗ T, and (A ⊗ T)coH. We then investigate à from the point of view of non-commutative projective geometry. We examine its point modules, line modules, and a certain quotient ... := Ã/(Θ,Θ') where Θ and Θ' are homogeneous central elements of degree two. In doing this we show that à differs from A in interesting ways. For example, the point modules for A are parametrized by E and 4 more points, whereas à has exactly 20 point modules. Although ... is not a twisted homogeneous coordinate ring in the sense of Artin and Van den Bergh, a certain quotient of the category of graded ...-modules is equivalent to the category of quasi-coherent sheaves on the curve E/E[2] where E[2] is the 2-torsion subgroup. We construct line modules for à that are parametrized by the disjoint union (E/(ξ1)) ⊔ (E/(ξ2)) ⊔ (E/(ξ3)) of the quotients of E by its three subgroups of order 2. [ABSTRACT FROM AUTHOR]

The classical problem of whether mth-powers with or without zero in a finite field Fq form a difference set has been extensively studied, and is related to many topics, such as flag transitive finite projective planes. In this paper new necessary and sufficient conditions are established including those via a system of polynomial equations on Gauss sums. The author thereby solves the problem for even q which is neglected in the literature, and extends the nonexistence list for even m up to 22. Moreover, conjectures toward the complete classification are posed. [ABSTRACT FROM AUTHOR]

In this paper, we describe a relation between the embedding degree of an elliptic curve over a prime field Fp and the inertial degree of the primes above p in a certain ring class field. From this relation, we conclude that the embedding degree divides the class number of a group of binary quadratic forms of a fixed discriminant.In this paper, we describe a relation between the embedding degree of an elliptic curve over a prime field Fp and the inertial degree of the primes above p in a certain ring class field. From this relation, we conclude that the embedding degree divides the class number of a group of binary quadratic forms of a fixed discriminant. [ABSTRACT FROM AUTHOR]

Let K be a number field with ring of integers 𝓞K, and let {fk}k∈N be a sequence of monic polynomials in 𝓞K[x] such that for every n ∈ ℕ, the composition f(n) = f1 ◦ f2 ◦ . . . ◦ fn is irreducible. In this paper we show that if the size of the Galois group of f(n) is large enough (in a precise sense) as a function of n, then the set of primes p ⊆ 𝓞K such that every f(n) is irreducible modulo p has density zero. Moreover, we prove that the subset of polynomial sequences such that the Galois group of f(n) is large enough has density 1, in an appropriate sense, within the set of all polynomial sequences. [ABSTRACT FROM AUTHOR]

In this paper, we compute the genus of the variety of linear series of rank r and degree d on a general curve of genus g, with ramification at least α and β at two given points, when that variety is 1-dimensional. Our proof uses degenerations and limit linear series along with an analysis of random staircase paths in Young tableaux, and produces an explicit scheme-theoretic description of the limit linear series of fixed rank and degree on a generic chain of elliptic curves when that scheme is itself a curve. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to lay the foundations of a theory of invariants in étale cohomology for smooth Artin stacks. We compute the invariants in the case of the stack of elliptic curves, and we use the theory we developed to get some results regarding Brauer groups of algebraic spaces. [ABSTRACT FROM AUTHOR]

Watkins' conjecture asserts that for a rational elliptic curve E the degree of the modular parametrization is divisible by 2r, where r is the rank of E. In this paper, we prove that if the modular degree is odd, then E has rank zero. Moreover, we prove that the conjecture holds for all rank two rational elliptic curves of prime conductor and positive discriminant. [ABSTRACT FROM AUTHOR]

The Torsion Anomalous Conjecture states that an irreducible variety V embedded in a semi-abelian variety contains only finitely many maximal V -torsion anomalous varieties. In this paper we consider an irreducible variety embedded in a product of elliptic curves. Our main result provides a totally explicit bound for the N'eron-Tate height of all maximal V-torsion anomalous points of relative codimension one in the non-CM case, and an analogous effective result in the CM case. As an application, we obtain the finiteness of such points. In addition, we deduce some new explicit results in the context of the effective Mordell-Lang Conjecture; in particular we bound the N'eron-Tate height of the rational points of an explicit family of curves of increasing genus. [ABSTRACT FROM AUTHOR]

Isogeny volcanoes are graphs whose vertices are elliptic curves and whose edges areℓ-isogenies. Algorithms allowing to travel on these graphs were developed by Kohel in his thesis (1996) and later on, by Fouquet and Morain (2001). However, up to now, no method was known, to predict, before taking a step on the volcano, the direction of this step. Hence, in Kohel's and Fouquet- Morain's algorithms, many steps are taken before choosing the right direction. In particular, ascending or horizontal isogenies are usually found using a trialand- error approach. In this paper, we propose an alternative method that efficiently finds all points P of order ℓ such that the subgroup generated by P is the kernel of a horizontal or an ascending isogeny. In many cases, our method is faster than previous methods. This is an extended version of a paper published in the proceedings of ANTS 2010. In addition, we treat the case of 2-isogeny volcanoes and we derive from the group structure of the curve and the pairing a new invariant of the endomorphism class of an elliptic curve. Our benchmarks show that the resulting algorithm for endomorphism ring computation is faster than Kohel's method for computing the ℓ-adic valuation of the conductor of the endomorphism ring for small ℓ. [ABSTRACT FROM AUTHOR]

This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve E defined over a number field K with E(K)[2] ≃ Z/2Z. We show that if E does not have a cyclic 4-isogeny defined over K(E[2]) with kernel containing E(K)[2], then subject only to constant 2-Selmer parity, each non-negative integer appears infinitely often as the 2-Selmer rank of a quadratic twist of E. If E has a cyclic 4-isogeny with kernel containing E(K)[2] defined over K(E[2]) but not over K, then we prove the same result for 2-Selmer ranks greater than or equal to r2, the number of complex places of K. We also obtain results about the minimum number of twists of E with rank 0 and, subject to standard conjectures, the number of twists with rank 1, provided E does not have a cyclic 4-isogeny defined over K. [ABSTRACT FROM AUTHOR]

We construct infinite families of elliptic curves with given torsion group structures over quartic number fields. In a 2006 paper, the first two authors and Park determined all of the group structures which occur infinitely often as the torsion of elliptic curves over quartic number fields. Our result presents explicit examples of their theoretical result. This paper also presents an efficient way of finding such families of elliptic curves with prescribed torsion group structures over quadratic or quartic number fields. [ABSTRACT FROM AUTHOR]

The modular variety of nonsingular and complete hyperelliptic curves with level-two structure of genus $ 3$ It has 36 irreducible (isomorphic) components. One of the purposes of this paper will be to describe the equations of one of these components. Two further models use the fact that hyperelliptic curves of genus three can be obtained as coverings of a projective line with $ 8$, uses the semistable degenerated point configurations in $ (P^1)^8$ $ Y=\overline{\mathcal{B}/\Gamma[1-{\textrm i}]}.$ We use the standard notation $ \bar M_{0,8}$}[rr]& &X\;.} \end{displaymath} --> $\displaystyle \xymatrix{ &\bar M_{0,8}\ar[dl]\ar[dr]&\\ Y\ar@{-->}[rr]& &X\;.}$ The horizontal arrow is only birational but not everywhere regular. In this paper we find another realization of this triangle which uses the fact that there are graded algebras (closely related to algebras of modular forms) $ A,B$ $ X=\mathop{\rm proj}\nolimits(A), Y=\mathop{\rm proj}\nolimits(B).$ [ABSTRACT FROM AUTHOR]

In this paper we construct infinite families of elliptic curves with given torsion group structures over cubic number fields. This result provides explicit examples of the theoretical result recently developed by the first two authors and A. Schweizer; they determined all the group structures which occur infinitely often as the torsion of elliptic curves over cubic number fields. In fact, this paper presents an efficient way of constructing such families of elliptic curves with prescribed torsion group structures over cubic number fields. [ABSTRACT FROM AUTHOR]

This paper gives a general survey of ranks of elliptic curves over the field of rational numbers. The rank is a measure of the size of the set of rational points. The paper includes discussions of the Birch and SwinnertonDyer Conjecture, the Parity Conjecture, ranks in families of quadratic twists, and ways to search for elliptic curves of large rank. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism groups are cyclic of order coprime to the characteristic of their ground field. We give an explicit and effectively computable description of this obstruction. Along the way, we obtain an arithmetic criterion for the existence of a so-called hyperelliptic descent. We define homogeneous dihedral invariants for general hyperelliptic curves, and show how the obstruction can be expressed in terms of these invariants. If this obstruction vanishes, then the homogeneous dihedral invariants can also be used to explicitly construct a model over the field of moduli of the curve; if not, then one still obtains a hyperelliptic model over a degree 2 extension of the field of moduli. [ABSTRACT FROM AUTHOR]

Isogenies are the morphisms between elliptic curves and are, accordingly, a topic of interest in the subject. As such, they have been well studied, and have been used in several cryptographic applications. V'elu's formulas show how to explicitly evaluate an isogeny, given a specification of the kernel as a list of points. However, V'elu's formulas only work for elliptic curves specified by a Weierstrass equation. This paper presents formulas similar to V'elu's that can be used to evaluate isogenies on Edwards curves and Huff curves, which are normal forms of elliptic curves that provide an alternative to the traditional Weierstrass form. Our formulas are not simply compositions of V'elu's formulas with mappings to and from Weierstrass form. Our alternate derivation yields efficient formulas for isogenies with lower algebraic complexity than such compositions. In fact, these formulas have lower algebraic complexity than V'elu's formulas on Weierstrass curves. [ABSTRACT FROM AUTHOR]

In this paper we construct infinite families of elliptic curves with given torsion group structures over cyclic cubic number fields. [ABSTRACT FROM AUTHOR]

This paper completes the classification of Artin-Schelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree one. We show that there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism, just as in the case where the algebras are assumed to be generated in degree one. In particular, we study the elliptic algebras $A(+)$, $A(-)$, and $A({\mathbf{a}})$, where ${\mathbf{a}}\in \BbP^{2}$, which were first defined in an earlier paper. We omit a set $S\subset \BbP^2$ consisting of 11 specified points where the algebras $A({\mathbf{a}})$ become too degenerate to be regular. \begin{thm} Let $A$ represent $A(+)$, $A(-)$ or $A({\mathbf{a}})$, where ${\mathbf{a}} \in \BbP^2\setminus S$. Then $A$ is an Artin-Schelter regular algebra of global dimension three. Moreover, $A$ is a Noetherian domain with the same Hilbert series as the (appropriately graded) commutative polynomial ring in three variables. \end{thm} This, combined with our earlier results, completes the classification. [ABSTRACT FROM AUTHOR]

We use the Wilf-Zeilberger method to prove identities between Mahler measures of polynomials. In particular, we offer a new proof of a formula due to Lal'ın, and we show how to translate the identity into a formula involving elliptic dilogarithms. This work settles a challenge problem proposed by Kontsevich and Zagier in their paper, "Periods". [ABSTRACT FROM AUTHOR]

In this paper, we present a probabilistic algorithm to compute the coefficients of modular forms of level one. Focusing on Ramanujan's tau function, we give the explicit complexity of the algorithm. From a practical viewpoint, the algorithm is particularly well suited for implementations. [ABSTRACT FROM AUTHOR]

In this paper, we will present a new method for evaluating high order divided differences for certain classes of analytic, possibly, operator-valued functions. This is a classical problem in numerical mathematics but also arises in new applications such as the use of generalized convolution quadrature to solve retarded potential integral equations. The functions which we will consider are allowed to grow exponentially to the left complex half-plane, polynomially to the right half-plane and have an oscillatory behaviour with increasing imaginary part. The interpolation points are scattered in a large real interval. Our approach is based on the representation of divided differences as contour integral and we will employ a subtle parameterization of the contour in combination with a quadrature approximation by the trapezoidal rule. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve defined over ℚ. The aim of this paper is to make it possible to compute Heegner L-functions and anticyclotomic Λ-adic regulators of E, which were studied by Mazur-Rubin and Howard. We generalize results of Cohen and Watkins and thereby compute Heegner points of non-fundamental discriminant. We then prove a relationship between the denominator of a point of E defined over a number field and the leading coefficient of the minimal polynomial of its x-coordinate. Using this relationship, we recast earlier work of Mazur, Stein, and Tate to produce effective algorithms to compute p-adic heights of points of E defined over number fields. These methods enable us to give the first explicit examples of Heegner L-functions and anticyclotomic Λ-adic regulators. [ABSTRACT FROM AUTHOR]

This is the third in a series of papers in which we study the n- Selmer group of an elliptic curve, with the aim of representing its elements as curves of degree n in ℙn-1. The methods we describe are practical in the case n = 3 for elliptic curves over the rationals, and have been implemented in MAGMA. One important ingredient of our work is an algorithm for trivialising central simple algebras. This is of independent interest; for example, it could be used for parametrising Brauer-Severi surfaces. [ABSTRACT FROM AUTHOR]

This paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients. The basis functions are solutions of local problems on vertex patches. The error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches. Hence, on a uniform mesh of size H, patches of diameter H log(1/H) are sufficient to preserve a linear rate of convergence in H without pre-asymptotic or resonance effects. The analysis does not rely on regularity of the solution or scale separation in the coefficient. This result motivates new and justifies old classes of variational multiscale methods. [ABSTRACT FROM AUTHOR]

In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in GL2(...). In (2010), Greicius developed necessary and sufficient criteria for determining when this representation is actually surjective and exhibits such an example. However, verifying these criteria turns out to be difficult in practice; Greicius describes tests for them that apply only to semistable elliptic curves over a specific class of cubic number fields. In this paper, we extend Greicius's methods in several directions. First, we consider the analogous problem for elliptic curves with full 2-torsion. Following Greicius, we obtain necessary and sufficient conditions for the associated adelic representation to be maximal and also develop a battery of computationally effective tests that can be used to verify these conditions. We are able to use our tests to construct an infinite family of curves over ℚ(α) with maximal image, where α is the real root of x³ + x + 1. Next, we extend Greicius's tests to more general settings, such as non-semistable elliptic curves over arbitrary cubic number fields. Finally, we give a general discussion concerning such problems for arbitrary torsion subgroups. [ABSTRACT FROM AUTHOR]

This paper introduces EECM-MPFQ, a fast implementation of the elliptic-curve method of factoring integers. EECM-MPFQ uses fewer modular multiplications than the well-known GMP-ECM software, takes less time than GMP-ECM, and finds more primes than GMP-ECM. The main improvements above the modular-arithmetic level are as follows: (1) use Edwards curves instead of Montgomery curves; (2) use extended Edwards coordinates; (3) use signed-sliding-window addition-subtraction chains; (4) batch primes to increase the window size; (5) choose curves with small parameters and base points; (6) choose curves with large torsion. [ABSTRACT FROM AUTHOR]

Given natural numbers n and k, with n > k, the Prouhet-Tarry- Escott (PTE) problem asks for distinct subsets of ℤ, say X = {x1,...xn} and Y = {y1,...yn}, such that xi1 + ... + xin = yi1 + ... + yin for i = 1, ... ,k. Many partial solutions to this problem were found in the late 19th century and early 20th century. When n = k - 1, we call a solution X =n-1 Y ideal. This is considered to be the most interesting case. Ideal solutions have been found using elementary methods, elliptic curves, and computational techniques. In 2007, Alpers and Tijdeman gave examples of solutions to the PTE problem over the Gaussian integers. This paper extends the framework of the problem to this setting. We prove generalizations of results from the literature, and use this information along with computational techniques to find ideal solutions to the PTE problem in the Gaussian integers. [ABSTRACT FROM AUTHOR]

In a 1998 paper, Kaneko and Zagier explain unpublished work of Atkin which exhibits an infinite sequence of polynomials with the property that when suitable polynomials are reduced mod p for a prime p, one gets the locus of supersingular elliptic curves. Here we generalize this phenomenon by considering the continued fraction expansions of modular and quasimodular forms. [ABSTRACT FROM AUTHOR]