Affine Curve Research Articles

Euclidean and affine curve reconstruction

Euclidean and affine curve reconstruction

Constructing abelian varieties from rank 2 Galois representations

Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let ${\mathbb {L}}$ be a rank $2$, geometrically irreducible lisse $\overline {{\mathbb {Q}}}_\ell$-sheaf on $U$ with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field $E\subset \overline {\mathbb {Q}}_{\ell }$, and has bad, infinite reduction at some closed point $x$ of $X\setminus U$. We show that ${\mathbb {L}}$ occurs as a summand of the cohomology of a family of abelian varieties over $U$. The argument follows the structure of the proof of a recent theorem of Snowden and Tsimerman, who show that when $E=\mathbb {Q}$, then ${\mathbb {L}}$ is isomorphic to the cohomology of an elliptic curve $E_U\rightarrow U$.

Schur type comparison theorems for affine curves

Schur type comparison theorems for affine curves

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

The number of rational points of a class of superelliptic curves

In this paper, we study the number of Fqn-rational points on the affine curve Xd,a,b given by the equationyd=axTr(x)+b, where Tr denote the trace function from Fqn to Fq and d is a positive integer. In particular, we present bounds for the number of Fq-rational points on Xd,a,b and, for the cases where d satisfies a natural condition, explicit formulas for the number of rational points are obtained. Particularly, a complete characterization is given for the case d=2. As a consequence of our results, we compute the number of elements α in Fqn such that α and Tr(α) are quadratic residues in Fqn.

Algebraic Points of any Given Degree on the Affine Curves y 2 = x(x + 2p)(x + 4p)(x 2 − 8p 2 )

Algebraic Points of any Given Degree on the Affine Curves y 2 = x(x + 2p)(x + 4p)(x 2 − 8p 2 )

Betti sequence of the projective closure of affine monomial curves

We introduce the notion of star gluing of numerical semigroups and show that this preserves the arithmetically Cohen-Macaulay and Gorenstein properties of the projective closure. Next, we give a sufficient condition involving Gröbner basis for the matching of Betti sequences of the affine curve and its projective closure. We also study the effect of simple gluing on Betti sequences of the projective closure. Finally, we construct numerical semigroups by gluing, such that for every positive integer n, the last Betti number of the corresponding affine curve and its projective closure are both n.

Partial zeta functions, partial exponential sums, and p-adic estimates

Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation their rationality is surprising, and even simple examples are delicate to compute. For instance, we give a detailed description of the partial zeta function of an affine curve where the number of unit poles varies, a property different from classical zeta functions. On the other hand, they do retain some properties similar to the classical case. To this end, we give Chevalley-Warning type bounds for partial zeta functions and L-functions associated to partial exponential sums.

Vanishing of higher order Alexander‐type invariants of plane curves

AbstractThe higher order degrees are Alexander‐type invariants of complements to an affine plane curve. In this paper, we characterize the vanishing of such invariants for a curve C given as a transversal union of plane curves and in terms of the finiteness and the vanishing properties of the invariants of and , and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial is a power of , and we characterize when in terms of the defining equations of and . Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves.

Abstract 2924: Target identification, selectivity profiling and mechanistic insights of a Cdk9 inhibitor using complementary proteomics methods

Abstract In an effort to map the selectivity and understand the mode of action of a CDK9 inhibitor (compound 1), we employed several orthogonal proteomics methods. Our results show a clear selectivity against the CDK family of kinases, with the highest specific affinity towards CDK9. In addition, our multi-method approach allows us to also map the protein binding sites of the inhibitor, identify non-kinase (off-) targets as well as detect the cellular molecular responses to the added inhibitor. Here we describe the methods used, their strengths and weaknesses, how they can be used in the drug discovery pipeline and how they synergize to provide mechanistic insights of compounds of interest. We have used chemoproteomics, kinase affinity tools (kinobeads), Cellular Thermal Shift Assay (CETSA) and Limited Proteolysis (LiP). The results obtained clearly show that CDK9 is the primary target of Compound 1, with affinity curves highly correlated between the different target deconvolution techniques. The chemoproteomic approach rely on a compound derivate, able to bind to a sepharose bead. Subsequently, the binding competition assays are performed on lysed cell material. The choice of a mild lysis buffer allowed us to identify, not only CDK9, but also it’s molecular partners in the p-Tefb complex (Cyclin T1, Cyclin T2 and Aff4) with similar concentration response behavior. The results for the kinases identified in the study were strikingly similar when also profiling the compound without the chemical modification using the kinobeads assay. In the CETSA experiments, where both lysed cells and intact cells were profiled, the lysate experiment most closely resembles that of the previous pull-downs. Here, only direct binders of Compound 1 show a thermal shift, for example several of the pulled down kinases but not the p-Tefb complex partners that were co-competed previously. In the intact cell version of CETSA, not only the direct binders of the compound show stability shifts, but also downstream events and other secondary modulatory effects leave thermal traces in the cell. For example, Compound 1 also binds to GSK3A/B, causing their melting temperature to increase. Inhibition of GSK3 further affects the phosphorylation state and cellular location of FOXK1, which in turn is identified as a destabilized protein. Finally, Limited Proteolysis was used to identify target protein and using the LiP-Quant approach their LiP scores were assigned. Further, out of the identified CDK targets, mapping of peptide cleavage pattern was performed for the members of the CDK family for which structural data is published. The result identified the peptides to be directly adjacent to the ATP binding pocket of CDK9 or regions of high homology. The use of complementary techniques, based on unique biological and biochemical processes, allow robust and confident characterization of inhibitor compounds. Citation Format: Adam Hendricks, Nigel Beaton, Alexey Chernobrovkin, Eric Miele, Ghaith Hamza, Piero Ricchiuto, Ronald Tomlinson, Tomas Friman, Cassandra Borenstein, Bernard Barlaam, Sudhir Hande, Chris De Savi, Rick Davies, Martin Main, Joakim Hellner, Kristina Beeler, Yuehan Feng, Roland Bruderer, Lukas Reiter, Daniel Martinez Molina, Maria Paola Castaldi. Target identification, selectivity profiling and mechanistic insights of a Cdk9 inhibitor using complementary proteomics methods [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2022; 2022 Apr 8-13. Philadelphia (PA): AACR; Cancer Res 2022;82(12_Suppl):Abstract nr 2924.

Affine monomial curves

Affine monomial curves

Urata's theorem in the logarithmic case and applications to integral points

Urata showed that a pointed compact hyperbolic variety admits only finitely many maps from a pointed curve. We extend Urata's theorem to the setting of (not necessarily compact) hyperbolically embeddable varieties. As an application, we show that a hyperbolically embeddable variety over a number field K $K$ with only finitely many O L , T $\mathcal {O}_{L,T}$ -points for any number field L / K $L/K$ and any finite set of finite places T $T$ of L $L$ has, in fact, only finitely many points in any given Z $\mathbb {Z}$ -finitely generated integral domain of characteristic zero. We use this latter result in combination with Green's criterion for hyperbolic embeddability to obtain novel finiteness results for integral points on symmetric self-products of smooth affine curves and on complements of large divisors in projective varieties. Finally, we use a partial converse to Green's criterion to further study hyperbolic embeddability (or its failure) in the case of symmetric self-products of curves. As a by-product of our results, we obtain the first example of a smooth affine Brody-hyperbolic threefold over C $\mathbb {C}$ which is not hyperbolically embeddable.

Points of bounded height on curves and the dimension growth conjecture over Fq[t]$\mathbb {F}_q[t

In this article, we prove several new uniform upper bounds on the number of points of bounded height on varieties over F q [ t ] $\mathbb {F}_q[t]$ . For projective curves, we prove the analogue of Walsh' result with polynomial dependence on q $q$ and the degree d $d$ of the curve. For affine curves, this yields an improvement to bounds by Sedunova, and Cluckers, Forey and Loeser. In higher dimensions, we prove a version of dimension growth for hypersurfaces of degree d ⩾ 64 $d\geqslant 64$ , building on work by Castryck, Cluckers, Dittmann and Nguyen in characteristic zero. These bounds depend polynomially on q $q$ and d $d$ , and it is this dependence which simplifies the treatment of the dimension growth conjecture.

Finite presentation of the tame fundamental group

Let p be a prime number, and let k be an algebraically closed field of characteristic p. We show that the tame fundamental group of a smooth affine curve over k is a projective profinite group. We prove that the fundamental group of a smooth projective variety over k is finitely presented; more generally, the tame fundamental group of a smooth quasi-projective variety over k, which admits a good compactification, is finitely presented.

P-Adic estimates of abelian ArtinL-functions on curves

AbstractThe purpose of this article is to prove a ‘Newton over Hodge’ result for finite characters on curves. LetXbe a smooth proper curve over a finite field$\mathbb {F}_q$of characteristic$p\geq 3$and let$V \subset X$be an affine curve. Consider a nontrivial finite character$\rho :\pi _1^{et}(V) \to \mathbb {C}^{\times }$. In this article, we prove a lower bound on the Newton polygon of theL-function$L(\rho ,s)$. The estimate depends on monodromy invariants of$\rho $: the Swan conductor and the local exponents. Under certain nondegeneracy assumptions, this lower bound agrees with the irregular Hodge filtration introduced by Deligne. In particular, our result further demonstrates Deligne’s prediction that the irregular Hodge filtration would forcep-adic bounds onL-functions. As a corollary, we obtain estimates on the Newton polygon of a curve with a cyclic action in terms of monodromy invariants.

An upper bound for the number of $S$-integral points on curves of genus zero

We consider affine plane algebraic curves of genus zero defined over number fields with three discrete valuations at infinity, and we determine an upper bound for the number of their $S$-integral points.

Algebraic Points of Degree at Most 5 on the Affine Curve y\(^{2}\) = x\(^{5}\) - 243

In this work, we determine the set of algebraic points of degree at most 5 on the ane curve y2 = x5 - 243. This result extends a result of J.TH Mulholland who described in [4] the set of \(\mathbb{Q}\)- rational points i.e the set of points of degree one over \(\mathbb{Q}\) on this curve.

On the torsion values for sections of an elliptic scheme

Abstract We shall consider sections of a complex elliptic scheme ℰ {{{\mathcal{E}}}} over an affine base curve B, and study the points of B where the section takes a torsion value. In particular, we shall relate the distribution in B of these points with the canonical height of the section, proving an integral formula involving a measure on B coming from the so-called Betti map of the section. We shall show that this measure is the same one which appears in dynamical issues related to the section. This analysis will also involve the multiplicity with which a torsion value is attained, which is an independent problem. We shall prove finiteness theorems for the points where the multiplicity is higher than expected. Such multiplicity has also a relation with Diophantine Approximation and quasi-integral points on ℰ {{{\mathcal{E}}}} (over the affine ring of B), and in Sections 5 and 6 of the paper we shall exploit this viewpoint, proving an effective result in the spirit of Siegel’s theorem on integral points.

Structural Evolution and Electronic Properties of Intermediate Sized Tin(n=33--60) Clusters

AbstractThe structural evolution of Tin () clusters has been investigated by using the systematic cluster growth method together with the many‐body Gupta potential. The lowest energy structures are further refined by using density functional theory computations within the accuracy of the Perdew‐Burke‐Ernzerhof (PBE) exchange‐correlation functional and a plane‐wave basis set. Using this approach, the structures and stability trends of Tin clusters in the size range previously reported, are reproduced. New clusters with enhanced stability are found for , 38, 41, 43, 46, 48, 50, 53, 55, and 58 sizes. Tightly packed structures dominate principally for these clusters in which the shape of the structures can be understood based on the encapsulated motif. The geometries of the clusters from Ti33 up to Ti60 adopt oblate forms which is in contrast to the smaller clusters. The vertical ionization potential (vIP) and vertical electron affinity (vEA) curves show a step behavior depending of the size range for , while the chemical hardness decreases monotonically with the size, which is consistent with the d‐band center parameter. In contrast to previous reports, for it is found that a distorted Mackay icosahedron is more stable than the regular one. The theoretical electronic properties for the most stable isomers of Tin clusters are in good agreement with the available experimental data.

Recovering affine curves over finite fields from L-functions

Let $K$ be the function field of a curve over a finite field of odd characteristic. We investigate using $L$-functions of Galois extensions of $K$ to effectively recover $K$. When $K$ is the function field of the projective line with four rational points removed, we show how to use $L$-functions of a ray class field to effectively recover the removed points up to automorphisms of the projective line. When $K$ is the function field of a plane curve, we show how to effectively recover the equation of that curve using $L$-functions of Artin-Schreier extensions of $K$.