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In this article, we are interested in modular forms with non-vanishing central critical values and linear independence of Fourier coefficients of modular forms. The main ingredient is a generalization of a theorem due to VanderKam to modular symbols of higher weights. We prove that for sufficiently large primes p, Hecke operators T 1 , T 2 , ... , T D act linearly independently on the winding elements inside the space of weight 2k cuspidal modular symbol S 2 k (Γ 0 (p)) with k ≥ 1 for D 2 ≪ p . This gives a bound on the number of newforms with non-vanishing arithmetic L-functions at their central critical points and linear independence on the reductions of these modular forms for prime modulo l ≠ p . [ABSTRACT FROM AUTHOR]

We compute the Mordell–Weil groups of the modular Jacobian varieties of hyperelliptic modular curves X 1 (M , M N) over every composite field of some quadratic number fields. Also we prove criteria for the existence of elliptic curves over such number fields with prescribed torsion points generalizing the results for quadratic number fields of Kamienny and Najman. [ABSTRACT FROM AUTHOR]

Let N be a positive integer. Let H be a subgroup of GL (2 , Z / N Z) of level N and let E be an elliptic curve defined over the rationals with j E ≠ 0 , 1728 . Then the image ρ ¯ E , N Gal Q ¯ / Q , of the mod-N Galois representation attached to E, is conjugate to a subgroup of H if and only if E corresponds to a non-cuspidal rational point on the modular curve X H generated by H. In this article, we are interested when ρ ¯ E , N Gal Q ¯ / Q is conjugate to H. More precisely, we classify all subgroups H of GL (2 , Z / N Z) that are direct products of groups of distinct prime-power level for which X H contains infinitely many non-cuspidal rational points but there is no elliptic curve E / Q such that ρ ¯ E , N Gal Q ¯ / Q is conjugate to H itself. [ABSTRACT FROM AUTHOR]

In this paper we inspect from closer the local and global points of the twists of the Klein quartic. For the local ones we use geometric arguments, while for the global ones we strongly use the modular interpretation of the twists. The main result is providing families with (conjecturally infinitely many) twists of the Klein quartic that are counterexamples to the Hasse Principle. [ABSTRACT FROM AUTHOR]

Let $X$ be a smooth geometrically connected projective curve over the field of fractions of a discrete valuation ring $R$ , and $\mathfrak {m}$ a modulus on $X$ , given by a closed subscheme of $X$ which is geometrically reduced. The generalized Jacobian $J_\mathfrak {m}$ of $X$ with respect to $\mathfrak {m}$ is then an extension of the Jacobian of $X$ by a torus. We describe its Néron model, together with the character and component groups of the special fibre, in terms of a regular model of $X$ over $R$. This generalizes Raynaud's well-known description for the usual Jacobian. We also give some computations for generalized Jacobians of modular curves $X_0(N)$ with moduli supported on the cusps. [ABSTRACT FROM AUTHOR]

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]

Let d ≥ 1 be an integer and let p be a rational prime. Recall that p is a torsion prime of degree d if there exists an elliptic curve E over a degree d number field K such that E has a K-rational point of order p. Derickx et al. (Algebra Number Theory 17(2):267–308, 2023) have computed the torsion primes of degrees 4, 5, 6 and 7. We verify that the techniques used in Derickx et al. (Algebra Number Theory 17(2):267–308, 2023) can be extended to determine the torsion primes of degree 8. [ABSTRACT FROM AUTHOR]

We determine all modular curves X 0 (N) with infinitely many quartic points. To do this, we define a pairing that induces a quadratic form representing all possible degrees of a rational morphism from X 0 (N) to a positive rank elliptic curve. [ABSTRACT FROM AUTHOR]

We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over ℚ with 1 2 -torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen and Fisher where it is assumed that the underlying isomorphism of 1 2 -torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrize such pairs of elliptic curves. A key ingredient in the proof is to construct simple (algebraic) conditions for the 2 , 3 or 4 -torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the j -invariants of the pair of elliptic curves. [ABSTRACT FROM AUTHOR]

In this paper we determine the \mathbb {Q}-gonalities of the modular curves X_0(N) for all N<145. We determine the \mathbb {C}-gonality of many of these curves and the \mathbb {Q}-gonalities and \mathbb {C}-gonalities for many larger values of N. Using these results and some further work, we determine all the modular curves X_0(N) of gonality 4, 5 and 6 over \mathbb {Q}. We also find the first known instances of pentagonal curves X_0(N) over \mathbb {C}. [ABSTRACT FROM AUTHOR]

Let p and q be distinct prime numbers with q ≡ 1 (mod 12) . Let N be a positive integer that is coprime to pq. We prove a formula relating the Hasse–Weil zeta function of the modular curve X 0 (q N) F q to the Ihara zeta function of the p-isogeny graph of supersingular elliptic curves defined over F q ¯ equipped with a Γ 0 (N) -level structure. When N = 1 , this recovers a result of Sugiyama. [ABSTRACT FROM AUTHOR]

Let C be a curve defined over a number field k. We say a closed point x ∈ C of degree d is isolated if it does not belong to an infinite family of degree d points parametrized by the projective line or a positive rank abelian subvariety of the curve's Jacobian. Building on work of Bourdon et al. (Adv Math 357(33):106824, 2019), we characterize elliptic curves with rational j-invariant which give rise to an isolated point of odd degree on X 1 (N) / Q for some positive integer N. [ABSTRACT FROM AUTHOR]

We describe an algorithm to compute the number of points over finite fields on a broad class of modular curves: we consider quotients X H / W for H a subgroup of GL 2 (Z / n Z) such that for each prime p dividing n , the subgroup H at p is either a Borel subgroup, a Cartan subgroup, or the normalizer of a Cartan subgroup of GL 2 (Z / p e Z) , and for W any subgroup of the Atkin-Lehner involutions of X H. We applied our algorithm to more than ten thousand curves of genus up to 50, finding more than one hundred record-breaking curves, namely curves X / F q with genus g that improve the previously known lower bound for the maximum number of points over F q of a curve with genus g. As a key technical tool for our computations, we prove the generalization of Chen's isogeny to all the Cartan modular curves of composite level. [ABSTRACT FROM AUTHOR]

In this paper we discuss applications of our earlier work in studying certain Galois groups and splitting fields of rational functions in Q (X 0 (N)) using Hilbert's irreducibility theorem and modular forms. We also consider computational aspect of the problem using MAGMA and SAGE. [ABSTRACT FROM AUTHOR]

Let ℰ be a ℚ -isogeny class of elliptic curves defined over ℚ. The isogeny graph associated to ℰ is a graph which has a vertex for each elliptic curve over ℚ of ℰ and an edge for each ℚ -isogeny of prime degree that maps one elliptic curve in ℰ to another elliptic curve in ℰ , with the degree of the isogeny recorded as a label of the edge. The isogeny-torsion graph associated to ℰ is the isogeny graph associated to ℰ where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over ℚ of the corresponding elliptic curve. The main result of the article is a classification of the 2 -adic Galois image at each vertex of the isogeny-torsion graphs whose associated ℚ -isogeny class consists of elliptic curves over ℚ with complex multiplication. [ABSTRACT FROM AUTHOR]

Bruin and Najman [LMS J. Comput. Math. 18 (2015), pp. 578–602], Ozman and Siksek [Math. Comp. 88 (2019), pp. 2461–2484], and Box [Math. Comp. 90 (2021), pp. 321–343] described all the quadratic points on the modular curves of genus 2\leq g(X_0(n)) \leq 5. Since all the hyperelliptic curves X_0(n) are of genus \leq 5 and as a curve can have infinitely many quadratic points only if it is either of genus \leq 1, hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves X_0(n) naturally arises; this question has recently also been posed by Mazur. We answer Mazur's question completely and describe the quadratic points on all the bielliptic modular curves X_0(n) for which this has not been done already. The values of n that we deal with are n=60, 62, 69, 79, 83, 89, 92, 94, 95, 101, 119 and 131; the curves X_0(n) are of genus up to 11. We find all the exceptional points on these curves and show that they all correspond to CM elliptic curves. The two main methods we use are Box's relative symmetric Chabauty method and an application of a moduli description of \mathbb {Q}-curves of degree d with an independent isogeny of degree m, which reduces the problem to finding the rational points on several quotients of modular curves. [ABSTRACT FROM AUTHOR]

We give a new proof of Faltings's $p$ -adic Eichler–Shimura decomposition of the modular curves via Bernstein–Gelfand–Gelfand (BGG) methods and the Hodge–Tate period map. The key property is the relation between the Tate module and the Faltings extension, which was used in the original proof. Then we construct overconvergent Eichler–Shimura maps for the modular curves providing 'the second half' of the overconvergent Eichler–Shimura map of Andreatta, Iovita and Stevens. We use higher Coleman theory on the modular curve developed by Boxer and Pilloni to show that the small-slope part of the Eichler–Shimura maps interpolates the classical $p$ -adic Eichler–Shimura decompositions. Finally, we prove that overconvergent Eichler–Shimura maps are compatible with Poincaré and Serre pairings. [ABSTRACT FROM AUTHOR]

We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus $g>1$ whose Jacobians have Mordell–Weil rank $g$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients $X_0^+(N)$ of prime level $N$ , the curve $X_{S_4}(13)$ , as well as a few other curves relevant to Mazur's Program B. We also compute the set of rational points on the genus 6 non-split Cartan modular curve $X_{\scriptstyle \mathrm { ns}} ^+ (17)$. [ABSTRACT FROM AUTHOR]

Purpose – The authors have determined whether the points fixed by all the full and the partial Atkin–Lehner involutions WQ on X0(N) for N ≤ 50 are Weierstrass points or not. Design/methodology/approach – The design is by using Lawittes's and Schoeneberg's theorems. Findings – Finding all Weierstrass points on X0(N) fixed by some Atkin–Lehner involutions. Besides, the authors have listed them in a table. Originality/value – The Weierstrass points have played an important role in algebra. For example, in algebraic number theory, they have been used by Schwartz and Hurwitz to determine the group structure of the automorphism groups of compact Riemann surfaces of genus g ≥ 2. Whereas in algebraic geometric coding theory, if one knows a Weierstrass nongap sequence of a Weierstrass point, then one is able to estimate parameters of codes in a concrete way. Finally, the set of Weierstrass points is useful in studying arithmetic and geometric properties of X0(N).

Let E/Q be an elliptic curve, let ... be a fixed algebraic closure of Q, and let GQ D Gal(...) be the absolute Galois group of Q. The action of GQ on the adelic Tate module of E induces the adelic Galois representation ... . The goal of this paper is to explain how the image of ρE can be smaller than expected. To this end, we offer a group theoretic categorization of different ways in which an entanglement between division fields can be explained and prove several results on elliptic curves (and more generally, principally polarized abelian varieties) over Q where the entanglement occurs over an abelian extension. [ABSTRACT FROM AUTHOR]

In this paper, we determine all modular Jacobian varieties J 1 (M , M N) over the cyclotomic fields Q (ζ M) with the Mordell–Weil rank zero assuming the Birch–Swinnerton-Dyer conjecture, following the method of Derickx, Etropolski, van Hoeij, Morrow, and Zureick-Brown. [ABSTRACT FROM AUTHOR]

In this paper we prove that there are finitely many modular curves that admit a smooth plane model. Moreover, if the degree of the model is greater than or equal to 19, no such curve exists. For modular curves of Shimura type we show that none can admit a smooth plane model of degree 5, 6 or 7. Further, if a modular curve of Shimura type admits a smooth plane model of degree 8 we show that it must be a twist of one of four curves. [ABSTRACT FROM AUTHOR]

Consider an elliptic curve E over a number field K. Suppose that E has supersingular reduction at some prime p of K lying above the rational prime p. We completely classify the valuations of the p n -torsion points of E by the valuation of a coefficient of the p th division polynomial. This classification corrects an error in earlier work of Lozano-Robledo. As an application, we find the minimum necessary ramification at p in order for E to have a point of exact order p n . Using this bound we show that sporadic points on the modular curve X 1 (p n) cannot correspond to supersingular elliptic curves without a canonical subgroup. We generalize our methods to X 1 (N) with N composite. [ABSTRACT FROM AUTHOR]

Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with monodromy PGL 2 (F q) or PSL 2 (F q) . We present a computational approach to enumerate Borel-type triangular modular curves of low genus, and we carry out this enumeration for prime level and small genus. [ABSTRACT FROM AUTHOR]

The Hilbert class polynomial has as roots the j-invariants of elliptic curves whose endomorphism ring is a given imaginary quadratic order. It can be used to compute elliptic curves over finite fields with a prescribed number of points. Since its coefficients are typically rather large, there has been continued interest in finding alternative modular functions whose corresponding class polynomials are smaller. Best known are Weber's functions, which reduce the size by a factor of 72 for a positive density subset of imaginary quadratic discriminants. On the other hand, Bröker and Stevenhagen showed that no modular function will ever do better than a factor of 100.83. We introduce a generalization of class polynomials, with reduction factors that are not limited by the Bröker–Stevenhagen bound. We provide examples matching Weber's reduction factor. For an infinite family of discriminants, their reduction factors surpass those of all previously known modular functions by a factor at least 2. [ABSTRACT FROM AUTHOR]

We study the rational torsion subgroup of the modular Jacobian J0(N) for N a square-free integer. We give a proof of a result of Ohta on a generalization of Ogg’s conjecture: For a prime number p∤6N, the p-primary part of the rational torsion subgroup equals that of the cuspidal subgroup. Whereas previous proofs of this result used explicit computations of the cardinalities of these groups, we instead use their structure as modules for the Hecke algebra. [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

We study the finiteness of low degree points on certain modular curves and their Atkin–Lehner quotients, and, as an application, prove the modularity of elliptic curves over all but finitely many totally real fields of degree 5. On the way, we prove a criterion for the finiteness of rational points of degree 5 on a curve of large genus over a number field using the results of Abramovich–Harris and Faltings on subvarieties of Jacobians. [ABSTRACT FROM AUTHOR]

W. Sun introduced the concept of g-frames which are generalized frames in Hilbert spaces. In this paper, we consider linear combination of g-frames with coefficients as bounded linear operators and construct some new g-frames from existing g-frames, considering the cases of orthogonal and alternate dual g-frame pair in a Hilbert space and give alternate proofs of some previously proved results. We use our result to construct Gabor frames. [ABSTRACT FROM AUTHOR]

We prove that every elliptic curve defined over a totally real number field of degree 4 not containing √5 is modular. To this end, we study the quartic points on four modular curves. [ABSTRACT FROM AUTHOR]

We study the splitting of primes in number fields generated by points on modular curves. Momose (Nagoya Math J 96:139–165, 1984) was the first to notice that quadratic points on X 1 (N) generate quadratic fields over which certain primes split in a particular way and his results were later expanded upon by Krumm (Quadratic Points on Modular Curves. PhD thesis, University of Georgia, 2013). We prove results about the splitting behaviour of primes in quadratic fields generated by points on the modular curves X 0 (N) which are hyperelliptic (except for N = 37 ) and in cubic fields generated by points on X 1 (2 , 14) . [ABSTRACT FROM AUTHOR]

In this paper, we study quadratic points on the non-split Cartan modular curves X n s (p) , for p = 7 , 1 1 , and 1 3. Recently, Siksek proved that all quadratic points on X n s (7) arise as pullbacks of rational points on X n s + (7). Using similar techniques for p = 1 1 , and employing a version of Chabauty for symmetric powers of curves for p = 1 3 , we show that the same holds for X n s (1 1) and X n s (1 3). As a consequence, we prove that certain classes of elliptic curves over quadratic fields are modular. [ABSTRACT FROM AUTHOR]

We construct Hida and Coleman theories for the degree 0 and 1 cohomology of automorphic line bundles on the modular curve, and we define a p-adic duality pairing between the theories in degree 0 and 1. [ABSTRACT FROM AUTHOR]

Let q > 2 be a prime number, χ a primitive Dirichletcharacter modulo q and f a~primitive holomorphic cusp form or a Hecke-Maass cusp formof level q and trivial nebentypus. We prove the subconvex bound L (1 / 2, f ⊗ χ) ≪ q 1 / 2 - 1 / 12 + ε, where the implicit constant depends only on the archimedean parameter of f and ε. The main input is a modifyingtrivial delta methoddeveloped in [1]. [ABSTRACT FROM AUTHOR]

W. Kohnen asked whether the values of the classical Dedekind sum are dense in the p-adic field Qp, and K. Girstmair solved this problem. In this paper, we solve the analogous problem for the Dedekind sum in function fields. [ABSTRACT FROM AUTHOR]

We show that there is essentially a unique elliptic curve E defined over a cubic Galois extension K of Q with a K-rational point of order 13 and such that E is not defined over Q. [ABSTRACT FROM AUTHOR]

For an extension K/Fq(T) of the rational function field over a finite field, we introduce the notion of virtually K-rational Drinfeld modules as a function field analogue of Q-curves. Our goal in this article is to prove that all virtually K-rational Drinfeld modules of rank two with no complex multiplication are parametrized up to isogeny by K-rational points of a quotient curve of the Drinfeld modular curve Y0(n) with some square-free level n. This is an analogue of Elkies' well-known result on Q-curves. [ABSTRACT FROM AUTHOR]

We describe an algorithm for computing a Q -rational model for the quotient of a modular curve by an automorphism group, under mild assumptions on the curve and the automorphisms, by determining q-expansions for a basis of the corresponding space of cusp forms. We also give a moduli interpretation for general morphisms between modular curves. [ABSTRACT FROM AUTHOR]

For a positive integer N , we say that ∞ is a Weierstrass point on the modular curve X 0 (N) if there is a non-zero cusp form of weight 2 on Γ 0 (N) which vanishes at ∞ to order greater than the genus of X 0 (N). If p is a prime with p ∤ N , Ogg proved that ∞ is not a Weierstrass point on X 0 (p N) if the genus of X 0 (N) is 0. We prove a similar result for even weights k ≥ 4. We also study the space of weight k cusp forms on Γ 0 (N) vanishing to order greater than the dimension. [ABSTRACT FROM AUTHOR]

This paper is a continuation of our previous works where we study maps from X 0 (N) , N ≥ 1 , into P 2 constructed via modular forms of the same weight and criteria that such a map is birational (see Muić in Monatsh Math 180(3):607–629, 2016). In the present paper, our approach is based on the theory of primitive elements in finite separable field extensions. We prove that in most of the cases the constructed maps are birational. We consider particular cases and their equations in P 2 . [ABSTRACT FROM AUTHOR]

We give a formula for divisors of modular units on X 1 (N) and use it to prove that the Q -gonality of the modular curve X 1 (N) is bounded above by [ 11 N 2 / 840 ] , where [ · ] denotes the nearest integer. [ABSTRACT FROM AUTHOR]

Bruin and Najman [LMS J. Comput. Math. 18 (2015), no. 1, 578-602] and Ozman and Siksek [Math. Comp. 88 (2019), no. 319, 2461-2484] have recently determined the quadratic points on each modular curve X0(N) of genus 2, 3, 4, or 5 whose Mordell-Weil group has rank 0. In this paper we do the same for the X0(N) of genus 2, 3, 4, and 5 and positive Mordell-Weil rank. The values of N are 37, 43, 53, 61, 57, 65, 67, and 73. The main tool used is a relative symmetric Chabauty method, in combination with the Mordell-Weil sieve. Often the quadratic points are not finite, as the degree 2 map X0(N) → X0(N)+ can be a source of infinitely many such points. In such cases, we describe this map and the rational points on X0(N)+, and we specify the exceptional quadratic points on X0(N) not coming from X0(N)+. In particular, we determine the j-invariants of the corresponding elliptic curves and whether they are Q-curves or have complex multiplication. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective geometrically irreducible curve over a perfect field k of positive characteristic p. Suppose G is a finite group acting faithfully on X such that G has non-trivial cyclic Sylow p -subgroups. We show that the decomposition of the space of holomorphic differentials of X into a direct sum of indecomposable k [ G ] -modules is uniquely determined by the lower ramification groups and the fundamental characters of closed points of X that are ramified in the cover X ⟶ X / G. We apply our method to determine the PSL (2 , F ℓ) -module structure of the space of holomorphic differentials of the reduction of the modular curve X (ℓ) modulo p when p and ℓ are distinct odd primes and the action of PSL (2 , F ℓ) on this reduction is not tamely ramified. This provides some non-trivial congruences modulo appropriate maximal ideals containing p between modular forms arising from isotypic components with respect to the action of PSL (2 , F ℓ) on X (ℓ). [ABSTRACT FROM AUTHOR]

Let N ≥ 1 be an square free integer and let W N be a non-trivial subgroup of the group of the Atkin-Lehner involutions of X 0 (N) such that the modular curve X 0 (N) / W N has genus at least two. We determine all pairs (N , W N) such that X 0 (N) / W N is a bielliptic curve and the pairs (N , W N) such that X 0 (N) / W N has an infinite number of quadratic points over Q. [ABSTRACT FROM AUTHOR]