We study the p-adic valuation of the values of normalised Hecke eigenforms attached to non-isotrivial elliptic curves defined over function fields of transcendence degree one over finite fields of characteristic p. We derive upper bounds on the smallest attained valuation in terms of the minimal discriminant under a certain assumption on the function field and provide examples to show that our estimates are optimal. As an application of our results we also prove the analogue of the degree conjecture unconditionally for strong Weil curves with square-free conductor defined over function fields satisfying the assumption mentioned above.
This note gives an explicit example of transcendental Brauer-Manin obstruction to weak approximation. It has two features which the only previously known example of such obstruction did not have: the class in the Brauer group which is responsible for the obstruction is divisible, and the underlying algebraic variety is an elliptic surface (a K3 surface).
We construct a smooth and projective surface over an arbitrary number field that is a counterexample to the Hasse principle but has the infinite etale Brauer-Manin set. We also construct a surface with a unique rational point and the infinite etale Brauer-Manin set. The key new ingredient is the arithmetic of singular projective curves.
We show that if over some number field there exists a certain diagonal plane cubic curve that is locally solvable everywhere, but that does not have points over any cubic galois extension of the number field, then the algebraic part of the Brauer-Manin obstruction is not the only obstruction to the Hasse principle for K3 surfaces.
Let C be a curve of genus at least 2 imbedded in a product of elliptic curves. We give an explicit upper bound for the points in the intersection of C with the union of all algebraic subgroups of a certain codimension. As a corollary we obtain an explicit upper bound for the height of the rational points for some families of curves and an explicit version of the Manin-Dem'janenko method in products of elliptic curves
In this paper, we propose a feasible algorithm to give an explicit basis of the space of regular differential forms on the nonsingular projective model of any given plane algebraic curve. The algorithm is demonstrated for concrete examples, with our implementation over the computer algebra system Magma. As an application, we also describe the Cartier-Manin matrix of the nonsingular projective curve with respect to the basis computed by the algorithm.
We wish to give a short elementary proof of S. Saito's result that the Brauer-Manin obstruction for zero-cycles of degree 1 is the only one for curves, supposing the finiteness of the Tate-Shafarevich-group $\sha^1(A)$ of the Jacobian variety. In fact we show that we only need a conjecturally finite part of the Brauer-group for this obstruction to be the only one. We also comment on the situation in higher dimensions
Let $\mathbb{F}$ be a finite field and $C,D$ smooth, geometrically irreducible proper curves over $\mathbb{F}$ and set $K = \mathbb{F}(D)$. We consider Brauer-Manin and abelian descent obstructions to the existence of rational points and to weak approximation for the curve $C \otimes_\mathbb{F} K$. In particular, we show that Brauer-Manin is the only obstruction to weak approximation and the Hasse principle in the case that the genus of $D$ is less than that of $C$. We also show that we can identify the points corresponding to non-constant maps $D \to C$ using Frobenius descents.
We construct a conic bundle over an elliptic curve over a real quadratic field that is a counterexample to the Hasse principle not explained by the étale Brauer-Manin obstruction. We also give simple examples of threefolds with the same property that are families of 2-dimensional quadrics, and discuss some other examples and general properties of the Brauer-Manin obstruction.
We study the intersection of an algebraic variety with the maximal compact subgroup of a universal vectorial extension of a product of elliptic curves. For this intersection we show a Manin-Mumford type statement. This answers some questions posed by Corvaja-Masser-Zannier which arose in connection with their investigation of the intersection of a curve with real analytic subgroups of various algebraic groups. They prove finiteness in the situation of a single elliptic curve. Using Khovanskii's zero-estimates combined with a stratification result of Gabrielov-Vorobjov and recent work of the au…
We conjecture that if C is a curve of genus >1 over a number field k such that C(k) is empty, then a method of Scharaschkin (equivalent to the Brauer-Manin obstruction in the context of curves) supplies a proof that C(k) is empty. As evidence, we prove a corresponding statement in which C(F_v) is replaced by a random subset of the same size in J(F_v) for each residue field F_v at a place v of good reduction for C, and the orders of Jacobians over finite fields are assumed to be smooth (in the sense of having only small prime divisors) as often as random integers of the same size. If our conjec…
We revisit the calculation of the strong Weil curve in an isogeny class of elliptic curves over Q, of the Manin constant and modular degree of an elliptic curve, using modular symbols as defined in [Pollack-Stevens], now implemented in Pari/GP. There is no innovation claim.
We establish Grothendieck's section conjecture for an open subset of the Reichardt-Lind curve, and introduce the notion of a Brauer-Manin obstruction for sections of the fundamental group extension.
We reinterpret Kim's non-abelian reciprocity maps for algebraic varieties as obstruction towers of mapping spaces of etale homotopy types, removing technical hypotheses such as global basepoints and cohomological constraints. We then extend the theory by considering alternative natural series of extensions, one of which gives an obstruction tower whose first stage is the Brauer--Manin obstruction, allowing us to determine when Kim's maps recover the Brauer-Manin locus. A tower based on relative completions yields non-trivial reciprocity maps even for Shimura varieties; for the stacky modular c…
We prove the Hasse-Weil inequality for genus 2 curves given by an equation of the form y^2 = f(x) with f a polynomial of degree 5, using arguments that mimic the elementary proof of the genus 1 case obtained by Yu. I. Manin in 1956.
Let $A$ be a simple abelian surface over an algebraically closed field $k$. Let $S\subset A(k)$ be the set of torsion points $x$ of $A$ such that there exists a genus $2$ curve $C$ and a map $f: C\to A$ such that $x$ is in the image of $f$, and $f$ sends a Weierstrass point of $C$ to the origin of $A$. The purpose of this note is to show that if $k$ has characteristic zero, then $S$ is finite -- this is in contrast to the situation where $k$ is the algebraic closure of a finite field, where $S=A(k)$, as shown by Bogomolov and Tschinkel. We deduce that if $k=\bar{\mathbb{Q}}$, the Kummer surfac…
For a curve over a global field we consider for which integers d the d-primary part of the Brauer group can obstruct the existence of rational points. We give examples showing it is possible that there is a d-primary obstruction for infinitely many relatively coprime d, and also an example where the odd part of the Brauer group does not obstruct although there is a Brauer-Manin obstruction. These examples demonstrate that a slightly stronger form of a conjecture of Poonen is false, despite being supported by the same heuristic.
We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomolgy with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural algebro-geometric framework for higher genus quasi Jacobi forms of index zero and their differential equations which are given as vector fields. In the case of elliptic curves we compute explicitly the Gauss-Manin connection and such vector fields.
Let X be a smooth variety over a number field k embedded as a degree d subvariety of $\mathbb{P}^n$ and suppose that X is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction. We consider the question of whether the obstruction is given by the d-primary subgroup of the Brauer group, which would have both theoretic and algorithmic implications. We prove that this question has a positive answer in the case of torsors under abelian varieties, Kummer surfaces and (conditional on finiteness of Tate-Shafarevich groups) bielliptic surfaces. For Kummer varieties we show th…
We introduce a general strategy for proving quantitative and uniform bounds on the number of common points of height zero for a pair of inequivalent height functions on $\mathbb{P}^1(\overline{\mathbb{Q}}).$ We apply this strategy to prove a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds on the number of common torsion points of elliptic curves in the case of two Legendre curves over $\mathbb{C}$. As a consequence, we obtain two uniform bounds for a two-dimensional family of genus 2 curves: a uniform Manin-Mumford bound for the family over $\mathbb{C}$, and a uniform Bogom…
For an optimal modular parametrization $J_0(n) \twoheadrightarrow E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$, Manin conjectured the agreement of two natural $\mathbb{Z}$-lattices in the $\mathbb{Q}$-vector space $H^0(E, Ω^1)$. Multiple authors generalized his conjecture to higher dimensional newform quotients. We prove the Manin conjecture for semistable $E$, give counterexamples to all the proposed generalizations, and prove several semistable special cases of these generalizations. The proofs establish general relations between the integral $p$-adic etale and de Rham coho…
Since Poonen's construction of a variety $X$ defined over a number field $k$ for which $X(k)$ is empty and the étale Brauer--Manin set $X(\mathbf{A}_k)^\text{Br,et}$ is not, several other examples of smooth, projective varieties have been found for which the étale Brauer--Manin obstruction does not explain the failure of the Hasse principle. All known examples are constructed using "Poonen's trick", i.e. they have the distinctive feature of being fibrations over a higher genus curve; in particular, their Albanese variety is non-trivial. In this paper, we construct examples for which the Albane…
We study rational points on conic bundles over elliptic curves with positive rank over a number field. We show that the etale Brauer-Manin obstruction is insufficient to explain failures of the Hasse principle for such varieties. We then further consider properties of the distribution of the set of rational points with respect to its image in the rational points of the elliptic curve. In the process, we prove results on a local-to-global principle for torsion points on elliptic curves over $\mathbb{Q}$.
Let k be an algebraically closed field of characteristic p > 0. Every Artin-Schreier k-curve X has an equation of the form y^p - y = f(x) for some f(x) in k(x) such that p does not divide the least common multiple L of the orders of the poles of f(x). Under the condition that p is congruent to 1 mod L, Zhu proved that the Newton polygon of the L-function of X is determined by the Hodge polygon of f(x). In particular, the Newton polygon depends only on the orders of the poles of f(x) and not on the location of the poles or otherwise on the coefficients of f(x). In this paper, we prove an analog…
We prove that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture, in the situation where the ambient variety is an abelian variety defined over the function field of a smooth curve over a finite field and the relevant group is a finitely generated group. In particular, in the setting of the last sentence, we provide a proof of the Mordell-Lang conjecture, which does not depend on tools coming from model theory.
We study the sequence of zeta functions $Z(C_p,T)$ of a generic Picard curve $C:y^3=f(x)$ defined over $\mathbb{Q}$ at primes $p$ of good reduction for $C$. We define a degree 9 polynomial $ψ_f\in \mathbb{Q}[x]$ such that the splitting field of $ψ_f(x^3/2)$ is the $2$-torsion field of the Jacobian of $C$. We prove that, for all but a density zero subset of primes, the zeta function $Z(C_p,T)$ is uniquely determined by the Cartier-Manin matrix $A_p$ of $C$ modulo $p$ and the splitting behavior modulo $p$ of $f$ and $ψ_f$; we also show that for primes $\equiv 1 \pmod{3}$ the matrix $A_p$ suffice…
Consider a subgroup of finite index of modular group. We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian of the corresponding modular curve. By BelyI theorem, such a criterion would apply to any curve over a number field. Our main tool is the explicit description, in terms of modular symbols, of what we call Eisenstein cycles. The latter are representations of relative homology classes over which integration of any holomorphic differential forms vanishes. Our approach relies in an essential way on the specific case , where we can consider convenient generalized …
We consider the Gauss-Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallelly to themselves. We reduce these equations modulo a prime integer $p$ and construct polynomial solutions of the new differential equations as $p$-analogs of the initial hypergeometric integrals. In some cases we interpret the $p$-analogs of the hypergeometric integrals as sums over points of hypersurfaces defined over the finite field $F_p$. That interpretation is similar to the interpretation by Yu.I. Manin in [Ma] of the number of po…
In this paper, we construct three kinds of Châtelet surfaces, which have some given arithmetic properties with respect to field extensions of number fields. We then use these constructions to study the properties of weak approximation with Brauer-Manin obstruction and the Hasse principle with Brauer-Manin obstruction for $3$-folds, which are pencils of Châtelet surfaces parameterized by a curve, with respect to field extensions of number fields. We give general constructions (conditional on a conjecture of M. Stoll) to negatively answer some questions, and illustrate these constructions and so…
An elliptic curve defined over a number field possesses only a finite number of torsion points defined over the cyclotomic closure of its field of definition. In analogy to the relative version of the Manin-Mumford conjecture stated by Masser and Zannier, we propose a family version of the above statement and prove it under a suitable integrality condition.
Stevens conjectured that for every optimal parametrization $ϕ\colon X_1(n) \rightarrow E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$, the pullback of some Néron differential on $E$ is the differential associated to the normalized new eigenform that corresponds to the isogeny class of $E$. We prove this conjecture under the assumption that $E$ is semistable, the key novelty lying in the $2$-primary analysis when $n$ is even. For this analysis, we first relate the general case of the conjecture to a divisibility relation between $\mathrm{deg}\, ϕ$ and a certain congruence number…
The hypergeometric solutions of the KZ equations were constructed almost 30 years ago. The polynomial solutions of the KZ equations over the finite field $F_p$ with a prime number $p$ of elements were constructed recently. In this paper we consider the example of the KZ equations whose hypergeometric solutions are given by hyperelliptic integrals of genus $g$. It is known that in this case the total $2g$-dimensional space of holomorphic solutions is given by the hyperelliptic integrals. We show that the recent construction of the polynomial solutions over the field $F_p$ in this case gives onl…
The Manin constant $c$ of an elliptic curve $E$ over $\mathbb{Q}$ is the nonzero integer that scales the differential $ω_f$ determined by the normalized newform $f$ associated to $E$ into the pullback of a Néron differential under a minimal parametrization $ϕ\colon X_0(N)_{\mathbb{Q}} \twoheadrightarrow E$. Manin conjectured that $c = \pm 1$ for optimal parametrizations, and we prove that in general $c \mid \mathrm{deg}(ϕ)$ under a minor assumption at $2$ and $3$ that is not needed for cube-free $N$ or for parametrizations by $X_1(N)_{\mathbb{Q}}$. Since $c$ is supported at the additive reduct…
We prove that there are only finitely many complex numbers $a$ and $b$ with $4a^3+27b^2\not=0$ such that the three points $(1,*),(2,*),$ and $(3,*)$ are simultaneously torsion on the elliptic curve defined in Weierstrass form by $y^2=x^3+ax+b$. This gives an affirmative answer to a question raised by Masser and Zannier. We thus confirm a special case in two dimensions of the relative Manin-Mumford Conjecture formulated by Pink and Masser-Zannier.
We deduce an analogue of the Bogomolov conjecture for non-degenerate subvarieties in fibered products of families of elliptic curves from the author's recent theorem on equidistribution in families of abelian varieties. This generalizes results of DeMarco and Mavraki and improves certain results of Manin-Mumford type proven by Masser and Zannier to results of Bogomolov type, yielding the first results of this type for subvarieties of relative dimension $>1$ in families of abelian varieties with trivial trace.
Let $C$ be an algebraic curve embedded transversally in a power $E^N$ of an elliptic curve $E$. In this article we produce a good explicit bound for the height of all the algebraic points on $C$ contained in the union of all proper algebraic subgroups of $E^N$. The method gives a totally explicit version of the Manin-Dam'janenko Theorem in the elliptic case and it is a generalisation of previous results only proved when $E$ does not have Complex Multiplication.
In this article, we study obstructions to weak approximation for connected linear groups and homogeneous spaces with connected or abelian stabilizers over finite extensions of $\mathbb C((x,y))$ or function fields of curves over $\mathbb C((t))$. We show that for connected linear groups, the usual Brauer-Manin obstruction works as in the case of tori. However, this Brauer-Manin obstruction is not enough for homogeneous spaces, as shown by the examples we give. We then construct an obstruction using torsors under quasi-trivial tori that explains the failure of weak-approximation.
We study arithmetic of the algebraic varieties defined over number fields by applying Lagrange interpolation to fibrations. Assuming the finiteness of the Tate-Shafarevich group of a certain elliptic curve, we show, for Châtelet surface bundles over curves, that the violation of Hasse principle being accounted for by the Brauer-Manin obstruction is not invariant under an arbitrary finite extension of the ground field.
We introduce a procedure for constructing a generalized elliptic curve from a genus-one twisted curve, and we use this procedure to define an explicit, modular isomorphism between two compactifications of moduli stacks of elliptic curves with level structure.
We study the elliptic modular surface attached to the commutator subgroup of the modular group. This has an elliptic curve as base and only one singular fibre. We employ an algebraic approach and then consider some arithmetic questions.
We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to $C_{16}$, as well as all elliptic curves over cubic fields with a subgroup isomorphic to $C_2\times C_{14}$, are base changes of elliptic curves defined over $\Q$. We obtain these results by studying geometric properties of modular curves and maps between modular curves, and then obtaining a modular description of these curves and maps.
