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Search: MSC category 11G ( Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14Gxx, 14Kxx] )

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1. CJM Online first

Cojocaru, Alina Carmen; Shulman, Andrew Michael
 The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank Let $\psi$ be a generic Drinfeld module of rank $r \geq 2$. We study the first elementary divisor $d_{1, \wp}(\psi)$ of the reduction of $\psi$ modulo a prime $\wp$, as $\wp$ varies. In particular, we prove the existence of the density of the primes $\wp$ for which $d_{1, \wp} (\psi)$ is fixed. For $r = 2$, we also study the second elementary divisor (the exponent) of the reduction of $\psi$ modulo $\wp$ and prove that, on average, it has a large norm. Our work is motivated by the study of J.-P. Serre of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M.R. Murty. Keywords:Drinfeld modules, density theoremsCategories:11R45, 11G09, 11R58

2. CJM 2015 (vol 67 pp. 1046)

Dubickas, Arturas; Sha, Min; Shparlinski, Igor
 Explicit Form of Cassels' $p$-adic Embedding Theorem for Number Fields In this paper, we mainly give a general explicit form of Cassels' $p$-adic embedding theorem for number fields. We also give its refined form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over $\mathbb{Z}$, we give a general unconditional upper bound for the smallest prime number $p$ such that $f$ has a simple root modulo $p$. Keywords:number field, $p$-adic embedding, height, polynomial, cyclotomic fieldCategories:11R04, 11S85, 11G50, 11R09, 11R18

3. CJM Online first

Fité, Francesc; González, Josep; Lario, Joan Carles
 Frobenius distribution for quotients of Fermat curves of prime exponent Let $\mathcal{C}$ denote the Fermat curve over $\mathbb{Q}$ of prime exponent $\ell$. The Jacobian $\operatorname{Jac}(\mathcal{C})$ of~$\mathcal{C}$ splits over $\mathbb{Q}$ as the product of Jacobians $\operatorname{Jac}(\mathcal{C}_k)$, $1\leq k\leq \ell-2$, where $\mathcal{C}_k$ are curves obtained as quotients of $\mathcal{C}$ by certain subgroups of automorphisms of $\mathcal{C}$. It is well known that $\operatorname{Jac}(\mathcal{C}_k)$ is the power of an absolutely simple abelian variety $B_k$ with complex multiplication. We call degenerate those pairs $(\ell,k)$ for which $B_k$ has degenerate CM type. For a non-degenerate pair $(\ell,k)$, we compute the Sato-Tate group of $\operatorname{Jac}(\mathcal{C}_k)$, prove the generalized Sato-Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of $(\ell,k)$ being degenerate or not, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the $\ell$-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices. Keywords:Sato-Tate group, Fermat curve, Frobenius distributionCategories:11D41, 11M50, 11G10, 14G10

4. CJM 2015 (vol 67 pp. 654)

Lim, Meng Fai; Murty, V. Kumar
 Growth of Selmer groups of CM Abelian varieties Let $p$ be an odd prime. We study the variation of the $p$-rank of the Selmer group of Abelian varieties with complex multiplication in certain towers of number fields. Keywords:Selmer group, Abelian variety with complex multiplication, $\mathbb{Z}_p$-extension, $p$-Hilbert class towerCategories:11G15, 11G10, 11R23, 11R34

5. CJM Online first

Bonfanti, Matteo Alfonso; van Geemen,
 Abelian Surfaces with an Automorphism and Quaternionic Multiplication We construct one dimensional families of Abelian surfaces with quaternionic multiplication which also have an automorphism of order three or four. Using Barth's description of the moduli space of $(2,4)$-polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space which parametrizes Abelian surfaces with real multiplication by $\mathbf{Z}[\sqrt{2}]$. Keywords:abelian surfaces, moduli, shimura curvesCategories:14K10, 11G10, 14K20

6. CJM Online first

Chandee, Vorrapan; David, Chantal; Koukoulopoulos, Dimitris; Smith, Ethan
 The frequency of elliptic curve groups over prime finite fields Letting $p$ vary over all primes and $E$ vary over all elliptic curves over the finite field $\mathbb{F}_p$, we study the frequency to which a given group $G$ arises as a group of points $E(\mathbb{F}_p)$. It is well-known that the only permissible groups are of the form $G_{m,k}:=\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/mk\mathbb{Z}$. Given such a candidate group, we let $M(G_{m,k})$ be the frequency to which the group $G_{m,k}$ arises in this way. Previously, the second and fourth named authors determined an asymptotic formula for $M(G_{m,k})$ assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for $M(G_{m,k})$, pointwise and on average. In particular, we show that $M(G_{m,k})$ is bounded above by a constant multiple of the expected quantity when $m\le k^A$ and that the conjectured asymptotic for $M(G_{m,k})$ holds for almost all groups $G_{m,k}$ when $m\le k^{1/4-\epsilon}$. We also apply our methods to study the frequency to which a given integer $N$ arises as the group order $\#E(\mathbb{F}_p)$. Keywords:average order, elliptic curves, primes in short intervalsCategories:11G07, 11N45, 11N13, 11N36

7. CJM Online first

Stange, Katherine E.
 Integral Points on Elliptic Curves and Explicit Valuations of Division Polynomials Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ and non-torsion point $P \in E(\mathbb{Q})$, there is at most one integral multiple $[n]P$ such that $n \gt C$. The proof is a modification of a proof of Ingram giving an unconditional but not uniform bound. The new ingredient is a collection of explicit formulae for the sequence $v(\Psi_n)$ of valuations of the division polynomials. For $P$ of non-singular reduction, such sequences are already well described in most cases, but for $P$ of singular reduction, we are led to define a new class of sequences called \emph{elliptic troublemaker sequences}, which measure the failure of the NÃ©ron local height to be quadratic. As a corollary in the spirit of a conjecture of Lang and Hall, we obtain a uniform upper bound on $\widehat{h}(P)/h(E)$ for integer points having two large integral multiples. Keywords:elliptic divisibility sequence, Lang's conjecture, height functionsCategories:11G05, 11G07, 11D25, 11B37, 11B39, 11Y55, 11G50, 11H52

8. CJM 2014 (vol 67 pp. 198)

Murty, V. Kumar; Patankar, Vijay M.
 Tate Cycles on Abelian Varieties with Complex Multiplication We consider Tate cycles on an Abelian variety $A$ defined over a sufficiently large number field $K$ and having complex multiplication. We show that there is an effective bound $C = C(A,K)$ so that to check whether a given cohomology class is a Tate class on $A$, it suffices to check the action of Frobenius elements at primes $v$ of norm $\leq C$. We also show that for a set of primes $v$ of $K$ of density $1$, the space of Tate cycles on the special fibre $A_v$ of the NÃ©ron model of $A$ is isomorphic to the space of Tate cycles on $A$ itself. Keywords:Abelian varieties, complex multiplication, Tate cyclesCategories:11G10, 14K22

9. CJM 2013 (vol 66 pp. 826)

Kim, Byoung Du
 Signed-Selmer Groups over the $\mathbb{Z}_p^2$-extension of an Imaginary Quadratic Field Let $E$ be an elliptic curve over $\mathbb Q$ which has good supersingular reduction at $p\gt 3$. We construct what we call the $\pm/\pm$-Selmer groups of $E$ over the $\mathbb Z_p^2$-extension of an imaginary quadratic field $K$ when the prime $p$ splits completely over $K/\mathbb Q$, and prove they enjoy a property analogous to Mazur's control theorem. Furthermore, we propose a conjectural connection between the $\pm/\pm$-Selmer groups and Loeffler's two-variable $\pm/\pm$-$p$-adic $L$-functions of elliptic curves. Keywords:elliptic curves, Iwasawa theoryCategory:11Gxx

10. CJM 2013 (vol 66 pp. 1305)

Koskivirta, Jean-Stefan
 Congruence Relations for Shimura Varieties Associated with $GU(n-1,1)$ We prove the congruence relation for the mod-$p$ reduction of Shimura varieties associated to a unitary similitude group $GU(n-1,1)$ over $\mathbb{Q}$, when $p$ is inert and $n$ odd. The case when $n$ is even was obtained by T. Wedhorn and O. B?ltel, as a special case of a result of B. Moonen, when the $\mu$-ordinary locus of the $p$-isogeny space is dense. This condition fails in our case. We show that every supersingular irreducible component of the special fiber of $p\textrm{-}\mathscr{I}sog$ is annihilated by a degree one polynomial in the Frobenius element $F$, which implies the congruence relation. Keywords:Shimura varieties, congruence relationCategories:11G18, 14G35, 14K10

11. CJM 2013 (vol 66 pp. 924)

Stankewicz, James
 Twists of Shimura Curves Consider a Shimura curve $X^D_0(N)$ over the rational numbers. We determine criteria for the twist by an Atkin-Lehner involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan-LivnÃ© on $\mathbf{Q}_p$ points when $p\mid D$ and for the first time give criteria for $\mathbf{Q}_p$ points when $p\mid N$. We also give congruence conditions for roots modulo $p$ of Hilbert class polynomials. Keywords:Shimura curves, complex multiplication, modular curves, elliptic curvesCategories:11G18, 14G35, 11G15, 11G10

12. CJM 2013 (vol 66 pp. 1167)

Rotger, Victor; de Vera-Piquero, Carlos
 Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves The purpose of this note is introducing a method for proving the existence of no rational points on a coarse moduli space $X$ of abelian varieties over a given number field $K$, in cases where the moduli problem is not fine and points in $X(K)$ may not be represented by an abelian variety (with additional structure) admitting a model over the field $K$. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired on the work of Ellenberg and Skinner on the modularity of $\mathbb{Q}$-curves, is that to a point $P=[A]\in X(K)$ represented by an abelian variety $A/\bar K$ one may still attach a Galois representation of $\operatorname{Gal}(\bar K/K)$ with values in the quotient group $\operatorname{GL}(T_\ell(A))/\operatorname{Aut}(A)$, provided $\operatorname{Aut}(A)$ lies in the centre of $\operatorname{GL}(T_\ell(A))$. We exemplify our method in the cases where $X$ is a Shimura curve over an imaginary quadratic field or an Atkin-Lehner quotient over $\mathbb{Q}$. Keywords:Shimura curves, rational points, Galois representations, Hasse principle, Brauer-Manin obstructionCategories:11G18, 14G35, 14G05

13. CJM 2012 (vol 66 pp. 170)

Guitart, Xavier; Quer, Jordi
 Modular Abelian Varieties Over Number Fields The main result of this paper is a characterization of the abelian varieties $B/K$ defined over Galois number fields with the property that the $L$-function $L(B/K;s)$ is a product of $L$-functions of non-CM newforms over $\mathbb Q$ for congruence subgroups of the form $\Gamma_1(N)$. The characterization involves the structure of $\operatorname{End}(B)$, isogenies between the Galois conjugates of $B$, and a Galois cohomology class attached to $B/K$. We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illustration of the ways in which the general results of the paper can be applied we prove the strong modularity of some particular abelian surfaces belonging to that family, and we show how to find nontrivial examples of strongly modular varieties by twisting. Keywords:Modular abelian varieties, $GL_2$-type varieties, modular formsCategories:11G10, 11G18, 11F11

14. CJM 2012 (vol 65 pp. 403)

Van Order, Jeanine
 On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini-Darmon, Longo, Nekovar, Pollack-Weston and others. The construction has direct applications to Iwasawa main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated $p$-adic $L$-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique of Skinner and Urban. Keywords:Iwasawa theory, Hilbert modular forms, abelian varietiesCategories:11G10, 11G18, 11G40

15. CJM 2011 (vol 64 pp. 1248)

Gärtner, Jérôme
 Darmon's Points and Quaternionic Shimura Varieties In this paper, we generalize a conjecture due to Darmon and Logan in an adelic setting. We study the relation between our construction and Kudla's works on cycles on orthogonal Shimura varieties. This relation allows us to conjecture a Gross-Kohnen-Zagier theorem for Darmon's points. Keywords:elliptic curves, Stark-Heegner points, quaternionic Shimura varietiesCategories:11G05, 14G35, 11F67, 11G40

16. CJM 2011 (vol 64 pp. 588)

Nekovář, Jan
 Level Raising and Anticyclotomic Selmer Groups for Hilbert Modular Forms of Weight Two In this article we refine the method of Bertolini and Darmon and prove several finiteness results for anticyclotomic Selmer groups of Hilbert modular forms of parallel weight two. Keywords:Hilbert modular forms, Selmer groups, Shimura curvesCategories:11G40, 11F41, 11G18

17. CJM 2011 (vol 64 pp. 282)

Dahmen, Sander R.; Yazdani, Soroosh
 Level Lowering Modulo Prime Powers and Twisted Fermat Equations We discuss a clean level lowering theorem modulo prime powers for weight $2$ cusp forms. Furthermore, we illustrate how this can be used to completely solve certain twisted Fermat equations $ax^n+by^n+cz^n=0$. Keywords:modular forms, level lowering, Diophantine equationsCategories:11D41, 11F33, 11F11, 11F80, 11G05

18. CJM 2011 (vol 64 pp. 301)

Hurlburt, Chris; Thunder, Jeffrey Lin
 Hermite's Constant for Function Fields We formulate an analog of Hermite's constant for function fields over a finite field and state a conjectural value for this analog. We prove our conjecture in many cases, and prove slightly weaker results in all other cases. Category:11G50

19. CJM 2011 (vol 64 pp. 151)

Miller, Steven J.; Wong, Siman
 Moments of the Rank of Elliptic Curves Fix an elliptic curve $E/\mathbb{Q}$ and assume the Riemann Hypothesis for the $L$-function $L(E_D, s)$ for every quadratic twist $E_D$ of $E$ by $D\in\mathbb{Z}$. We combine Weil's explicit formula with techniques of Heath-Brown to derive an asymptotic upper bound for the weighted moments of the analytic rank of $E_D$. We derive from this an upper bound for the density of low-lying zeros of $L(E_D, s)$ that is compatible with the random matrix models of Katz and Sarnak. We also show that for any unbounded increasing function $f$ on $\mathbb{R}$, the analytic rank and (assuming in addition the Birch and Swinnerton-Dyer conjecture) the number of integral points of $E_D$ are less than $f(D)$ for almost all $D$. Keywords:elliptic curve, explicit formula, integral point, low-lying zeros, quadratic twist, rankCategories:11G05, 11G40

20. CJM 2011 (vol 63 pp. 992)

Bruin, Nils; Doerksen, Kevin
 The Arithmetic of Genus Two Curves with (4,4)-Split Jacobians In this paper we study genus $2$ curves whose Jacobians admit a polarized $(4,4)$-isogeny to a product of elliptic curves. We consider base fields of characteristic different from $2$ and $3$, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their $4$-torsion, and we derive the relation their absolute invariants satisfy. As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus $2$ curves, where previously only Richelot isogenies with kernels that are pointwise defined over the base field were considered. Our main tool is a Galois theoretic characterization of genus $2$ curves admitting multiple Richelot isogenies. Keywords:Genus 2 curves, isogenies, split Jacobians, elliptic curvesCategories:11G30, 14H40

21. CJM 2011 (vol 63 pp. 826)

Errthum, Eric
 Singular Moduli of Shimura Curves The $j$-function acts as a parametrization of the classical modular curve. Its values at complex multiplication (CM) points are called singular moduli and are algebraic integers. A Shimura curve is a generalization of the modular curve and, if the Shimura curve has genus~$0$, a rational parameterizing function exists and when evaluated at a CM point is again algebraic over~$\mathbf{Q}$. This paper shows that the coordinate maps given by N.~Elkies for the Shimura curves associated to the quaternion algebras with discriminants $6$ and $10$ are Borcherds lifts of vector-valued modular forms. This property is then used to explicitly compute the rational norms of singular moduli on these curves. This method not only verifies conjectural values for the rational CM points, but also provides a way of algebraically calculating the norms of CM points with arbitrarily large negative discriminant. Categories:11G18, 11F12

22. CJM 2011 (vol 63 pp. 616)

Lee, Edward
 A Modular Quintic Calabi-Yau Threefold of Level 55 In this note we search the parameter space of Horrocks-Mumford quintic threefolds and locate a Calabi-Yau threefold that is modular, in the sense that the $L$-function of its middle-dimensional cohomology is associated with a classical modular form of weight 4 and level 55. Keywords: Calabi-Yau threefold, non-rigid Calabi-Yau threefold, two-dimensional Galois representation, modular variety, Horrocks-Mumford vector bundleCategories:14J15, 11F23, 14J32, 11G40

23. CJM 2011 (vol 63 pp. 481)

Baragar, Arthur
 The Ample Cone for a K3 Surface In this paper, we give several pictorial fractal representations of the ample or KÃ¤hler cone for surfaces in a certain class of $K3$ surfaces. The class includes surfaces described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a sufficiently large number field $K$ that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be $1.296 \pm .010$. Keywords:Fractal, Hausdorff dimension, K3 surface, Kleinian groups, dynamicsCategories:14J28, , , , 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05

24. CJM 2010 (vol 62 pp. 1155)

Young, Matthew P.
 Moments of the Critical Values of Families of Elliptic Curves, with Applications We make conjectures on the moments of the central values of the family of all elliptic curves and on the moments of the first derivative of the central values of a large family of positive rank curves. In both cases the order of magnitude is the same as that of the moments of the central values of an orthogonal family of $L$-functions. Notably, we predict that the critical values of all rank $1$ elliptic curves is logarithmically larger than the rank $1$ curves in the positive rank family. Furthermore, as arithmetical applications, we make a conjecture on the distribution of $a_p$'s amongst all rank $2$ elliptic curves and show how the Riemann hypothesis can be deduced from sufficient knowledge of the first moment of the positive rank family (based on an idea of Iwaniec) Categories:11M41, 11G40, 11M26

25. CJM 2010 (vol 62 pp. 1060)

Darmon, Henri; Tian, Ye
 Heegner Points over Towers of Kummer Extensions Let $E$ be an elliptic curve, and let $L_n$ be the Kummer extension generated by a primitive $p^n$-th root of unity and a $p^n$-th root of $a$ for a fixed $a\in \mathbb{Q}^\times-\{\pm 1\}$. A detailed case study by Coates, Fukaya, Kato and Sujatha and V. Dokchitser has led these authors to predict unbounded and strikingly regular growth for the rank of $E$ over $L_n$ in certain cases. The aim of this note is to explain how some of these predictions might be accounted for by Heegner points arising from a varying collection of Shimura curve parametrisations. Categories:11G05, 11R23, 11F46
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