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

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1. 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

2. CJM 2014 (vol 67 pp. 893)

Mok, Chung Pang; Tan, Fucheng
 Overconvergent Families of Siegel-Hilbert Modular Forms We construct one-parameter families of overconvergent Siegel-Hilbert modular forms. This result has applications to construction of Galois representations for automorphic forms of non-cohomological weights. Keywords:p-adic automorphic form, rigid analytic geometryCategories:11F46, 14G22

3. 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

4. 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

5. CJM 2013 (vol 65 pp. 1125)

Vandenbergen, Nicolas
 On the Global Structure of Special Cycles on Unitary Shimura Varieties In this paper, we study the reduced loci of special cycles on local models of the Shimura variety for $\operatorname{GU}(1,n-1)$. Those special cycles are defined by Kudla and Rapoport. We explicitly compute the irreducible components of the reduced locus of a single special cycle, as well as of an arbitrary intersection of special cycles, and their intersection behaviour in terms of Bruhat-Tits theory. Furthermore, as an application of our results, we prove the connectedness of arbitrary intersections of special cycles, as conjectured by Kudla and Rapoport. Keywords:Shimura varieties, local models, special cyclesCategory:14G35

6. 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

7. 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

8. CJM 2011 (vol 64 pp. 1122)

 $p$-adic $L$-functions and the Rationality of Darmon Cycles Darmon cycles are a higher weight analogue of Stark--Heegner points. They yield local cohomology classes in the Deligne representation associated with a cuspidal form on $\Gamma _{0}( N)$ of even weight $k_{0}\geq 2$. They are conjectured to be the restriction of global cohomology classes in the Bloch--Kato Selmer group defined over narrow ring class fields attached to a real quadratic field. We show that suitable linear combinations of them obtained by genus characters satisfy these conjectures. We also prove $p$-adic Gross--Zagier type formulas, relating the derivatives of $p$-adic $L$-functions of the weight variable attached to imaginary (resp. real) quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express the second derivative of the Mazur--Kitagawa $p$-adic $L$-function of the weight variable in terms of a global cycle defined over a quadratic extension of $\mathbb{Q}$. Categories:11F67, 14G05

9. CJM 2010 (vol 63 pp. 86)

Chen, Xi
 On Vojta's $1+\varepsilon$ Conjecture We give another proof of Vojta's $1+\varepsilon$ conjecture. Keywords:Vojta, 1+epsilonCategories:14G40, 14H15

10. CJM 2010 (vol 62 pp. 668)

Vollaard, Inken
 The Supersingular Locus of the Shimura Variety for GU(1,s) In this paper we study the supersingular locus of the reduction modulo $p$ of the Shimura variety for $GU(1,s)$ in the case of an inert prime $p$. Using DieudonnÃ© theory we define a stratification of the corresponding moduli space of $p$-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat--Tits building of a unitary group. In the case of $GU(1,2)$, we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour. Categories:14G35, 11G18, 14K10

11. CJM 2009 (vol 61 pp. 1118)

Pontreau, Corentin
 Petits points d'une surface Pour toute sous-vari\'et\'e g\'eom\'etriquement irr\'eductible $V$ du grou\-pe multiplicatif $\mathbb{G}_m^n$, on sait qu'en dehors d'un nombre fini de translat\'es de tores exceptionnels inclus dans $V$, tous les points sont de hauteur minor\'ee par une certaine quantit\'e $q(V)^{-1}>0$. On conna\^it de plus une borne sup\'erieure pour la somme des degr\'es de ces translat\'es de tores pour des valeurs de $q(V)$ polynomiales en le degr\'e de $V$. Ceci n'est pas le cas si l'on exige une minoration quasi-optimale pour la hauteur des points de $V$, essentiellement lin\'eaire en l'inverse du degr\'e. Nous apportons ici une r\'eponse partielle \a ce probl\eme\,: nous donnons une majoration de la somme des degr\'es de ces translat\'es de sous-tores de codimension $1$ d'une hypersurface $V$. Les r\'esultats, obtenus dans le cas de $\mathbb{G}_m^3$, mais compl\etement explicites, peuvent toutefois s'\'etendre \a $\mathbb{G}_m^n$, moyennant quelques petites complications inh\'erentes \a la dimension $n$. Keywords:Hauteur normalisÃ©e, groupe multiplicatif, problÃ¨me de Lehmer, petits pointsCategories:11G50, 11J81, 14G40

12. CJM 2009 (vol 61 pp. 828)

Howard, Benjamin
 Twisted Gross--Zagier Theorems The theorems of Gross--Zagier and Zhang relate the N\'eron--Tate heights of complex multiplication points on the modular curve $X_0(N)$ (and on Shimura curve analogues) with the central derivatives of automorphic $L$-function. We extend these results to include certain CM points on modular curves of the form $X(\Gamma_0(M)\cap\Gamma_1(S))$ (and on Shimura curve analogues). These results are motivated by applications to Hida theory that can be found in the companion article "Central derivatives of $L$-functions in Hida families", Math.\ Ann.\ \textbf{399}(2007), 803--818. Categories:11G18, 14G35

13. CJM 2008 (vol 60 pp. 1267)

Blake, Ian F.; Murty, V. Kumar; Xu, Guangwu
 Nonadjacent Radix-$\tau$ Expansions of Integers in Euclidean Imaginary Quadratic Number Fields In his seminal papers, Koblitz proposed curves for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix-$\tau$ expansion of integers in the number fields $\Q(\sqrt{-3})$ and $\Q(\sqrt{-7})$. The (window) nonadjacent form of $\tau$-expansion of integers in $\Q(\sqrt{-7})$ was first investigated by Solinas. For integers in $\Q(\sqrt{-3})$, the nonadjacent form and the window nonadjacent form of the $\tau$-expansion were studied. These are used for efficient point multiplications on Koblitz curves. In this paper, we complete the picture by producing the (window) nonadjacent radix-$\tau$ expansions for integers in all Euclidean imaginary quadratic number fields. Keywords:algebraic integer, radix expression, window nonadjacent expansion, algorithm, point multiplication of elliptic curves, cryptographyCategories:11A63, 11R04, 11Y16, 11Y40, 14G50

14. CJM 2008 (vol 60 pp. 734)

Baba, Srinath; Granath, H\aa kan
 Genus 2 Curves with Quaternionic Multiplication We explicitly construct the canonical rational models of Shimura curves, both analytically in terms of modular forms and algebraically in terms of coefficients of genus 2 curves, in the cases of quaternion algebras of discriminant 6 and 10. This emulates the classical construction in the elliptic curve case. We also give families of genus 2 QM curves, whose Jacobians are the corresponding abelian surfaces on the Shimura curve, and with coefficients that are modular forms of weight 12. We apply these results to show that our $j$-functions are supported exactly at those primes where the genus 2 curve does not admit potentially good reduction, and construct fields where this potentially good reduction is attained. Finally, using $j$, we construct the fields of moduli and definition for some moduli problems associated to the Atkin--Lehner group actions. Keywords:Shimura curve, canonical model, quaternionic multiplication, modular form, field of moduliCategories:11G18, 14G35

15. CJM 2007 (vol 59 pp. 372)

Maisner, Daniel; Nart, Enric
 Zeta Functions of Supersingular Curves of Genus 2 We determine which isogeny classes of supersingular abelian surfaces over a finite field $k$ of characteristic $2$ contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus $2$. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta function and find formulas for the number of curves, up to $k$-isomorphism, leading to the same zeta function. Categories:11G20, 14G15, 11G10

16. CJM 2002 (vol 54 pp. 352)

Haines, Thomas J.
 On Connected Components of Shimura Varieties We study the cohomology of connected components of Shimura varieties $S_{K^p}$ coming from the group $\GSp_{2g}$, by an approach modeled on the stabilization of the twisted trace formula, due to Kottwitz and Shelstad. More precisely, for each character $\olomega$ on the group of connected components of $S_{K^p}$ we define an operator $L(\omega)$ on the cohomology groups with compact supports $H^i_c (S_{K^p}, \olbbQ_\ell)$, and then we prove that the virtual trace of the composition of $L(\omega)$ with a Hecke operator $f$ away from $p$ and a sufficiently high power of a geometric Frobenius $\Phi^r_p$, can be expressed as a sum of $\omega$-{\em weighted} (twisted) orbital integrals (where $\omega$-{\em weighted} means that the orbital integrals and twisted orbital integrals occuring here each have a weighting factor coming from the character $\olomega$). As the crucial step, we define and study a new invariant $\alpha_1 (\gamma_0; \gamma, \delta)$ which is a refinement of the invariant $\alpha (\gamma_0; \gamma, \delta)$ defined by Kottwitz. This is done by using a theorem of Reimann and Zink. Categories:14G35, 11F70

17. CJM 1999 (vol 51 pp. 771)

Flicker, Yuval Z.
 Stable Bi-Period Summation Formula and Transfer Factors This paper starts by introducing a bi-periodic summation formula for automorphic forms on a group $G(E)$, with periods by a subgroup $G(F)$, where $E/F$ is a quadratic extension of number fields. The split case, where $E = F \oplus F$, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups $H$ which occur in the case of standard conjugacy. The spectral side of the bi-period summation formula involves periods, namely integrals over the group of $F$-adele points of $G$, of cusp forms on the group of $E$-adele points on the group $G$. Our stabilization suggests that such cusp forms---with non vanishing periods---and the resulting bi-period distributions associated to periodic'' automorphic forms, are related to analogous bi-period distributions associated to periodic'' automorphic forms on the endoscopic symmetric spaces $H(E)/H(F)$. This offers a sharpening of the theory of liftings, where periods play a key role. The stabilization depends on the fundamental lemma'', which conjectures that the unit elements of the Hecke algebras on $G$ and $H$ have matching orbital integrals. Even in stating this conjecture, one needs to introduce a transfer factor''. A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case. Finally, the fundamental lemma is verified for $\SL(2)$. Categories:11F72, 11F70, 14G27, 14L35

18. CJM 1997 (vol 49 pp. 749)

Howe, Lawrence
 Twisted Hasse-Weil $L$-functions and the rank of Mordell-Weil groups Following a method outlined by Greenberg, root number computations give a conjectural lower bound for the ranks of certain Mordell-Weil groups of elliptic curves. More specifically, for $\PQ_{n}$ a \pgl{{\bf Z}/p^{n}{\bf Z}}-extension of ${\bf Q}$ and $E$ an elliptic curve over {\bf Q}, define the motive $E \otimes \rho$, where $\rho$ is any irreducible representation of $\Gal (\PQ_{n}/{\bf Q})$. Under some restrictions, the root number in the conjectural functional equation for the $L$-function of $E \otimes \rho$ is easily computible, and a $-1$' implies, by the Birch and Swinnerton-Dyer conjecture, that $\rho$ is found in $E(\PQ_{n}) \otimes {\bf C}$. Summing the dimensions of such $\rho$ gives a conjectural lower bound of $$p^{2n} - p^{2n - 1} - p - 1$$ for the rank of $E(\PQ_{n})$. Categories:11G05, 14G10
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