Canadian Mathematical Society
Canadian Mathematical Society
  location:  Publicationsjournals
Search results

Search: MSC category 11 ( Number theory )

  Expand all        Collapse all Results 1 - 25 of 246

1. CJM Online first

Pan, Shu-Yen
$L$-Functoriality for Local Theta Correspondence of Supercuspidal Representations with Unipotent Reduction
The preservation principle of local theta correspondences of reductive dual pairs over a $p$-adic field predicts the existence of a sequence of irreducible supercuspidal representations of classical groups. Adams/Harris-Kudla-Sweet have a conjecture about the Langlands parameters for the sequence of supercuspidal representations. In this paper we prove modified versions of their conjectures for the case of supercuspidal representations with unipotent reduction.

Keywords:local theta correspondence, supercuspidal representation, preservation principle, Langlands functoriality
Categories:22E50, 11F27, 20C33

2. CJM Online first

Varma, Sandeep
On Residues of Intertwining Operators in Cases with Prehomogeneous Nilradical
Let $\operatorname{P} = \operatorname{M} \operatorname{N}$ be a Levi decomposition of a maximal parabolic subgroup of a connected reductive group $\operatorname{G}$ over a $p$-adic field $F$. Assume that there exists $w_0 \in \operatorname{G}(F)$ that normalizes $\operatorname{M}$ and conjugates $\operatorname{P}$ to an opposite parabolic subgroup. When $\operatorname{N}$ has a Zariski dense $\operatorname{Int} \operatorname{M}$-orbit, F. Shahidi and X. Yu describe a certain distribution $D$ on $\operatorname{M}(F)$ such that, for irreducible unitary supercuspidal representations $\pi$ of $\operatorname{M}(F)$ with $\pi \cong \pi \circ \operatorname{Int} w_0$, $\operatorname{Ind}_{\operatorname{P}(F)}^{\operatorname{G}(F)} \pi$ is irreducible if and only if $D(f) \neq 0$ for some pseudocoefficient $f$ of $\pi$. Since this irreducibility is conjecturally related to $\pi$ arising via transfer from certain twisted endoscopic groups of $\operatorname{M}$, it is of interest to realize $D$ as endoscopic transfer from a simpler distribution on a twisted endoscopic group $\operatorname{H}$ of $\operatorname{M}$. This has been done in many situations where $\operatorname{N}$ is abelian. Here, we handle the `standard examples' in cases where $\operatorname{N}$ is nonabelian but admits a Zariski dense $\operatorname{Int} \operatorname{M}$-orbit.

Keywords:induced representation, intertwining operator, endoscopy
Categories:22E50, 11F70

3. CJM Online first

Xu, Bin
On Moeglin's parametrization of Arthur packets for p-adic quasisplit $Sp(N)$ and $SO(N)$
We give a survey on Moeglin's construction of representations in the Arthur packets for $p$-adic quasisplit symplectic and orthogonal groups. The emphasis is on comparing Moeglin's parametrization of elements in the Arthur packets with that of Arthur.

Keywords:symplectic and orthogonal group, Arthur packet, endoscopy
Categories:22E50, 11F70

4. CJM Online first

Zydor, Michał
La variante infinitésimale de la formule des traces de Jacquet-Rallis pour les groupes unitaires
We establish an infinitesimal version of the Jacquet-Rallis trace formula for unitary groups. Our formula is obtained by integrating a truncated kernel à la Arthur. It has a geometric side which is a sum of distributions $J_{\mathfrak{o}}$ indexed by classes of elements of the Lie algebra of $U(n+1)$ stable by $U(n)$-conjugation as well as the "spectral side" consisting of the Fourier transforms of the aforementioned distributions. We prove that the distributions $J_{\mathfrak{o}}$ are invariant and depend only on the choice of the Haar measure on $U(n)(\mathbb{A})$. For regular semi-simple classes $\mathfrak{o}$, $J_{\mathfrak{o}}$ is a relative orbital integral of Jacquet-Rallis. For classes $\mathfrak{o}$ called relatively regular semi-simple, we express $J_{\mathfrak{o}}$ in terms of relative orbital integrals regularised by means of zêta functions.

Keywords:formule des traces relative
Categories:11F70, 11F72

5. CJM Online first

Papikian, Mihran; Rabinoff, Joseph
Optimal quotients of Jacobians with toric reduction and component groups
Let $J$ be a Jacobian variety with toric reduction over a local field $K$. Let $J \to E$ be an optimal quotient defined over $K$, where $E$ is an elliptic curve. We give examples in which the functorially induced map $\Phi_J \to \Phi_E$ on component groups of the Néron models is not surjective. This answers a question of Ribet and Takahashi. We also give various criteria under which $\Phi_J \to \Phi_E$ is surjective, and discuss when these criteria hold for the Jacobians of modular curves.

Keywords:Jacobians with toric reduction, component groups, modular curves
Categories:11G18, 14G22, 14G20

6. CJM Online first

Ganguly, Arijit; Ghosh, Anish
Dirichlet's theorem in function fields
We study metric Diophantine approximation for function fields specifically the problem of improving Dirichlet's theorem in Diophantine approximation.

Keywords:Dirichlet's theorem, Diophantine approximation, positive characteristic
Categories:11J83, 11K60, 37D40, 37A17, 22E40

7. CJM 2016 (vol 68 pp. 1120)

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 Néron 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 functions
Categories:11G05, 11G07, 11D25, 11B37, 11B39, 11Y55, 11G50, 11H52

8. CJM Online first

Günther, Christian; Schmidt, Kai-Uwe
$L^q$ norms of Fekete and related polynomials
A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all of its coefficients in $\{-1,1\}$. There are various old unsolved problems, mostly due to Littlewood and Erdős, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small $L^q$ norm on the complex unit circle. We consider the Fekete polynomials \[ f_p(z)=\sum_{j=1}^{p-1}(j\,|\,p)\,z^j, \] where $p$ is an odd prime and $(\,\cdot\,|\,p)$ is the Legendre symbol (so that $z^{-1}f_p(z)$ is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of $L^q$ and $L^2$ norm of $f_p$ when $q$ is an even positive integer and $p\to\infty$. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many $q$. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the $L^4$ norm of these polynomials.

Keywords:character polynomial, Fekete polynomial, $L^q$ norm, Littlewood polynomial
Categories:11B83, 42A05, 30C10

9. CJM 2016 (vol 68 pp. 961)

Greenberg, Matthew; Seveso, Marco
$p$-adic Families of Cohomological Modular Forms for Indefinite Quaternion Algebras and the Jacquet-Langlands Correspondence
We use the method of Ash and Stevens to prove the existence of small slope $p$-adic families of cohomological modular forms for an indefinite quaternion algebra $B$. We prove that the Jacquet-Langlands correspondence relating modular forms on $\textbf{GL}_2/\mathbb{Q}$ and cohomomological modular forms for $B$ is compatible with the formation of $p$-adic families. This result is an analogue of a theorem of Chenevier concerning definite quaternion algebras.

Keywords:modular forms, p-adic families, Jacquet-Langlands correspondence, Shimura curves, eigencurves
Categories:11F11, 11F67, 11F85

10. CJM Online first

Sikirić, Mathieu Dutour
The seven dimensional perfect Delaunay polytopes and Delaunay simplices
For a lattice $L$ of $\mathbb{RR}^n$, a sphere $S(c,r)$ of center $c$ and radius $r$ is called empty if for any $v\in L$ we have $\Vert v - c\Vert \geq r$. Then the set $S(c,r)\cap L$ is the vertex set of a {\em Delaunay polytope} $P=\operatorname{conv}(S(c,r)\cap L)$. A Delaunay polytope is called {\em perfect} if any affine transformation $\phi$ such that $\phi(P)$ is a Delaunay polytope is necessarily an isometry of the space composed with an homothety. Perfect Delaunay polytopes are remarkable structure that exist only if $n=1$ or $n\geq 6$ and they have shown up recently in covering maxima studies. Here we give a general algorithm for their enumeration that relies on the Erdahl cone. We apply this algorithm in dimension $7$ which allow us to find that there are only two perfect Delaunay polytopes: $3_{21}$ which is a Delaunay polytope in the root lattice $\mathsf{E}_7$ and the Erdahl Rybnikov polytope. We then use this classification in order to get the list of all types Delaunay simplices in dimension $7$ and found $11$ types.

Keywords:Delaunay polytope, enumeration, polyhedral methods
Categories:11H06, 11H31

11. CJM Online first

Brandes, Julia; Parsell, Scott T.
Simultaneous additive equations: Repeated and differing degrees
We obtain bounds for the number of variables required to establish Hasse principles, both for existence of solutions and for asymptotic formulæ, for systems of additive equations containing forms of differing degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning--Heath-Brown, which give weak results when specialized to the diagonal situation, this is the first result on such "hybrid" systems. We also obtain specialised results for systems of quadratic and cubic forms, where we are able to take advantage of some of the stronger methods available in that setting. In particular, we achieve essentially square root cancellation for systems consisting of one cubic and $r$ quadratic equations.

Keywords:equations in many variables, counting solutions of Diophantine equations, applications of the Hardy-Littlewood method
Categories:11D72, 11D45, 11P55

12. CJM Online first

Mauduit, Christian; Rivat, Joël; Sárközy, András
On the digits of sumsets
Let $\mathcal A$, $\mathcal B$ be large subsets of $\{1,\ldots,N\}$. We study the number of pairs $(a,b)\in\mathcal A\times\mathcal B$ such that the sum of binary digits of $a+b$ is fixed.

Categories:11A63, 11B13

13. CJM Online first

Brasca, Riccardo
Eigenvarieties for cuspforms over PEL type Shimura varieties with dense ordinary locus
Let $p \gt 2$ be a prime and let $X$ be a compactified PEL Shimura variety of type (A) or (C) such that $p$ is an unramified prime for the PEL datum and such that the ordinary locus is dense in the reduction of $X$. Using the geometric approach of Andreatta, Iovita, Pilloni, and Stevens we define the notion of families of overconvergent locally analytic $p$-adic modular forms of Iwahoric level for $X$. We show that the system of eigenvalues of any finite slope cuspidal eigenform of Iwahoric level can be deformed to a family of systems of eigenvalues living over an open subset of the weight space. To prove these results, we actually construct eigenvarieties of the expected dimension that parameterize finite slope systems of eigenvalues appearing in the space of families of cuspidal forms.

Keywords:$p$-adic modular forms, eigenvarieties, PEL-type Shimura varieties
Categories:11F55, 11F33

14. CJM 2016 (vol 68 pp. 721)

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 intervals
Categories:11G07, 11N45, 11N13, 11N36

15. CJM 2016 (vol 68 pp. 908)

Sugiyama, Shingo; Tsuzuki, Masao
Existence of Hilbert Cusp Forms with Non-vanishing $L$-values
We develop a derivative version of the relative trace formula on $\operatorname{PGL}(2)$ studied in our previous work, and derive an asymptotic formula of an average of central values (derivatives) of automorphic $L$-functions for Hilbert cusp forms. As an application, we prove the existence of Hilbert cusp forms with non-vanishing central values (derivatives) such that the absolute degrees of their Hecke fields are arbitrarily large.

Keywords:automorphic representations, relative trace formulas, central $L$-values, derivatives of $L$-functions
Categories:11F67, 11F72

16. CJM 2016 (vol 68 pp. 571)

Gras, Georges
Les $\theta$-régulateurs locaux d'un nombre algébrique : Conjectures $p$-adiques
Let $K/\mathbb{Q}$ be Galois and let $\eta\in K^\times$ be such that $\operatorname{Reg}_\infty (\eta) \ne 0$. We define the local $\theta$-regulators $\Delta_p^\theta(\eta) \in \mathbb{F}_p$ for the $\mathbb{Q}_p\,$-irreducible characters $\theta$ of $G=\operatorname{Gal}(K/\mathbb{Q})$. A linear representation ${\mathcal L}^\theta\simeq \delta \, V_\theta$ is associated with $\Delta_p^\theta (\eta)$ whose nullity is equivalent to $\delta \geq 1$. Each $\Delta_p^\theta (\eta)$ yields $\operatorname{Reg}_p^\theta (\eta)$ modulo $p$ in the factorization $\prod_{\theta}(\operatorname{Reg}_p^\theta (\eta))^{\varphi(1)}$ of $\operatorname{Reg}_p^G (\eta) := \frac{ \operatorname{Reg}_p(\eta)}{p^{[K : \mathbb{Q}\,]} }$ (normalized $p$-adic regulator). From $\operatorname{Prob}\big (\Delta_p^\theta(\eta) = 0 \ \& \ {\mathcal L}^\theta \simeq \delta \, V_\theta\big ) \leq p^{- f \delta^2}$ ($f \geq 1$ is a residue degree) and the Borel-Cantelli heuristic, we conjecture that, for $p$ large enough, $\operatorname{Reg}_p^G (\eta)$ is a $p$-adic unit or that $p^{\varphi(1)} \parallel \operatorname{Reg}_p^G (\eta)$ (a single $\theta$ with $f=\delta=1$); this obstruction may be lifted assuming the existence of a binomial probability law confirmed through numerical studies (groups $C_3$, $C_5$, $D_6$). This conjecture would imply that, for all $p$ large enough, Fermat quotients, normalized $p$-adic regulators are $p$-adic units and that number fields are $p$-rational. We recall some deep cohomological results that may strengthen such conjectures.

Keywords:$p$-adic regulators, Leopoldt-Jaulent conjecture, Frobenius group determinants, characters, Fermat quotient, Abelian $p$-ramification, probabilistic number theory
Categories:11F85, 11R04, 20C15, 11C20, 11R37, 11R27, 11Y40

17. CJM 2016 (vol 68 pp. 361)

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 distribution
Categories:11D41, 11M50, 11G10, 14G10

18. CJM 2016 (vol 68 pp. 422)

Kohen, Daniel; Pacetti, Ariel
Heegner Points on Cartan Non-split Curves
Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$, and let $K$ be an imaginary quadratic field such that the root number of $E/K$ is $-1$. Let $\mathscr{O}$ be an order in $K$ and assume that there exists an odd prime $p$, such that $p^2 \mid\mid N$, and $p$ is inert in $\mathscr{O}$. Although there are no Heegner points on $X_0(N)$ attached to $\mathscr{O}$, in this article we construct such points on Cartan non-split curves. In order to do that we give a method to compute Fourier expansions for forms on Cartan non-split curves, and prove that the constructed points form a Heegner system as in the classical case.

Keywords:Cartan curves, Heegner points
Categories:11G05, 11F30

19. CJM 2016 (vol 68 pp. 395)

Garibaldi, Skip; Nakano, Daniel K.
Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups
The representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the question of whether a given complex representation is symplectic or orthogonal has been solved since at least the 1950s. Similar results for Weyl modules of split reductive groups over fields of characteristic different from 2 hold by using similar proofs. This paper considers analogues of these results for simple, induced and tilting modules of split reductive groups over fields of prime characteristic as well as a complete answer for Weyl modules over fields of characteristic 2.

Keywords:orthogonal representations, symmetric tensors, alternating forms, characteristic 2, split reductive groups
Categories:20G05, 11E39, 11E88, 15A63, 20G15

20. CJM 2015 (vol 68 pp. 179)

Takeda, Shuichiro
Metaplectic Tensor Products for Automorphic Representation of $\widetilde{GL}(r)$
Let $M=\operatorname{GL}_{r_1}\times\cdots\times\operatorname{GL}_{r_k}\subseteq\operatorname{GL}_r$ be a Levi subgroup of $\operatorname{GL}_r$, where $r=r_1+\cdots+r_k$, and $\widetilde{M}$ its metaplectic preimage in the $n$-fold metaplectic cover $\widetilde{\operatorname{GL}}_r$ of $\operatorname{GL}_r$. For automorphic representations $\pi_1,\dots,\pi_k$ of $\widetilde{\operatorname{GL}}_{r_1}(\mathbb{A}),\dots,\widetilde{\operatorname{GL}}_{r_k}(\mathbb{A})$, we construct (under a certain technical assumption, which is always satisfied when $n=2$) an automorphic representation $\pi$ of $\widetilde{M}(\mathbb{A})$ which can be considered as the ``tensor product'' of the representations $\pi_1,\dots,\pi_k$. This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place $v$, $\pi_v$ is equivalent to the local metaplectic tensor product of $\pi_{1,v},\dots,\pi_{k,v}$ defined by Mezo. Then we show that if all of $\pi_i$ are cuspidal (resp. square-integrable modulo center), then the metaplectic tensor product is cuspidal (resp. square-integrable modulo center). We also show that (both locally and globally) the metaplectic tensor product behaves in the expected way under the action of a Weyl group element, and show the compatibility with parabolic inductions.

Keywords:automorphic forms, representations of covering groups

21. CJM Online first

Levin, Aaron; Wang, Julie Tzu-Yueh
On non-Archimedean curves omitting few components and their arithmetic analogues
Let $\mathbf{k}$ be an algebraically closed field complete with respect to a non-Archimedean absolute value of arbitrary characteristic. Let $D_1,\dots, D_n$ be effective nef divisors intersecting transversally in an $n$-dimensional nonsingular projective variety $X$. We study the degeneracy of non-Archimedean analytic maps from $\mathbf{k}$ into $X\setminus \cup_{i=1}^nD_i$ under various geometric conditions. When $X$ is a rational ruled surface and $D_1$ and $D_2$ are ample, we obtain a necessary and sufficient condition such that there is no non-Archimedean analytic map from $\mathbf{k}$ into $X\setminus D_1 \cup D_2$. Using the dictionary between non-Archimedean Nevanlinna theory and Diophantine approximation that originated in earlier work with T. T. H. An, % we also study arithmetic analogues of these problems, establishing results on integral points on these varieties over $\mathbb{Z}$ or the ring of integers of an imaginary quadratic field.

Keywords:non-Archimedean Picard theorem, non-Archimedean analytic curves, integral points
Categories:11J97, 32P05, 32H25

22. CJM 2015 (vol 67 pp. 1326)

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 theorems
Categories:11R45, 11G09, 11R58

23. 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 field
Categories:11R04, 11S85, 11G50, 11R09, 11R18

24. 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 tower
Categories:11G15, 11G10, 11R23, 11R34

25. CJM 2015 (vol 68 pp. 24)

Bonfanti, Matteo Alfonso; van Geemen, Bert
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 curves
Categories:14K10, 11G10, 14K20
   1 2 3 4 ... 10    

© Canadian Mathematical Society, 2016 :