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26. 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-Livné 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 curves
Categories:11G18, 14G35, 11G15, 11G10

27. 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 obstruction
Categories:11G18, 14G35, 14G05

28. CJM 2013 (vol 66 pp. 844)

Kuo, Wentang; Liu, Yu-Ru; Zhao, Xiaomei
Multidimensional Vinogradov-type Estimates in Function Fields
Let $\mathbb{F}_q[t]$ denote the polynomial ring over the finite field $\mathbb{F}_q$. We employ Wooley's new efficient congruencing method to prove certain multidimensional Vinogradov-type estimates in $\mathbb{F}_q[t]$. These results allow us to apply a variant of the circle method to obtain asymptotic formulas for a system connected to the problem about linear spaces lying on hypersurfaces defined over $\mathbb{F}_q[t]$.

Keywords:Vinogradov's mean value theorem, function fields, circle method
Categories:11D45, 11P55, 11T55

29. CJM 2013 (vol 66 pp. 566)

Choiy, Kwangho
Transfer of Plancherel Measures for Unitary Supercuspidal Representations between $p$-adic Inner Forms
Let $F$ be a $p$-adic field of characteristic $0$, and let $M$ be an $F$-Levi subgroup of a connected reductive $F$-split group such that $\Pi_{i=1}^{r} SL_{n_i} \subseteq M \subseteq \Pi_{i=1}^{r} GL_{n_i}$ for positive integers $r$ and $n_i$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M(F)$ is identically transferred under the local Jacquet-Langlands type correspondence between $M$ and its $F$-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of Muić and Savin (2000) for Siegel Levi subgroups of the groups $SO_{4n}$ and $Sp_{4n}$ under the local Jacquet-Langlands correspondence. It can be applied to a simply connected simple $F$-group of type $E_6$ or $E_7$, and a connected reductive $F$-group of type $A_{n}$, $B_{n}$, $C_n$ or $D_n$.

Keywords:Plancherel measure, inner form, local to global global argument, cuspidal automorphic representation, Jacquet-Langlands correspondence
Categories:22E50, 11F70, 22E55, 22E35

30. 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 forms
Categories:11G10, 11G18, 11F11

31. CJM 2012 (vol 65 pp. 1201)

Cho, Peter J.; Kim, Henry H.
Application of the Strong Artin Conjecture to the Class Number Problem
We construct unconditionally several families of number fields with the largest possible class numbers. They are number fields of degree 4 and 5 whose Galois closures have the Galois group $A_4, S_4$ and $S_5$. We first construct families of number fields with smallest regulators, and by using the strong Artin conjecture and applying zero density result of Kowalski-Michel, we choose subfamilies of $L$-functions which are zero free close to 1. For these subfamilies, the $L$-functions have the extremal value at $s=1$, and by the class number formula, we obtain the extreme class numbers.

Keywords:class number, strong Artin conjecture
Categories:11R29, 11M41

32. CJM 2012 (vol 65 pp. 544)

Deitmar, Anton; Horozov, Ivan
Iterated Integrals and Higher Order Invariants
We show that higher order invariants of smooth functions can be written as linear combinations of full invariants times iterated integrals. The non-uniqueness of such a presentation is captured in the kernel of the ensuing map from the tensor product. This kernel is computed explicitly. As a consequence, it turns out that higher order invariants are a free module of the algebra of full invariants.

Keywords:higher order forms, iterated integrals
Categories:14F35, 11F12, 55D35, 58A10

33. 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 varieties
Categories:11G10, 11G18, 11G40

34. CJM 2012 (vol 65 pp. 171)

Lyall, Neil; Magyar, Ákos
Optimal Polynomial Recurrence
Let $P\in\mathbb Z[n]$ with $P(0)=0$ and $\varepsilon\gt 0$. We show, using Fourier analytic techniques, that if $N\geq \exp\exp(C\varepsilon^{-1}\log\varepsilon^{-1})$ and $A\subseteq\{1,\dots,N\}$, then there must exist $n\in\mathbb N$ such that \[\frac{|A\cap (A+P(n))|}{N}\gt \left(\frac{|A|}{N}\right)^2-\varepsilon.\] In addition to this we also show, using the same Fourier analytic methods, that if $A\subseteq\mathbb N$, then the set of $\varepsilon$-optimal return times \[R(A,P,\varepsilon)=\left\{n\in \mathbb N \,:\,\delta(A\cap(A+P(n)))\gt \delta(A)^2-\varepsilon\right\}\] is syndetic for every $\varepsilon\gt 0$. Moreover, we show that $R(A,P,\varepsilon)$ is dense in every sufficiently long interval, in particular we show that there exists an $L=L(\varepsilon,P,A)$ such that \[\left|R(A,P,\varepsilon)\cap I\right| \geq c(\varepsilon,P)|I|\] for all intervals $I$ of natural numbers with $|I|\geq L$ and $c(\varepsilon,P)=\exp\exp(-C\,\varepsilon^{-1}\log\varepsilon^{-1})$.

Keywords:Sarkozy, syndetic, polynomial return times

35. CJM 2012 (vol 64 pp. 1019)

Fiorilli, Daniel
On a Theorem of Bombieri, Friedlander, and Iwaniec
In this article, we show to which extent one can improve a theorem of Bombieri, Friedlander and Iwaniec by using Hooley's variant of the divisor switching technique. We also give an application of the theorem in question, which is a Bombieri-Vinogradov type theorem for the Tichmarsh divisor problem in arithmetic progressions.

Keywords:primes in arithmetic progressions, Titchmarsh divisor problem

36. CJM 2012 (vol 64 pp. 254)

Bell, Jason P.; Hare, Kevin G.
Corrigendum to ``On $\mathbb{Z}$-modules of Algebraic Integers''
We fix a mistake in the proof of Theorem 1.6 in the paper in the title.

Keywords:Pisot numbers, algebraic integers, number rings, Schmidt subspace theorem
Categories:11R04, 11R06

37. CJM 2012 (vol 64 pp. 497)

Li, Wen-Wei
Le lemme fondamental pondéré pour le groupe métaplectique
Dans cet article, on énonce une variante du lemme fondamental pondéré d'Arthur pour le groupe métaplectique de Weil, qui sera un ingrédient indispensable de la stabilisation de la formule des traces. Pour un corps de caractéristique résiduelle suffisamment grande, on en donne une démonstration à l'aide de la méthode de descente, qui est conditionnelle: on admet le lemme fondamental pondéré non standard sur les algèbres de Lie. Vu les travaux de Chaudouard et Laumon, on s'attend à ce que cette condition soit ultérieurement vérifiée.

Keywords:fundamental lemma, metaplectic group, endoscopy, trace formula
Categories:11F70, 11F27, 22E50

38. CJM 2011 (vol 64 pp. 1201)

Aistleitner, Christoph; Elsholtz, Christian
The Central Limit Theorem for Subsequences in Probabilistic Number Theory
Let $(n_k)_{k \geq 1}$ be an increasing sequence of positive integers, and let $f(x)$ be a real function satisfying \begin{equation} \tag{1} f(x+1)=f(x), \qquad \int_0^1 f(x) ~dx=0,\qquad \operatorname{Var_{[0,1]}} f \lt \infty. \end{equation} If $\lim_{k \to \infty} \frac{n_{k+1}}{n_k} = \infty$ the distribution of \begin{equation} \tag{2} \frac{\sum_{k=1}^N f(n_k x)}{\sqrt{N}} \end{equation} converges to a Gaussian distribution. In the case $$ 1 \lt \liminf_{k \to \infty} \frac{n_{k+1}}{n_k}, \qquad \limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \lt \infty $$ there is a complex interplay between the analytic properties of the function $f$, the number-theoretic properties of $(n_k)_{k \geq 1}$, and the limit distribution of (2). In this paper we prove that any sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} = 1$ contains a nontrivial subsequence $(m_k)_{k \geq 1}$ such that for any function satisfying (1) the distribution of $$ \frac{\sum_{k=1}^N f(m_k x)}{\sqrt{N}} $$ converges to a Gaussian distribution. This result is best possible: for any $\varepsilon\gt 0$ there exists a sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \leq 1 + \varepsilon$ such that for every nontrivial subsequence $(m_k)_{k \geq 1}$ of $(n_k)_{k \geq 1}$ the distribution of (2) does not converge to a Gaussian distribution for some $f$. Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

Keywords:central limit theorem, lacunary sequences, linear Diophantine equations, Ramsey type theorem
Categories:60F05, 42A55, 11D04, 05C55, 11K06

39. CJM 2011 (vol 65 pp. 22)

Blomer, Valentin; Brumley, Farrell
Non-vanishing of $L$-functions, the Ramanujan Conjecture, and Families of Hecke Characters
We prove a non-vanishing result for families of $\operatorname{GL}_n\times\operatorname{GL}_n$ Rankin-Selberg $L$-functions in the critical strip, as one factor runs over twists by Hecke characters. As an application, we simplify the proof, due to Luo, Rudnick, and Sarnak, of the best known bounds towards the Generalized Ramanujan Conjecture at the infinite places for cusp forms on $\operatorname{GL}_n$. A key ingredient is the regularization of the units in residue classes by the use of an Arakelov ray class group.

Keywords:non-vanishing, automorphic forms, Hecke characters, Ramanujan conjecture
Categories:11F70, 11M41

40. CJM 2011 (vol 64 pp. 1036)

Koh, Doowon; Shen, Chun-Yen
Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields
In this paper we study the extension problem, the averaging problem, and the generalized Erdős-Falconer distance problem associated with arbitrary homogeneous varieties in three dimensional vector spaces over finite fields. In the case when the varieties do not contain any plane passing through the origin, we obtain the best possible results on the aforementioned three problems. In particular, our result on the extension problem modestly generalizes the result by Mockenhaupt and Tao who studied the particular conical extension problem. In addition, investigating the Fourier decay on homogeneous varieties enables us to give complete mapping properties of averaging operators. Moreover, we improve the size condition on a set such that the cardinality of its distance set is nontrivial.

Keywords:extension problems, averaging operator, finite fields, Erdős-Falconer distance problems, homogeneous polynomial
Categories:42B05, 11T24, 52C17

41. 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 varieties
Categories:11G05, 14G35, 11F67, 11G40

42. 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 curves
Categories:11G40, 11F41, 11G18

43. CJM 2011 (vol 64 pp. 1122)

Seveso, Marco Adamo
$p$-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

44. CJM 2011 (vol 64 pp. 935)

McIntosh, Richard J.
The H and K Families of Mock Theta Functions
In his last letter to Hardy, Ramanujan defined 17 functions $F(q)$, $|q|\lt 1$, which he called mock $\theta$-functions. He observed that as $q$ radially approaches any root of unity $\zeta$ at which $F(q)$ has an exponential singularity, there is a $\theta$-function $T_\zeta(q)$ with $F(q)-T_\zeta(q)=O(1)$. Since then, other functions have been found that possess this property. These functions are related to a function $H(x,q)$, where $x$ is usually $q^r$ or $e^{2\pi i r}$ for some rational number $r$. For this reason we refer to $H$ as a ``universal'' mock $\theta$-function. Modular transformations of $H$ give rise to the functions $K$, $K_1$, $K_2$. The functions $K$ and $K_1$ appear in Ramanujan's lost notebook. We prove various linear relations between these functions using Appell-Lerch sums (also called generalized Lambert series). Some relations (mock theta ``conjectures'') involving mock $\theta$-functions of even order and $H$ are listed.

Keywords:mock theta function, $q$-series, Appell-Lerch sum, generalized Lambert series
Categories:11B65, 33D15

45. 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 equations
Categories:11D41, 11F33, 11F11, 11F80, 11G05

46. 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.


47. CJM 2011 (vol 64 pp. 345)

McKee, James; Smyth, Chris
Salem Numbers and Pisot Numbers via Interlacing
We present a general construction of Salem numbers via rational functions whose zeros and poles mostly lie on the unit circle and satisfy an interlacing condition. This extends and unifies earlier work. We then consider the ``obvious'' limit points of the set of Salem numbers produced by our theorems and show that these are all Pisot numbers, in support of a conjecture of Boyd. We then show that all Pisot numbers arise in this way. Combining this with a theorem of Boyd, we produce all Salem numbers via an interlacing construction.

Keywords:Salem numbers, Pisot numbers

48. CJM 2011 (vol 64 pp. 81)

David, C.; Wu, J.
Pseudoprime Reductions of Elliptic Curves
Let $E$ be an elliptic curve over $\mathbb Q$ without complex multiplication, and for each prime $p$ of good reduction, let $n_E(p) = | E(\mathbb F_p) |$. For any integer $b$, we consider elliptic pseudoprimes to the base $b$. More precisely, let $Q_{E,b}(x)$ be the number of primes $p \leq x$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$, and let $\pi_{E, b}^{\operatorname{pseu}}(x)$ be the number of compositive $n_E(p)$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$ (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address the problem of finding upper bounds for $Q_{E,b}(x)$ and $\pi_{E, b}^{\operatorname{pseu}}(x)$, generalising some of the literature for the classical pseudoprimes to this new setting.

Keywords:Rosser-Iwaniec sieve, group order of elliptic curves over finite fields, pseudoprimes
Categories:11N36, 14H52

49. CJM 2011 (vol 63 pp. 1328)

Gun, Sanoli; Murty, M. Ram; Rath, Purusottam
On a Conjecture of Chowla and Milnor
In this paper, we investigate a conjecture due to S. and P. Chowla and its generalization by Milnor. These are related to the delicate question of non-vanishing of $L$-functions associated to periodic functions at integers greater than $1$. We report on some progress in relation to these conjectures. In a different vein, we link them to a conjecture of Zagier on multiple zeta values and also to linear independence of polylogarithms.

Categories:11F20, 11F11

50. 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, rank
Categories:11G05, 11G40
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