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1. 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

2. 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

3. 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

4. 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

5. CJM 2011 (vol 63 pp. 1284)

Dewar, Michael
 Non-Existence of Ramanujan Congruences in Modular Forms of Level Four Ramanujan famously found congruences like $p(5n+4)\equiv 0 \operatorname{mod} 5$ for the partition function. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on $\Gamma_{1}(4)$ that is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored $F$-partitions. Keywords:modular form, Ramanujan congruence, generalized Frobenius partition, overpartition, crankCategories:11F33, 11P83

6. 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