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Two abelian varieties A_1, A_2 over a number field K are strongly iso-Kummerian if the torsion fields K(A_1[d]) and K(A_2[d]) coincide for all d \geq 1. For all g \geq 4 we construct geometrically simple, strongly iso-Kummerian g-dimensional abelian varieties over number fields that are not geometrically isogenous. We also discuss related examples and put significant constraints on any further iso-Kummerian pair. [ABSTRACT FROM AUTHOR]

In this paper, we consider representations of the absolute Galois group Gal(Q/Q) attached to modular forms for noncongruence subgroups of SL2(Z). When the underlying modular curves have a model over Q, these representations are constructed by Scholl in [Invent. Math. 99 (1985), pp. 49-77] and are referred to as Scholl representations, which form a large class of motivic Galois representations. In particular, by a result of Belyi, Scholl representations include the Galois actions on the Jacobian varieties of algebraic curves defined over Q. As Scholl representations are motivic, they are expected to correspond to automorphic representations according to the Langlands philosophy. Using recent developments on automorphy lifting theorem, we obtain various automorphy and potential automorphy results for potentially GL2-type Galois representations associated to noncongruence modular forms. Our results are applied to various kinds of examples. In particular, we obtain potential automorphy results for Galois representations attached to an infinite family of spaces of weight 3 noncongruence cusp forms of arbitrarily large dimensions. [ABSTRACT FROM AUTHOR]

We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute EndK(A) when A is the Jacobian of a nice genus-2 curve over a number field K. We use this algorithm to confirm that the description of the structure of the geometric endomorphism ring of Jac(C) given in the LMFDB (L-functions and modular forms database) is correct for all the genus-2 curves C currently listed in it. We also discuss the determination of the field of definition of the endomorphisms in some special cases. [ABSTRACT FROM AUTHOR]

This article is an introduction to the early life and work of Robert P. Langlands, creator and founder of the Langlands program. The story is, to a large extent, told by Langlands himself, in his own words. Our focus is on two of Langlands' major discoveries: automorphic L-functions and the principle of functoriality. It was Langlands' desire to communicate his excitement about his newly discovered objects that resulted in his famous letter to Andr'e Weil. This article is aimed at a general mathematical audience and we have purposely not included the more technical aspects of Langlands' work. [ABSTRACT FROM AUTHOR]

Let X be a complete smooth variety defined over a number field K and let i be an integer. The absolute Galois group GalK of K acts on the ith étale cohomology group Hièt(XK¯ ,Ql) for all primes l, producing a system of l-adic representations {Фl}l. The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of Фl admits a reductive Q-form that is independent of l if X is projective. Denote by Γl and Gl respectively the monodromy group and the algebraic monodromy group of Фssl, the semisimplification of Фl. Assuming that Gl0 satisfies some group theoretic conditions for some prime l0, we construct a connected quasi-split Q-reductive group GQ which is a common Q-form of Gol for all sufficiently large l. Let GscQ be the universal cover of the derived group of GQ. As an application, we prove that the monodromy group Γl is big in the sense that Γscll = GscQ (Zl) for all sufficiently large l. [ABSTRACT FROM AUTHOR]

Let F be a totally ramified extension of Qp. We consider supersingular representations of GL2(F) whose socles as GL2(OF )-modules are of a certain form that is expected to appear in the mod p local Langlands correspondence and establish a condition under which they are irreducible. [ABSTRACT FROM AUTHOR]

Let F be a totally ramified extension of Qp. We consider supersingular representations of GL2(F) whose socles as GL2(OF )-modules are of a certain form that is expected to appear in the mod p local Langlands correspondence and establish a condition under which they are irreducible. [ABSTRACT FROM AUTHOR]