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Let N be a positive integer. Let H be a subgroup of GL (2 , Z / N Z) of level N and let E be an elliptic curve defined over the rationals with j E ≠ 0 , 1728 . Then the image ρ ¯ E , N Gal Q ¯ / Q , of the mod-N Galois representation attached to E, is conjugate to a subgroup of H if and only if E corresponds to a non-cuspidal rational point on the modular curve X H generated by H. In this article, we are interested when ρ ¯ E , N Gal Q ¯ / Q is conjugate to H. More precisely, we classify all subgroups H of GL (2 , Z / N Z) that are direct products of groups of distinct prime-power level for which X H contains infinitely many non-cuspidal rational points but there is no elliptic curve E / Q such that ρ ¯ E , N Gal Q ¯ / Q is conjugate to H itself. [ABSTRACT FROM AUTHOR]

We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over ℚ with 1 2 -torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen and Fisher where it is assumed that the underlying isomorphism of 1 2 -torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrize such pairs of elliptic curves. A key ingredient in the proof is to construct simple (algebraic) conditions for the 2 , 3 or 4 -torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the j -invariants of the pair of elliptic curves. [ABSTRACT FROM AUTHOR]

Let E be a Q -isogeny class of elliptic curves defined over Q. The isogeny graph associated to E is a graph which has a vertex for each element of E and an edge for each Q -isogeny of prime degree that maps one element of E to another element of E , with the degree recorded as a label of the edge. The isogeny-torsion graph associated to E is the isogeny graph associated to E where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over Q of the corresponding elliptic curve. The main result of the article is a determination of which isogeny-torsion graphs associated to Q -isogeny classes of elliptic curves defined over Q correspond to infinitely many j -invariants. [ABSTRACT FROM AUTHOR]

We study the Galois groups of polynomials arising from a compatible family of representations with big orthogonal monodromy. We show that the Galois groups are usually as large as possible given the constraints imposed on them by a functional equation and discriminant considerations. As an application, we consider the Frobenius polynomials arising from the middle étale cohomology of hypersurfaces in ℙ q 2 n + 1 of degree at least three. We also consider the L -functions of quadratic twists of fixed degree of an elliptic curve over a function field q (t). To determine the typical Galois group in the elliptic curve setting requires using some known cases of the Birch and Swinnerton-Dyer conjecture. This extends and generalizes work of Chavdarov, Katz and Jouve. [ABSTRACT FROM AUTHOR]

For a given elliptic curve E defined over the rationals, we study the density of primes p satisfying gcd (# E (F p) , p − 1) = 1 and give a conjectural value for this density with strong heuristic evidence for most elliptic curves, in an appropriate sense. [ABSTRACT FROM AUTHOR]

Let F be a newform of weight 2k on Γ 0 (N) with an odd integer N and a positive integer k, and ℓ be a prime larger than or equal to 5 with (ℓ , N) = 1 . For each fundamental discriminant D, let χ D be a quadratic character associated with quadratic field Q (D) . Assume that for each D, the ℓ -adic valuation of the algebraic part of L (F ⊗ χ D , k) is non-negative. Let W ℓ + (resp. W ℓ - ) be the set of positive (resp. negative) fundamental discriminants D with (D , N) = 1 such that the ℓ -adic valuation of the algebraic part of L (F ⊗ χ D , k) is zero. We prove that for each sign ϵ , if W ℓ ϵ is a non-empty finite set, then W ℓ ϵ ⊂ 1 , (- 1) ℓ - 1 2 ℓ. By this result, we prove that if ϵ is the sign of (- 1) k , then k ≥ ℓ - 1 or k = ℓ - 1 2. These are applied to obtain a lower bound for # { D ∈ W ℓ ϵ : | D | ≤ X } and the indivisibility of the order of the Shafarevich–Tate group of an elliptic curve over Q . To prove these results, first we refine Waldspurger's formula on the Shimura correspondence for general odd levels N. Next we study mod ℓ modular forms of half-integral weight with few non-vanishing coefficients. To do this, we use the filtration of mod ℓ modular forms and mod ℓ Galois representations. [ABSTRACT FROM AUTHOR]

Inspired by the work of Lang–Trotter on the densities of primes with fixed Frobenius traces for elliptic curves defined over Q and by the subsequent generalization of Cojocaru–Davis–Silverberg–Stange to generic abelian varieties, we study the analogous question for abelian surfaces isogenous to products of non-CM elliptic curves over Q that are not Q ¯ -isogenous. We formulate the corresponding conjectural asymptotic, provide upper bounds, and explicitly compute (when the elliptic curves lie outside a thin set) the arithmetically significant constants appearing in the asymptotic. This allows us to provide computational evidence for the conjecture. [ABSTRACT FROM AUTHOR]

We describe a systematic investigation into the existence of congruences between the mod p torsion modules of elliptic curves defined over Q, including methods to determine the symplectic type of such congruences. We classify the existence and symplectic type of mod p congruences between twisted elliptic curves over number fields, giving global symplectic criteria that apply in situations where the available local methods may fail. We report on the results of applying our methods for all primes p≥7 to the elliptic curves in the LMFDB database, which currently includes all elliptic curves of conductor less than 500000. We also show that while such congruences exist for each p≤17, there are none for p≥19 in the database, in line with a strong form of the Frey-Mazur conjecture. [ABSTRACT FROM AUTHOR]

Let F be a totally real number field which is unramified at 3 , 5 , and 7. We give explicit conditions on a finite place v 0 of F such that any elliptic curve over F with potential good reduction at v 0 is modular. The main point is to prove that there exist no elliptic curves with reducible mod 5, and 7 representations (or mod 3, 5, and 7 representations) under our assumptions. [ABSTRACT FROM AUTHOR]

In this paper we prove that for every integer d ≥ 1 , there exists an explicit constant B d such that the following holds. Let K be a number field of degree d , let q > max { d − 1 , 5 } be any rational prime that is totally inert in K and E any elliptic curve defined over K such that E has potentially multiplicative reduction at the prime above q. Then for every rational prime p > B d , E has an irreducible mod p Galois representation. This result has Diophantine applications within the "modular method". We present one such application in the form of an Asymptotic version of Fermat's Last Theorem that has not been covered in the existing literature. [ABSTRACT FROM AUTHOR]

We prove a local–global compatibility result in the mod $p$ Langlands program for $\mathrm {GL}_2(\mathbf {Q}_{p^f})$. Namely, given a global residual representation $\bar {r}$ appearing in the mod $p$ cohomology of a Shimura curve that is sufficiently generic at $p$ and satisfies a Taylor–Wiles hypothesis, we prove that the diagram occurring in the corresponding Hecke eigenspace of mod $p$ completed cohomology is determined by the restrictions of $\bar {r}$ to decomposition groups at $p$. If these restrictions are moreover semisimple, we show that the $(\varphi ,\Gamma)$ -modules attached to this diagram by Breuil give, under Fontaine's equivalence, the tensor inductions of the duals of the restrictions of $\bar {r}$ to decomposition groups at $p$. [ABSTRACT FROM AUTHOR]

Drinfeld modules and A-motives s are the function field analogs of elliptic curves and abelian varieties. For both Drinfeld modules and A -motives, one can construct their l -adic Galois representations and ask whether the images are open. For Drinfeld modules, this question has been answered by Richard Pink and his co-authors; however, this question has not been addressed for A -motives. Here, we clarify the rank-one case for A-motives and show that the image of Galois is open if and only if the virtual dimension is prime to the characteristic of the ground field. [ABSTRACT FROM AUTHOR]

In this paper, we study some ternary Diophantine equations of the form A a n + B b n = C c m , where m ∈ { 2 , 3 } , and n ⩾ 7 is a prime. In fact, we completely solve some particular cases: A = 5 α , B = 64 , C = 3 , when m = 2 ; A = 2 α , B = 27 , C ∈ { 7 , 13 } , when m = 3 . [ABSTRACT FROM AUTHOR]

We determine explicit birational models over Q for the modular surfaces parametrising pairs of N-congruent elliptic curves in all cases where this surface is an elliptic surface. In each case we also determine the rank of the Mordell–Weil lattice and the geometric Picard number. [ABSTRACT FROM AUTHOR]

In this paper, we present a description of the ℓ -adic Galois representation attached to an elliptic curve defined over a 2 -adic field K , in the case where the image of inertia is non-abelian. There are two possibilities for the image of inertia, namely Q 8 and SL 2 (𝔽 3) , and in each case, we need to distinguish whether the inertia degree of K over ℚ 2 is even or odd. The results presented here are being implemented in an algorithm to compute explicitly the Galois representation in these four cases. [ABSTRACT FROM AUTHOR]

Abstract We present a strategy to obtain explicit equations for the modular double covers associated respectively to both a split and a non-split Cartan subgroup of GL 2 (F p) with p prime. Then we apply it successfully to the level 13 case. [ABSTRACT FROM AUTHOR]

We develop methods to study 2-dimensional 2-adic Galois representations ρ of the absolute Galois group of a number field K , unramified outside a known finite set of primes S of K , which are presented as Black Box representations, where we only have access to the characteristic polynomials of Frobenius automorphisms at a finite set of primes. Using suitable finite test sets of primes, depending only on K and S , we show how to determine the determinant det ⁡ ρ , whether or not ρ is residually reducible, and further information about the size of the isogeny graph of ρ whose vertices are homothety classes of stable lattices. The methods are illustrated with examples for K = Q , and for K imaginary quadratic, ρ being the representation attached to a Bianchi modular form. These results form part of the first author's thesis [2] . [ABSTRACT FROM AUTHOR]

Let A be a g -dimensional abelian variety over Q whose adelic Galois representation has open image in GSp 2 g Z ˆ . We investigate the Frobenius fields Q ( π p ) = End ( A p ) ⊗ Q of the reduction of A modulo primes p at which this reduction is ordinary and simple. We obtain conditional and unconditional asymptotic upper bounds on the number of primes at which Q ( π p ) is a specified number field and, when A is two-dimensional, at which Q ( π p ) contains a specified real quadratic number field. These investigations continue the investigations of variants of the Lang–Trotter Conjectures on elliptic curves. [ABSTRACT FROM AUTHOR]

A theorem of Mazur gives the set of possible prime degrees for rational isogenies between elliptic curves. In this paper, we are working on a similar problem in the case of abelian surfaces of -type over with quaternionic multiplication (over ) endowed with a level structure. We prove the following result: for a fixed indefinite quaternion algebra of discriminant and a fixed quadratic imaginary field , there exists an effective bound such that for a prime number , not dividing the conductor of the order , there do not exist abelian surfaces such that is a maximal order of and is endowed with a level structure. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic char ( K ) ≠ 2 , 3 and let E be an elliptic curve defined over K . Let m be a positive integer, prime with char ( K ) if char ( K ) ≠ 0 ; we denote by E [ m ] the m -torsion subgroup of E and by K m : = K ( E [ m ] ) the field obtained by adding to K the coordinates of the points of E [ m ] . Let P i : = ( x i , y i ) ( i = 1 , 2 ) be a Z -basis for E [ m ] ; then K m = K ( x 1 , y 1 , x 2 , y 2 ) . We look for small sets of generators for K m inside { x 1 , y 1 , x 2 , y 2 , ζ m } trying to emphasize the role of ζ m (a primitive m -th root of unity). In particular, we prove that K m = K ( x 1 , ζ m , y 2 ) , for any odd m ⩾ 5 . When m = p is prime and K is a number field we prove that the generating set { x 1 , ζ p , y 2 } is often minimal, while when the classical Galois representation Gal ( K p / K ) → GL 2 ( Z / p Z ) is not surjective we are sometimes able to further reduce the set of generators. We also describe explicit generators, degree and Galois groups of the extensions K m / K for m = 3 and m = 4 . [ABSTRACT FROM AUTHOR]

Given $k\geqslant 2$, we show that there are at most finitely many rational numbers $x$ and $y\neq 0$ and integers $\ell \geqslant 2$ (with $(k,\ell )\neq (2,2)$) for which $$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$ In particular, if we assume that $\ell$ is prime, then all such triples $(x,y,\ell )$ satisfy either $y=0$ or $\ell <\exp (3^{k})$. [ABSTRACT FROM AUTHOR]

Let K be a number field and let E 1 , … , E n be elliptic curves over K , pairwise non-isogenous over K ‾ and without complex multiplication over K ‾ . We study the image G ∞ of the adelic representation of Gal ( K ‾ / K ) naturally attached to E 1 × ⋯ × E n . The main result is an explicit bound for the index of G ∞ in { ( x 1 , … , x n ) ∈ GL 2 ( Z ˆ ) n | det ⁡ x i = det ⁡ x j ∀ i , j } . [ABSTRACT FROM AUTHOR]

We show that a two dimensional ℓ -adic representation of the absolute Galois group of a number field which is locally potentially equivalent to a GL ( 2 ) - ℓ -adic representation ρ at a set of places of K of positive upper density is potentially equivalent to ρ . As an application, for E 1 and E 2 defined over a number field K , with at least one of them without complex multiplication, we prove that the set of places v of K of good reduction such that the corresponding Frobenius fields are equal has positive upper density if and only if E 1 and E 2 are isogenous over some extension of K . For an elliptic curve E defined over a number field K , we show that the set of finite places of K such that the Frobenius field F ( E , v ) at v equals a fixed imaginary quadratic field F has positive upper density if and only if E has complex multiplication by F . [ABSTRACT FROM AUTHOR]

Let E and E' be elliptic curves over an algebraic number field. We show that systems of l-adic representations associated with E and E' are cohomologically coprime, in the sense that the Galois cohomology groups corresponding to respective fields of division points are all trivial. This provides a generalization of some known results about the vanishing of the cohomology groups associated with the l-adic Tate module of an elliptic curve. [ABSTRACT FROM AUTHOR]

In this work we compute the framed deformation functor associated to a reducible representation given as a direct sum of 2-dimensional representations associated to elliptic curves with appropriate local conditions. Such conditions arise in the works of Schoof and correspond to reduction properties of modular elliptic curves. [ABSTRACT FROM AUTHOR]

Given an elliptic curve E / Q , the torsion points of E give rise to a natural Galois representation ρ E : Gal ( Q ¯ / Q ) → GL 2 ( Z ˆ ) associated to E . In 1972, Serre showed that [ GL 2 ( Z ˆ ) : Im ρ E ] ≥ 2 for all non-CM elliptic curves. The main goal of this paper is to exhibit an elliptic surface such that the Galois representations associated to almost all of the rational specializations have maximal image. Further, we find an explicit set S ⊂ Q , such that if t ∉ S , then the Galois representation associated to the specialization at t has maximal image, with a bounded number of exceptions. [ABSTRACT FROM AUTHOR]

Serre's uniformity problem asks whether there exists a bound k such that for any p > k, the Galois representation associated to the p-torsion of an elliptic curve E/Q is surjective independent of the choice of E. Serre showed that if this representation is not surjective, then it has to be contained in either a Borel subgroup, the normalizer of a split Cartan subgroup, the normalizer of a non-split Cartan subgroup, or one of a finite list of "exceptional" subgroups. We will focus on the case when the image is contained in the normalizer of a split Cartan subgroup. In particular, we will show that the only elliptic curves whose Galois representation at 11 is contained in the normalizer of a split Cartan have complex multiplication. To prove this we compute Xs+ (11) using modular units, use the methods of Poonen and Schaefer to compute its Jacobian, and then use the method of Chabauty and Coleman to show that the only points on this curve correspond to CM elliptic curves. [ABSTRACT FROM AUTHOR]

We prove modularity of some two-dimensional, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$-adic Galois representations over a totally real field that are nearly ordinary at all places above $2$ and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles, using Hida families, together with the $2$-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above $2$. [ABSTRACT FROM PUBLISHER]

Abstract: Using a multi-dimensional large sieve inequality, we prove that, for almost all pairs (or indeed almost all k-tuples) of elliptic curves, the associated Galois representation on their torsion has maximal image. This generalizes the authorʼs previous work and provides evidence for an affirmative answer to a higher-dimensional analogue of Serreʼs uniformity question for single elliptic curves. Furthermore, as a consequence of our main theorem, one deduces the triviality of the Brauer group of the Kummer surface attached to almost all pairs of elliptic curves. [Copyright &y& Elsevier]

Abstract: Let K be a fixed number field and its absolute Galois group. We give a bound , depending only on the degree, the class number and the discriminant of K, such that for any elliptic curve E defined over K and any prime number p strictly larger than , the image of the representation of attached to the p-torsion points of E contains a subgroup of homotheties of index smaller than 12. [Copyright &y& Elsevier]

Let E be an elliptic curve defined over a number field K. We say that a prime number p is reducible for (E, K) if E admits a p-isogeny defined over K. The so-called reducible set of all such prime numbers is finite if and only if E does not have complex multiplication over K. In this paper, we prove that the reducible set is included in the set of prime divisors of an explicit list of integers (depending on E and K), infinitely many of them being non-zero. It provides an efficient algorithm for computing it in the finite case. Other less general but rather useful criteria are given, as well as many numerical examples. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve over the rationals. In 1988, Koblitz conjectured an asymptotic for the number of primes p for which the cardinality of the group of 픽p-points of E is prime. However, the constant occurring in his asymptotic does not take into account that the distributions of the |E(픽p)| need not be independent modulo distinct primes. We shall describe a corrected constant. We also take the opportunity to extend the scope of the original conjecture to ask how often |E(픽p)|/t is an integer and prime for a fixed positive integer t, and to consider elliptic curves over arbitrary number fields. Several worked out examples are provided to supply numerical evidence for the new conjecture. [ABSTRACT FROM AUTHOR]

Recently, Bruinier and Ono investigated the arithmetic of the coefficients of Ramanujan’s mock theta function ω( q). In Ramanujan J. () they obtained congruences with respect to the modulus 512. Here we show that ω( q) modulo 5 is dictated by an elliptic curve. [ABSTRACT FROM AUTHOR]

Abstract: We derive upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree . Our main tool is the adelic openness of the image of Galois representations, due to Serre, Pink and Rütsche. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules. [Copyright &y& Elsevier]

Let K be a number field. In this paper, we are interested in the following problem: does there exist a constant cK, which depends only on K, such that for any semi-stable elliptic curve defined over K, the Galois representation in its p-torsion points is irreducible whenever p is a prime number greater than cK? In case the answer is positive, how can we get such a constant? We prove that if a certain condition is satisfied by K, the answer is positive and we obtain cK explicitly. Furthermore, we prove that this condition is realized in many situations. [ABSTRACT FROM AUTHOR]