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In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves X_0(N) of genus up to 8, and genus up to 10 with N prime, for which they were previously unknown. The values of N we consider are contained in the set \begin{equation*} \mathcal {L}=\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \}. \end{equation*} We obtain that all the non-cuspidal quadratic points on X_0(N) for N\in \mathcal {L} are complex multiplication (CM) points, except for one pair of Galois conjugate points on X_0(103) defined over \mathbb {Q}(\sqrt {2885}). We also compute the j-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings. [ABSTRACT FROM AUTHOR]

In this paper, we construct a compactification of the space of Bridgeland stability conditions on a smooth projective curve, as an analogue of Thurston compactifications in Teichmüller theory. In the case of elliptic curves, we compare our results with the classical one of the torus via homological mirror symmetry and give the Nielsen–Thurston classification of autoequivalences using the compactification. Furthermore, we observe an interesting phenomenon in the case of the projective line. [ABSTRACT FROM AUTHOR]

Building on Mazur's 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over \mathbb {Q}. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields \mathbb {Q}(\sqrt {d}) with |d| < 10^4 we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over 19 quadratic fields, including \mathbb {Q}(\sqrt {213}) and \mathbb {Q}(\sqrt {-2289}). To make this procedure work, we determine all of the finitely many quadratic points on the modular curves X_0(125) and X_0(169), which may be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, we study moments of Hurwitz class numbers associated to imaginary quadratic orders restricted into fixed arithmetic progressions. In particular, we fix t in an arithmetic progression t\equiv m\ \, \left (\operatorname {mod} \, M \right) and consider the ratio of the 2k-th moment to the zeroeth moment for H(4n-t^2) as one varies n. The special case n=p^r yields as a consequence asymptotic formulas for moments of the trace t\equiv m\ \, \left (\operatorname {mod} \, M \right) of Frobenius on elliptic curves over finite fields with p^r elements. [ABSTRACT FROM AUTHOR]

Given an elliptic curve E defined over \mathbb {C}, let E^{\times } be an open subset of E obtained by removing a point. In this paper, we show that the i-th Betti number of the unordered configuration space \mathrm {Conf}^{n}(E^{\times }) of n points on E^{\times } appears as a coefficient of an explicit rational function in two variables. We also compute its Hodge numbers as coefficients of another explicit rational function in four variables. Our result is interesting because these rational functions resemble the generating function of the \mathbb {F}_{q}-point counts of \mathrm {Conf}^{n}(E^{\times }), which can be obtained from the zeta function of E over any fixed finite field \mathbb {F}_{q}. We show that the mixed Hodge structure of the i-th singular cohomology group H^{i}(\mathrm {Conf}^{n}(E^{\times })) with complex coefficients is pure of weight w(i), an explicit integer we provide in this paper. This purity statement implies our main result about the Betti numbers and the Hodge numbers. Our proof uses Totaro's spectral sequence computation that describes the weight filtration of the mixed Hodge structure on H^{i}(\mathrm {Conf}^{n}(E^{\times })). [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve over \mathbb {Q} with discriminant \Delta _E. For primes p of good reduction, let N_p be the number of points modulo p and write N_p=p+1-a_p. In 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies \begin{equation*} \lim _{x\to \infty }\frac {1}{\log x}\sum _{\substack {p\leq x\\ p\nmid \Delta _{E}}}\frac {a_p\log p}{p}=-r+\frac {1}{2}, \end{equation*} where r is the order of the zero of the L-function L_{E}(s) of E at s=1, which is predicted to be the Mordell-Weil rank of E(\mathbb {Q}). We show that if the above limit exits, then the limit equals -r+1/2. We also relate this to Nagao's conjecture. This paper also includes an appendix by Andrew V. Sutherland which gives evidence for the convergence of the above-mentioned limit. [ABSTRACT FROM AUTHOR]

In this paper, we initiate a systematic study of entanglements of division fields from a group theoretic perspective. For a positive integer n and a subgroup G\subseteq GL_2(\mathbb {Z}/n\mathbb {Z}) with surjective determinant, we provide a definition for G to represent an (a,b)-entanglement and give additional criteria for G to represent an explained or unexplained (a,b)-entanglement. Using these new definitions, we determine the tuples ((p,q),T), with p

Let A be a principally polarized abelian variety of dimension g over a number field K. Assume that the image of the adelic Galois representation of A is an open subgroup of GSp_{2g}(\hat {\mathbb {Z}}). Then there exists a positive integer m so that the Galois image of A is the full preimage of its reduction modulo m. The least m with this property, denoted m_A, is called the image conductor of A. Jones [Pacific J. Math. 308 (2020), pp. 307–331] recently established an upper bound for m_A, in terms of standard invariants of A, in the case that A is an elliptic curve without complex multiplication. In this paper, we generalize the aforementioned result to provide an analogous bound in arbitrary dimension. [ABSTRACT FROM AUTHOR]

We provide some minor corrections to our paper "Statistics for Iwasawa invariants of elliptic curves". [ABSTRACT FROM AUTHOR]