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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]

A recent paper of Shekhar compares the ranks of elliptic curves E1 and E2 for which there is an isomorphism E1[p] ∼ E2[p]asGal(.../Q)- modules, where p is a prime of good ordinary reduction for both curves. In this paper we prove an analogous result in the case where p is a prime of good supersingular reduction. [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field of characteristic different from 2, let g be a positive integer, let ƒ(x) \in K[x] be a degree 2g+1 monic polynomial without multiple roots, let Cƒ: y2 = ƒ(x) be the corresponding genus g hyperelliptic curve over K, and let J be the Jacobian of Cƒ. We identify Cƒ with the image of its canonical embedding into J (the infinite point of Cƒ goes to the zero of the group law on J). It is known [Izv. Math. 83 (2019), pp. 501-520] that if g ≥ 2, then Cƒ(K) contains no points of orders lying between 3 and 2g. In this paper we study torsion points of order 2g + 1 on Cƒ(K). Despite the striking difference between the cases of g = 1 and g ≥ 2, some of our results may be viewed as a generalization of well-known results about points of order 3 on elliptic curves. E.g., if p = 2g + 1 is a prime that coincides with char(K), then every odd degree genus g hyperelliptic curve contains at most two points of order p. If g is odd and ƒ(x) has real coefficients, then there are at most two real points of order 2g + 1 on Cƒ. If ƒ(x) has rational coefficients and g ≤ 51, then there are at most two rational points of order 2g+1 on Cƒ. (However, there exist odd degree genus 52 hyperelliptic curves over Q that have at least four rational points of order 105.) [ABSTRACT FROM AUTHOR]

Bruin and Najman [LMS J. Comput. Math. 18 (2015), pp. 578–602], Ozman and Siksek [Math. Comp. 88 (2019), pp. 2461–2484], and Box [Math. Comp. 90 (2021), pp. 321–343] described all the quadratic points on the modular curves of genus 2\leq g(X_0(n)) \leq 5. Since all the hyperelliptic curves X_0(n) are of genus \leq 5 and as a curve can have infinitely many quadratic points only if it is either of genus \leq 1, hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves X_0(n) naturally arises; this question has recently also been posed by Mazur. We answer Mazur's question completely and describe the quadratic points on all the bielliptic modular curves X_0(n) for which this has not been done already. The values of n that we deal with are n=60, 62, 69, 79, 83, 89, 92, 94, 95, 101, 119 and 131; the curves X_0(n) are of genus up to 11. We find all the exceptional points on these curves and show that they all correspond to CM elliptic curves. The two main methods we use are Box's relative symmetric Chabauty method and an application of a moduli description of \mathbb {Q}-curves of degree d with an independent isogeny of degree m, which reduces the problem to finding the rational points on several quotients of modular curves. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve over a number field K, cv the Tamagawa number of E at v, and let cE = ∏v cv. Lorenzini proved that v13(cE) is positive for all elliptic curves over quadratic fields with a point of order 13. Krumm conjectured, based on extensive computation, that the 13-adic valuation of cE is even for all such elliptic curves. In this note we prove this conjecture and furthermore prove that there is a unique such curve satisfying v13(cE)=2. [ABSTRACT FROM AUTHOR]