arXiv subjects

Kenneth A. Ribet

Publications and source records attributed to Kenneth A. Ribet.

9 recordsLinked to original sources

arXiv2025

Generalized Jacobians of graphs

We define a generalized Jacobian $\mathrm{J}_\mathfrak{m}(\mathit{Gr})$ and a generalized Picard group $\mathrm{P}_\mathfrak{m}(\mathit{Gr})$ of a graph $\mathit{Gr}$ with respect to a modulus $ \mathfrak{m}=\sum_{i=1}^s m_iw_i$ with $w_i$ vertices of $\mathit{Gr}$ and $m_i\geq 1$. These groups occur as the component groups of Néron models of generalized Jacobians. We prove a universal mapping property for $\mathrm{J}_\mathfrak{m}(\mathit{Gr})$ and show that an Abel-Jacobi map in this context induces an isomorphism from $\mathrm{P}_\frak{m}(\mathit{Gr})$ to $\mathrm{J}_\mathfrak{m}(\mathit{Gr})$. We also reinterpret $\mathrm{P}_\mathfrak{m}(\mathit{Gr})$ in terms of sheaves on the geometric realization $\left| \mathit{Gr}\right|$ of $\mathit{Gr}$, making a connection with tropical geometry.

math.CO↗

arXiv2023

Modular curves and Néron models of generalized Jacobians

Let $X$ be a smooth geometrically connected projective curve over the field of fractions of a discrete valuation ring $R$, and $\mathfrak{m}$ a modulus on $X$, given by a closed subscheme of $X$ which is geometrically reduced. The generalized Jacobian $J_\mathfrak{m}$ of $X$ with respect to $\mathfrak{m}$ is then an extension of the Jacobian of $X$ by a torus. We describe its Néron model, together with the character and component groups of the special fibre, in terms of a regular model of $X$ over $R$. This generalizes Raynaud's well-known description for the usual Jacobian. We also give some computations for generalized Jacobians of modular curves $X_0(N)$ with moduli supported on the cusps.

math.AG↗

arXiv2022

Another look at rational torsion of modular Jacobians

We study the rational torsion subgroup of the modular Jacobian $J_0(N)$ for $N$ a square-free integer. We give a new proof of a result of Ohta on a generalization of Ogg's conjecture: for a prime number $p \nmid 6N$, the $p$-primary part of the rational torsion subgroup equals that of the cuspidal subgroup. Whereas previous proofs of this result used explicit computations of the cardinalities of these groups, we instead use their structure as modules for the Hecke algebra.

math.NT↗

arXiv2003

Torsion points on modular curves and Galois theory

We outline the proof of a theorem of M. Baker and A. Tamagawa that gives a complete description of the torsion points on a modular curve embedded in its Jacobian using the notion of an `almost rational torsion point.'

math.NT↗

arXiv2002

Galois theory and torsion points on curves

In this paper, we survey some Galois-theoretic techniques for studying torsion points on curves. In particular, we give new proofs of some results of A. Tamagawa and the present authors for studying torsion points on curves with "ordinary good" or "ordinary semistable" reduction at a given prime. We also give new proofs of: (1) The Manin-Mumford conjecture: There are only finitely many torsion points lying on a curve of genus at least 2 embedded in its Jacobian by an Albanese map; and (2) The Coleman-Kaskel-Ribet conjecture: If p is a prime number which is at least 23, then the only torsion points lying on the curve X_0(p), embedded in its Jacobian by a cuspidal embedding, are the cusps (together with the hyperelliptic branch points when X_0(p) is hyperelliptic and p is not 37). In an effort to make the exposition as useful as possible, we provide references for all of the facts about modular curves which are needed for our discussion.

math.NT↗

arXiv2001

Letter about component groups of elliptic curves

This is the text of a letter written by K. Ribet to J.-F. Mestre in November, 1987. The letter proves three formulas that had been conjectured by Mestre and Oesterlé. These formulas concern component groups of elliptic curves that occur as optimal quotients of certain Jacobians of modular curves, and relations between the component groups and the character group of the toric component of the Jacobian. (The "toric component" and component groups are computed in positive characteristic.)

math.AG↗

arXiv1995

On the equation $a^p + 2^alpha b^p + c^p =0$

We discuss the equation $a^p + 2^\a b^p + c^p =0$ in which $a$, $b$, and $c$ are non-zero relatively prime integers, $p$ is an odd prime number, and $\a$ is a positive integer. The technique used to prove Fermat's Last Theorem shows that the equation has no solutions with $\a>1$ or $b$ even. When $\a=1$ and $b$ is odd, there are the two trivial solutions $(\pm 1, \mp 1, \pm 1)$. In 1952, Dénes conjectured that these are the only ones. Using methods of Darmon, we prove this conjecture for $p\equiv1$ mod~4. We link the case $p\equiv3$ mod~4 to conjectures of Frey and Darmon about elliptic curves over~$\Q$ with isomorphic mod~$p$ Galois representations.

math.NT↗

arXiv1995

Galois representations and modular forms

In this article, I discuss material which is related to the recent proof of Fermat's Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey's construction, and the conjectures of Serre and of Taniyama--Shimura.

math.NT↗

arXiv1992

Abelian varieties over Q and modular forms

This paper gives a conjectural characterization of those elliptic curves over the field of complex numbers which "should" be covered by standard modular curves. The elliptic curves in question all have algebraic j-invariant, so they can be viewed as curves over Q-bar, the field of algebraic numbers. The condition that they satisfy is that they must be isogenous to all their Galois conjugates. Borrowing a term from B.H. Gross, "Arithmetic on elliptic curves with complex multiplication," we say that the elliptic curves in question are "Q-curves." Since all complex multiplication elliptic curves are Q-curves (with this definition), and since they are all uniformized by modular forms (Shimura), we consider only non-CM curves for the remainder of this abstract. We prove: 1. Let C be an elliptic curve over Q-bar. Then C is a Q-curve if and only if C is a Q-bar simple factor of an abelian variety A over Q whose algebra of Q-endomorphisms is a number field of degree dim(A). (We say that abelian varieties A/Q with this property are of "GL(2) type.") 2. Suppose that Serre's conjecture on mod p modular forms are correct (Ref: Duke Journal, 1987). Then an abelian variety A over Q is of GL(2)-type if and only if it is a simple factor (over Q) of the Jacobian J_1(N) for some integer N\ge1. (The abelian variety J_1(N) is the Jacobian of the standard modular

alg-geom↗