Abelian subvarieties of Drinfeld Jacobians and congruences modulo the characteristic

Papikian, Mihran

doi:10.1007/s00208-006-0029-3

Cited by 3 publications

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“…If n is irreducible, this is [Pa1, Theorem 1.2]. More generally, in the terminology of[Pa2], if E is not Frobenius minimal, then by [Pa2, Theorem 1.1] we have p ∈ C (p), but C (p) = ∅ since S(n) p-old = 0. 2…”

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confidence: 90%

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Strong Weil curves over Fq(T) with small conductor

Schweizer¹

2011

Journal of Number Theory

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We continue work of Gekeler and others on elliptic curves over F q (T ) with conductor ∞ · n where n ∈ F q [T ] has degree 3. Because of the Frobenius isogeny there are infinitely many curves in each isogeny class, and we discuss in particular which of these curves is the strong Weil curve with respect to the uniformization by the Drinfeld modular curve X 0 (n). As a corollary we obtain that the strong Weil curve E/F q (T ) always gives a rational elliptic surface over F q .The first paper that systematically used Drinfeld modular curves and the Bruhat-Tits tree in order to classify elliptic curves over a function field was [Ge1].With an eye towards feasibility of practical calculations, what we are talking about here are elliptic curves over a rational function field F q (T ) and Drinfeld modular curves X 0 (n) for n ∈ F q [T ]. Every elliptic curve over F q (T ) that is modular, that is, covered by some X 0 (n), must have split multiplicative reduction at the place ∞ (= pole divisor of T ). Conversely, every elliptic curve over F q (T ) with conductor ∞ · n and split multiplicative reduction at ∞ is an isogeny factor of the Jacobian of X 0 (n).

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“…More precisely, they are obtained from finitely many curves by repeated application of the Frobenius isogeny. Papikian [Pa1,Pa2] has shown that in certain situations the strong Weil curve is not the Frobenius of another curve over F q (T ), but from examples it is known that this is not a general phenomenon.…”

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confidence: 92%

Strong Weil curves over Fq(T) with small conductor

Schweizer¹

2011

Journal of Number Theory

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A combinatorial Li–Yau inequality and rational points on curves

2014

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Abstract. We present a method to control gonality of nonarchimedean curves based on graph theory.Let k denote a complete nonarchimedean valued field. We first prove a lower bound for the gonality of a curve over the algebraic closure of k in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinement of the dual graph of the stable model of the curve.Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some "volume" of the original graph; this can be seen as a substitute for graphs of the Li-Yau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs.Finally, we apply the results to give a lower bound for the gonality of arbitrary Drinfeld modular curves over finite fields and for general congruence subgroups Γ of Γ(1) that is linear in the index [Γ(1) : Γ], with a constant that only depends on the residue field degree and the degree of the chosen "infinite" place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian.

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Graph Laplacians, component groups and Drinfeld modular curves

Papikian¹

2016

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Abelian subvarieties of Drinfeld Jacobians and congruences modulo the characteristic

Abstract: Abstract.We prove a level lowering result over rational function fields, with the congruence prime being the characteristic of the field. We apply this result to show that semi-stable optimal elliptic curves are not Frobenius conjugates of other curves defined over the same field.

Cited by 3 publications

References 20 publications

Strong Weil curves over Fq(T) with small conductor

Strong Weil curves over Fq(T) with small conductor

A combinatorial Li–Yau inequality and rational points on curves

Graph Laplacians, component groups and Drinfeld modular curves

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