## Elliptic curves with large rank over function fields (2001)

Citations: | 28 - 1 self |

### BibTeX

@MISC{Ulmer01ellipticcurves,

author = {Douglas Ulmer},

title = {Elliptic curves with large rank over function fields},

year = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

We produce explicit elliptic curves over Fp(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related elliptic surfaces) and then use zeta functions to determine the rank. In contrast to earlier examples of Shafarevitch and Tate, our curves are not isotrivial. Asymptotically these curves have maximal rank for their conductor. Motivated by this fact, we make a conjecture about the growth of ranks of elliptic curves over number fields.

### Citations

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(Show Context)
Citation Context ...f this model is ∆ = t d (1 − 2 4 3 3 t d ) and j(E) = 1/∆. Thus E has good reduction at all places of K except t = 0, the divisors of (1 − 2 4 3 3 t d ) and possibly t = ∞. Applying Tate’s algorithm (=-=[Tat75]-=-), we see that at t = 0, E has split multiplicative reduction of type Id and all geometric components of the special fiber are rational over k. At places v dividing (1 − 2 4 3 3 t d ), E has multiplic... |

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60 |
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(Show Context)
Citation Context ... the largest proven analytic rank (i.e., order of vanishing of L-series at s = 1) is 3 ([GZ86]). 1.3. For fields of characteristic p, it suffices to consider the rational function field K = Fp(t). In =-=[TS67]-=-, Shafarevitch and Tate produced elliptic curves over K of arbitrarily large rank. They considered a supersingular curve E0 defined over Fp (viewed as a curve E over K in the obvious way) and showed t... |

58 |
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(Show Context)
Citation Context ...field K = Fq(C)), the Tate conjecture for X is equivalent to the Birch and Swinnerton-Dyer conjecture for E. (More precisely (T) for X implies that RankE(K) = ords=1 L(E/K, s). Moreover, Tate proved (=-=[Tat66]-=-) that when (T) holds the refined conjecture of Birch and Swinnerton-Dyer on the leading Taylor coefficient of L(E/K, s) is true up to a power of p. Milne showed ([Mil75]) that the full refined conjec... |

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(Show Context)
Citation Context ...q and C is an explicit constant depending only on g and q. (Here we ignore the finitely many elliptic curves over K with trivial conductor.) This is the function field analogue of a theorem of Mestre =-=[Mes86]-=- which says that if E is a modular elliptic curve over Q then, assuming a generalized Riemann hypothesis, ords=1 L(E/Q, s) = O(log N/ loglog N). Brumer’s bound is visibly sensitive to the field of con... |

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(Show Context)
Citation Context ...r ∗ acts by multiplication by q. This, together with the cohomological description of zeta functions, gives inequalities (6.2.1) RankNS(X) ≤ dimQℓ H2 (X) Fr=q ≤ − ords=1 ζ(X, s). The Tate conjecture (=-=[Tat65]-=-) asserts that these are all equalities. We will refer to this assertion as “(T) for X.” (It would be more precise to refer to this as (T1), since there are conjectures for cycles of every codimension... |

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27 |
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(Show Context)
Citation Context ... ǫ term accounts for the points on E with either x = 0 or y = 0. Proof. Part 1 follows from the existence of a dominant rational map Fd���E and well-known results on the Tate conjecture. (We refer to =-=[Tat94]-=-, especially Section 5 for these results.) Indeed, if X���Y is a dominant rational map, (T) for X implies (T) for Y . But (T) is trivial for curves, and its truth for two varieties implies it for thei... |

26 |
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(Show Context)
Citation Context ... use, e.g., for height computations. 7. The zeta function of a Fermat surface 7.1. The zeta functions of Fermat varieties were computed in terms of Gauss and Jacobi sums by Weil in his landmark paper =-=[Wei49]-=-. We will need a refinement of this calculation due to Shioda which takes into account the action of G, and we will need to make the relevant Jacobi sums explicit. Remarkably, an explicit calculation ... |

25 |
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(Show Context)
Citation Context ...K, s). Moreover, Tate proved ([Tat66]) that when (T) holds the refined conjecture of Birch and Swinnerton-Dyer on the leading Taylor coefficient of L(E/K, s) is true up to a power of p. Milne showed (=-=[Mil75]-=-) that the full refined conjecture is true at least if p ̸= 2. We only need the rank conjecture.) Still assuming that X is an elliptic surface, the cohomological expression 6.1.1 for the zeta function... |

23 |
An explicit algorithm for computing the Picard number of certain algebraic surfaces
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(Show Context)
Citation Context ...t pairing, looks like an interesting project. Do these Mordell-Weil lattices have high densities or other special properties? 1.10. It is a pleasure to thank Felipe Voloch for bringing Shioda’s paper =-=[Shi86]-=- to my attention, Pavlos Tzermias for his help with p-adic Gamma functions and the Gross-Koblitz formula, and Dinesh Thakur for a number of useful remarks. 2. Invariants of E 2.1. We work in somewhat ... |

21 |
On Fermat varieties
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(Show Context)
Citation Context ...nt rational map, (T) for X implies (T) for Y . But (T) is trivial for curves, and its truth for two varieties implies it for their product. Since a Fermat variety is dominated by a product of curves (=-=[SK79]-=-), (T) follows for Fermat varieties, and this implies (T) for E. The equivalence of (T) for E and the conjecture of Birch and Swinnerton-Dyer for E was already noted above. To prove 2, we use the geom... |

17 |
Heegner points and derivatives of L-series, Invent
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(Show Context)
Citation Context ... quite difficult to produce examples with large rank. At this writing, the largest known rank is 24 ([MM00]) and the largest proven analytic rank (i.e., order of vanishing of L-series at s = 1) is 3 (=-=[GZ86]-=-). 1.3. For fields of characteristic p, it suffices to consider the rational function field K = Fp(t). In [TS67], Shafarevitch and Tate produced elliptic curves over K of arbitrarily large rank. They ... |

12 | Crystalline cohomology, Dieudonné modules, and Jacoby sums. Automorphic forms, representation theory and arithmetic
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(Show Context)
Citation Context ... Fermat varieties of the same degree. This allows one to reduce to the case of curves. The proposition for Fermat curves is [Kat81, Cor. 2.4]. We will sketch another proof, closely related to that in =-=[Kat81]-=-, which works uniformly in all dimensions. For simplicity we will only discuss the case of Fermat surfaces. To that end, consider the finite morphism π : Fd → Fd/G ∼ = P 2 . The sheaf F = π∗Qℓ carries... |

5 | on algebraic cycles in l-adic cohomology, Motives - Conjectures - 1991 |

3 |
An elliptic curve over Q with rank at least 24, Preprint (2000), Posted to the Usenet newsgroup math.sci.nmbrthry by V. Miller on May 2, 2000. Available at http://listserv.nodak.edu/archives/nmbrthry.html
- Martin, McMillen
(Show Context)
Citation Context ...c zero, it obviously suffices to treat the case K = Q. Here the question is open and it seems to be quite difficult to produce examples with large rank. At this writing, the largest known rank is 24 (=-=[MM00]-=-) and the largest proven analytic rank (i.e., order of vanishing of L-series at s = 1) is 3 ([GZ86]). 1.3. For fields of characteristic p, it suffices to consider the rational function field K = Fp(t)... |

2 |
Numerical sections on elliptic surfaces
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(Show Context)
Citation Context ...direct sum, with respect to the intersection pairing, of L ⊗ Q and E(K) ⊗ Q.) The exact sequence is equivalent to the assertion that that s(E(K)) is a set of coset representatives for L in NS(E). See =-=[MP86]-=- for more details. The Shioda-Tate formula will allow us to compute the rank of E(K) in terms of NS(E), which we will eventually compute using the Tate conjecture. 3.6. We will want to consider the si... |