# Park City lectures on elliptic curves over function fields

@article{Ulmer2011ParkCL, title={Park City lectures on elliptic curves over function fields}, author={Douglas Ulmer}, journal={arXiv: Number Theory}, year={2011} }

These are the notes from a course of five lectures at the 2009 Park City Math Institute. The focus is on elliptic curves over function fields over finite fields. In the first three lectures, we explain the main classical results (mainly due to Tate) on the Birch and Swinnerton-Dyer conjecture in this context and its connection to the Tate conjecture about divisors on surfaces. This is preceded by a "Lecture 0" on background material. In the remaining two lectures, we discuss more recent… Expand

#### 5 Citations

A family of elliptic curves of large rank

- Mathematics
- 2012

Abstract Fix a finite field k, a positive integer d relatively prime to the characteristic of k, and an element a of k. In this article we study the elliptic curve E with equation x ( x − 1 ) ( y − a… Expand

If Riemann’s Zeta Function is True, it Contradicts Zeta’s Dirichlet Series, Causing "Explosion". If it is False, it Causes Unsoundness.

- Mathematics
- 2019

Riemann's "analytic continuation" produces a second definition of the Zeta function, that Riemann claimed is convergent throughout half-plane $s \in \mathbb{C}$, $\text{Re}(s)\le1$, (except at… Expand

Propositional Logic Applied to Three Contradictory Definitions of the Zeta Function, and to Conditionally Convergent Series

- 2019

The paper discusses the following contradictions: <br><br>(1) Contradictory definitions of the Zeta function. These three definitions of the Zeta function contradict each other: the Dirichlet series,… Expand

Explicit points on $y^2 + xy - t^d y = x^3$ and related character sums

- Mathematics
- 2014

Let $\mathbb{F}_q$ denote a finite field of characteristic $p \geq 5$ and let $d = q+1$. Let $E_d$ denote the elliptic curve over the function field $\mathbb{F}_{q^2}(t)$ defined by the equation $y^2… Expand

Quantum Electrodynamics (QED) Renormalization is a Logical Paradox, Zeta Function Regularization is Logically Invalid, and Both are Mathematically Invalid

- Mathematics
- 2020

Quantum Electrodynamics (QED) renormalizaion is a paradox. It uses the Euler-Mascheroni constant, which is defined by a conditionally convergent series. But Riemann's series theorem proves that any… Expand

#### References

SHOWING 1-10 OF 74 REFERENCES

Towers of surfaces dominated by products of curves and elliptic curves of large rank over function fields

- Mathematics
- 2008

Abstract With the goal of producing elliptic curves and higher-dimensional abelian varieties of large rank over function fields, we provide a geometric construction of towers of surfaces dominated by… Expand

Advanced Topics in the Arithmetic of Elliptic Curves

- Mathematics
- 1994

In The Arithmetic of Elliptic Curves, the author presented the basic theory culminating in two fundamental global results, the Mordell-Weil theorem on the finite generation of the group of rational… Expand

Geometric non-vanishing

- Mathematics
- 2005

We consider L-functions attached to representations of the Galois group of the function field of a curve over a finite field. Under mild tameness hypotheses, we prove non-vanishing results for twists… Expand

On Mordell–Weil groups of Jacobians over function fields

- Mathematics
- Journal of the Institute of Mathematics of Jussieu
- 2012

Abstract We study the arithmetic of abelian varieties over $K= k(t)$ where $k$ is an arbitrary field. The main result relates Mordell–Weil groups of certain Jacobians over $K$ to homomorphisms of… Expand

L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields

- Mathematics
- 2007

The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of… Expand

Arithmetic moduli of elliptic curves

- Mathematics
- 1985

This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova"… Expand

Elliptic fibers over non-perfect residue fields

- Mathematics
- 2004

Abstract Kodaira and Neron classified and described the geometry of the special fibers of the Neron model of an elliptic curve defined over a discrete valuation ring with a perfect residue field.… Expand

On the algebraic theory of elliptic modular functions

- Mathematics
- 1968

Let k denote an algebraically closed field over a prime field F (= Q or Z/~Z) and j a variable over k. Choose an elliptic curve A5 defined over F(j) with j as its absolute invariant. Two such… Expand

MINIMAL MODELS FOR ELLIPTIC CURVES

- Mathematics
- 2003

R. Moreover, it has an abstract minimal Weierstrass model over R that is unique up to unique R-isomorphism. It is natural to ask how the preferred models for elliptic curves are related to each… Expand

Endomorphisms of abelian varieties over finite fields

- Mathematics
- 1966

Almost all of the general facts about abelian varieties which we use without comment or refer to as "well known" are due to WEIL, and the references for them are [12] and [3]. Let k be a field, k its… Expand