Sample L^{A}TEX document

Thomas R. Shemanske Dartmouth College

July 7, 2000

This is dedicated to the one I love

Abstract This is a great paper. Read no further, because I don’t want you to hurt yourself,

but if you can’t help yourself, better strap in. It gets bumpy from here on in.

1 Introduction

The results which follow will dwarf all others that have come before. It amazes me that I have been able to write them down. I know that you too will be duely impressed.

2 Preliminaries

What! You don’t know what I’m talking about!!

Let’s try a little fraktur 𝔄𝔅ℭ. Let’s try a little black board bold ℤ, ℙ, ℚ, ℝ, ℂ. Let’s try some other symbols like x ≫ 0 or ⊗. How about M ⊗_{ℤ} N or ℤ^{ℤ^{ℤ}}?

How about p ∤ N or ⊞?

3 Some sample theorems

Lemma 3.1. Let’s put here exactly what we need to prove the next theorem.

Theorem 3.2. Let f be a nonzero element of S_{k/2}(4N, ψ). Then there exist an infinite number of square-free positive integers t such that S_{t}(f) ≠ 0.

Proof. If S_{t}(f) = 0 for all but a finite number of square-free positive integers t, then by Lemma 3.1 the Fourier coefficients of f are supported on only a finite number of square classes. By Theorem 3 of [3] the weight of f must be 1/2 of 3/2 and at weight 3/2 must be in the span of the theta series h_{ψ}, contrary to assumption.

1991 Mathematics Subject Classification. Primary 11Fxx; Secondary 11Fxx

Key Words and Phrases. Maximal Order, Central Simple Algebra, Bruhat–Tits Building

By Theorem 3.2, we see that it we can always find nonzero Shimura lifts.

Definition 3.3. A horse is a horse of course, of course, but noone can talk to a horse of course ….

Here we have some displayed and aligned equations. Here is an unnumbered displayed equation:

T (m)T (n) = ∑_{d|(m,n)} d^{k−1}χ(d)T (mn/d^{2}).

Here is a numbered displayed equation:

T (m)T (n) = ∑_{d|(m,n)} d^{k−1}χ(d)T (mn/d^{2}).  (3.1)

Here is the same expression, but inline and not displayed. Notice it is set smaller and the summation indeices are placed differently: T (m)T (n) = ∑_{d|(m,n)} d^{k−1}χ(d)T (mn/d^{2}). Note I need to use $ to surround my formula when in an inline mode.

For an aligned display we have

Λ_{N}(s; f) = (\frac{2π}{\sqrt{N}} )^{−s} Γ(s)L(s; f)
Λ_{M}(s; g) = (\frac{2π}{\sqrt{M}} )^{−s} Γ(s)L(s; g)

A numbered version is given by

Λ_{N}(s; f) = (\frac{2π}{\sqrt{N}} )^{−s} Γ(s)L(s; f)  (3.2)
Λ_{M}(s; g) = (\frac{2π}{\sqrt{M}} )^{−s} Γ(s)L(s; g)  (3.3)

A version with only one number associated to the group of equations is given by

\begin{aligned}Λ_{N}(s; f) & = (\frac{2π}{\sqrt{N}} )^{−s} Γ(s)L(s; f) \\ Λ_{M}(s; g) & = (\frac{2π}{\sqrt{M}} )^{−s} Γ(s)L(s; g) \end{aligned}  (3.4)

Something with cases

ϕ_{p}(s) = \begin{cases}(\frac{1−b(p)p^{−s}+ψ(p)p^{k−1−2s}}{1−a(p)p^{−s}+χ(p)p^{k−1−2s}} ) & if p | L \\ 1 & if p ∤ L. \end{cases}

Theorem 3.4. Suppose that N is an odd positive integer and ψ is an even Dirichlet character defined modulo 4N. Let F ∈ S_{k−1}^{+}(N, ψ^{2}) ∪ S_{k−1}^{+}(2N, ψ^{2}) be a normalized newform, and suppose that S_{k/2}(4M, ψ, F) ≠ 0 for some M | N. Then

1. M = N

2. S_{k/2}^{−}(4N, ψ) ∩ S_{k/2}(4N, ψ, F) = {0}, and so S_{k/2}(4N, ψ, F) ⊂ S_{k/2}^{+}(4N, ψ).

3. If N is square-free and ψ^{2} = 1, then S_{k/2}^{+}(4N, ψ)_{K} ⊂ S_{k/2}^{+}(4N, ψ).

Let’s get the other references in now. See [1] and [2].

References

[1] B. Cipra, On the Niwa-Shintani Theta-Kernel Lifting of Modular Forms, Nagoya Math. J., 91, (1983), 49–117.

[2] N. Koblitz, “Introduction to Elliptic Curves and Modular Forms”, Springer-Verlag, New York, 1984.

[3] J.-P. Serre and H. Stark, Modular Forms of Weight 1/2, In Lecture Notes in Math. 627, Springer-Verlag, Berlin and New York, (1977), 27–67.

Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755

E-mail address: Jumpin'.Jack.Flash@dartmouth.edu
URL: http://www.math.dartmouth.edu/~trs/