local version - University of Arizona Math
... Over number fields, one typically considers automorphic L-functions, since only these are known to have good analytic properties. Here, proofs of non-vanishing results necessarily use automorphic methods such as modular symbols, Fourier coefficients of half-integral weight forms, metaplectic Eisenst ...
... Over number fields, one typically considers automorphic L-functions, since only these are known to have good analytic properties. Here, proofs of non-vanishing results necessarily use automorphic methods such as modular symbols, Fourier coefficients of half-integral weight forms, metaplectic Eisenst ...
Chapter IV. Quotients by group schemes. When we work with group
... and det = δ. By direct computation one readily verifies that (i) the orbit of a diagonal matrix A = diag(λ, λ) is the single closed point A; (ii) the orbit of a diagonal matrix diag(λ1 , λ2 ) with λ1 '= λ2 equals N (λ1 + λ2 , λ1 λ2 ); (iii) the orbit of a matrix Jλ equals N (2λ, λ2 ) \ {diag(λ, λ)}; ...
... and det = δ. By direct computation one readily verifies that (i) the orbit of a diagonal matrix A = diag(λ, λ) is the single closed point A; (ii) the orbit of a diagonal matrix diag(λ1 , λ2 ) with λ1 '= λ2 equals N (λ1 + λ2 , λ1 λ2 ); (iii) the orbit of a matrix Jλ equals N (2λ, λ2 ) \ {diag(λ, λ)}; ...
Review for Chapter 1 Multiple Choice Identify the choice that best
... ____ 23. To which subsets of the real numbers does the number 1.68 belong? a. rational numbers b. natural numbers, whole numbers, integers, rational numbers c. rational numbers, irrational numbers d. none of the above ____ 24. To which subsets of the real numbers does the number 22 belong? a. whole ...
... ____ 23. To which subsets of the real numbers does the number 1.68 belong? a. rational numbers b. natural numbers, whole numbers, integers, rational numbers c. rational numbers, irrational numbers d. none of the above ____ 24. To which subsets of the real numbers does the number 22 belong? a. whole ...
a(x)
... are spaced on the average one every (ln n) integers • All even numbers can be immediately rejected, so it is 0.5 ln(n). • so in practice need only test 0.5 ln(n) numbers of size n to locate a prime • Eg: for numbers round 2^200 would check 0.5ln(2^200) = 69 numbers on average. • This is only an a ...
... are spaced on the average one every (ln n) integers • All even numbers can be immediately rejected, so it is 0.5 ln(n). • so in practice need only test 0.5 ln(n) numbers of size n to locate a prime • Eg: for numbers round 2^200 would check 0.5ln(2^200) = 69 numbers on average. • This is only an a ...
Constructible Sheaves, Stalks, and Cohomology
... S-schemes and the category of lcc sheaves on S. Proof. (Sketch) The functor is fully faithful by Yoneda’s Lemma. To check that it lands in the expected target we may work Zariski-locally on S so that the rank of each Xs is equal to an integer n ≥ 1. The diagonal X → X ×S X is étale and a closed imme ...
... S-schemes and the category of lcc sheaves on S. Proof. (Sketch) The functor is fully faithful by Yoneda’s Lemma. To check that it lands in the expected target we may work Zariski-locally on S so that the rank of each Xs is equal to an integer n ≥ 1. The diagonal X → X ×S X is étale and a closed imme ...
Structured Stable Homotopy Theory and the Descent Problem for
... is a weak equivalence of spectra. Here, for a spectrum X, Xl∧ denotes the l-adic completion of X, as defined in [6]. This is a relative result which asserts that the K-theory spectra of any two algebraically closed fields of a given characteristic are equivalent to each other after completion at a p ...
... is a weak equivalence of spectra. Here, for a spectrum X, Xl∧ denotes the l-adic completion of X, as defined in [6]. This is a relative result which asserts that the K-theory spectra of any two algebraically closed fields of a given characteristic are equivalent to each other after completion at a p ...
Moduli of elliptic curves
... over H, which is relatively algebraic in that it is defined by polynomial equations whose coefficients are holomorphic functions on H—the coefficients are even modular forms. From this beginning, one must be somewhat careful to prove the claim, but nothing too serious is involved. Before discussing ...
... over H, which is relatively algebraic in that it is defined by polynomial equations whose coefficients are holomorphic functions on H—the coefficients are even modular forms. From this beginning, one must be somewhat careful to prove the claim, but nothing too serious is involved. Before discussing ...
Modular functions and modular forms
... To construct a modular function, we have to construct a meromorphic function on H that is invariant under the action of .N /. This is difficult. It is easier to construct functions that transform in a certain way under the action of .N /; the quotient of two such functions of same type will then be ...
... To construct a modular function, we have to construct a meromorphic function on H that is invariant under the action of .N /. This is difficult. It is easier to construct functions that transform in a certain way under the action of .N /; the quotient of two such functions of same type will then be ...
Polynomials
... In mathematics, a polynomial is a finite length expression constructed from variables (also known as indeterminates) and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x ...
... In mathematics, a polynomial is a finite length expression constructed from variables (also known as indeterminates) and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x ...
12 Recognizing invertible elements and full ideals using finite
... invertible, thus invertible since G/htk i contains G/C as a finite index subgroup and Mn (Z[G/C]) has finitely detectable invertibles. We can send λk to the twisted group algebra Z[ωk ][G/C]ξ , where the product eg .eh = c(g, h)egh , with c the two-cocycle associated to the central extension C → G → ...
... invertible, thus invertible since G/htk i contains G/C as a finite index subgroup and Mn (Z[G/C]) has finitely detectable invertibles. We can send λk to the twisted group algebra Z[ωk ][G/C]ξ , where the product eg .eh = c(g, h)egh , with c the two-cocycle associated to the central extension C → G → ...