# Siegel Mass Formula

• ### nt.number theory - Siegel's Mass Formula for ternary ...

4. In his paper " On the theory of indefnite quadratic forms ", Siegel gives the formula (Thm. 1) μ ( S, T) = ∏ p α p ( S, T), where. S is an m × m non singular integral symmetric matrix of signature ( r, m − r), T is an n × n integral symmetric matrix, μ ( S, T) is the measure of …

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• ### [1001.5081] Siegel's mass formula and averages of ...

Jan 28, 2010· Let D be a square-free polynomial in F_q[t], where q is odd, and let G be a genus of definite ternary lattices over F_q[t] of determinant D. In this paper we give self-contained and relatively elementary proofs of Siegel's formulas for the weighted sum of primitive representations numbers over the classes of G and for the mass of G. Our proof of the mass formula shows …

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• ### 0 Summary

AN EXACT MASS FORMULA FOR QUADRATIC FORMS OVER NUMBER FIELDS Jonathan Hanke Princeton University Abstra ct. In this paper we give an explicit formula for the mass of a quadratic form in n ≥ 3 variables with respect to a maximal lattice over an arbitrary number ﬁeld k, and use this to ﬁnd the mass of many a-maximal lattices. We make the minor

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• ### Proof of a simple case of the Siegel-Weil formula

First, I confess I never understood Siegel’s arguments for his mass formula relating positive de nite quadratic forms and Eisenstein series. Of course, Siegel’s context did not separate things into local and global arguments. On the other hand, while current technique is arguably much more sophisticated, the questions addressed ...

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• ### Variations of mass formulas for definite division algebras ...

Jan 15, 2015· Section 3 compares these masses (Theorem 1.1) and deduces a mass formula (Theorem 1.2) in the case where R is a maximal order. In Section 4 we compute the local indices and prove Theorem 1.2. The last section discusses a mass formula for types of orders. 2. Definitions of masses2.1. Setting. Let K be a global field.

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• ### 7 Quadratic forms in n variables - OU Math

p generalizes Gauss’s genus theory and lead to Siegel’s mass formula, which is a generalization of Dirichlet’s mass formula to n-ary quadratic forms. The main “algebraic” question about quadratic forms is how they can be classiﬁed, up to equiv-alence. Deﬁnition 7.0.9. Let Q

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• ### Mass formula - Wikipedia

A mass formula is an equation or set of equations in physics which attempts to predict the mass or mass ratios of the subatomic particles.. An important step in high energy physics was the discovery of the Gell-Mann–Okubo mass formula predicting relationships between masses of the members of SU(3) multiplets.. The development of an accurate mass formula is one of several fundamental aspects ...

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• ### The Three Meanings Of E=mc^2, Einstein's Most Famous Equation

Jan 23, 2018· Mass/Energy conversion, and E=mc^2. There are only three parts to Einstein's most famous statement: E, or energy, which is the entirety of one …

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• ### Sums of Squares and Golden Gates - Institute for Advanced ...

C.L. Siegel: From his 1934 mass formula book: “ It is somewhat surprising that in a branch of mathematics as old as the theory of quadratic forms which originated among the ancient Babylonians and has been intensively studied during the last three centuries by a succession of mathematicians of the highest rank, including

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• ### The Bernoulli Number Page

The Minkowski-Siegel mass formula states for positive integers n = 2 k with 8 ∣ n that. where the sum runs over all even unimodular lattices Λ in dimension n and Aut (Λ) is the automorphism group of Λ. The products of (divided) Bernoulli numbers with explicit asymptotic constants (Kellner Kellner2009) are …

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• ### Weil’s Conjecture Revisited

Smith-Minkowski-Siegel mass formula. In the first lecture, I'll review the Siegel mass formula and explain a reformulation (due to Tamagawa and Weil) in terms of the volumes of certain adelic groups. This reformulation led Weil to conjecture a generalization of the mass formula, which applies to any (simply connected) semi-

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• ### MASS FORMULAS AND EISENSTEIN CONGRUENCES IN …

E=Q and q2 + qif qis inert in E=Q. For G= U(3), the spherical eigenvalue of 1 G for T q is q2 + q+ 1 if qis split in E=Q and q4 + qif qis inert in E=Q. In general, there are many local Hecke operators at q. See (3.9) and (3.13) for a general description of the unrami ed Hecke eigenvalues for 1 G. As a speci c example of the Bernoulli number divisibility condition, for any

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• ### The Siegel Mass Formula and Weil's Conjecture | Fields ...

Let L be a positive definite lattice. There are only finitely many positive definite lattices L' which are isomorphic to L modulo N for every N > 0: in fact, there is a formula for the number of such lattices, called the Siegel mass formula. In this talk, I'll review the Siegel mass formula and explain Weil's reformulation of it in terms of the volume of adelic groups.

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• ### [1001.5081] Siegel's mass formula and averages of ...

Jan 28, 2010· Let D be a square-free polynomial in F_q[t], where q is odd, and let G be a genus of definite ternary lattices over F_q[t] of determinant D. In this paper we give self-contained and relatively elementary proofs of Siegel's formulas for the weighted sum of primitive representations numbers over the classes of G and for the mass of G. Our proof of the mass formula shows an interesting relation ...

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• ### The Supersingular Loci and Mass Formulas on Siegel …

Keywords and Phrases: Siegel modular varieties, supersingular locus, superspecial locus, parahoric level structure, mass formula 1 1. Introduction In this paper we discuss some extensions of works of Katsura and Oort , and of Li and Oort  on the supersingular locus of a mod p Siegel modular variety.

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• ### f, - TAU

Siegel’s proofin IS1] also has this feature, butit relies heavily ontheta-functions andSiegelmodularforms. Our result (2.3) is valid in muchgreater generality, with the quadric Vk being

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• ### (PDF) Low-Dimensional Lattices IV: The Mass Formula

Siegel's mass formula provides a check that the list is complete. The mass formula is also used to verify that Niemeier's list of even lattices of dimension 24 is complete. Read more.

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• ### [1811.12828] Three applications of the Siegel mass formula

Nov 30, 2018· We present three applications of the Siegel mass formula. First we estimate the number of solutions of a quadratic system of equations. We also include estimates for the distribution of lattice points on caps of four dimensional spheres and for the number of non-congruent lattice tetrahedra.

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• ### Siegel–Weil formula - Wikipedia

In mathematics, the Siegel–Weil formula, introduced by Weil (1964, 1965) as an extension of the results of Siegel (1951, 1952), expresses an Eisenstein seriesas a weighted averageof theta seriesof lattices in a genus, where the weights are proportional to the inverse …

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• ### The Three Meanings Of E=mc², Einstein’s Most Famous ...

Jan 30, 2018· Only the gluons and photons are massless; everyone else, even the neutrinos, have a non-zero rest mass. Image credit: E. Siegel / Beyond The …

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• ### Slide 1

Siegel Mass Formula: Two lattices ... Speculation: Using results on the mass formula for lattices with nontrivial automorphism we can again prove that sequences {C. M} with a holographic dual are measure zero. Define a ``genus’’ to be an equivalence class under tensoring with a lattice theory of chiral scalar fields.

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• ### The Supersingular Loci and Mass Formulas on Siegel Modular ...

Keywords and Phrases: Siegel modular varieties, supersingular locus, superspecial locus, parahoric level structure, mass formula 1 1. Introduction In this paper we discuss some extensions of works of Katsura and Oort , and of Li and Oort  on the supersingular locus of a mod p Siegel modular variety.

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• ### Study group in number theory - City University of New York

Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel's mass formula) can be used to compute the representation numbers of certain ternary quadratic forms. Mar 6 : Alexandra Florea, Columbia University Non-vanishing for cubic L-functions

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• ### The Siegel Mass Formula and Weil's Conjecture | Department ...

Mar 12, 2018· There are only finitely many positive definite lattices L' which are isomorphic to L modulo N for every N > 0: in fact, there is a formula for the number of such lattices, called the Siegel mass formula. In this talk, I'll review the Siegel mass formula and explain Weil's reformulation of it in terms of the volume of adelic groups.

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• ### The Three Meanings Of E=mc², Einstein’s Most Famous ...

Jan 30, 2018· Only the gluons and photons are massless; everyone else, even the neutrinos, have a non-zero rest mass. Image credit: E. Siegel / Beyond The Galaxy. Even masses at rest have an energy inherent to ...

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• ### number theory - On Siegel mass formula - Mathematics Stack ...

Sep 29, 2017· He discusses the Siegel mass formula in a simple setting and explains how you can use can get formulas for representation numbers in special cases. In general, you can get "explicit formulas" for representation numbers in terms of Fourier coefficients of Hecke eigenforms.

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• ### Three applications of the Siegel mass formula

We rely on two new ingredients, incidence theory and Siegel's mass formula. View. Show abstract. A group-theoretic viewpoint on Erdos-Falconer problems and the Mattila integral. Article.

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