# Indefinite siegel theta function

Vignéras constructs non- holomorphic theta functions according to indefinite quadratic forms with arbitrary signature. holomorphic projections and ramanujan’ s mock theta functions. these indefinite zeta functions are defined as mellin transforms of indefinite theta functions in the sense of zwegers, which are in turn generalised to the siegel modular setting. siegel proved that ˆ( l; 2n) is a product of \ p- adic densities" ( which he did not compute) ; siegel also proved that ˆ( l; 2n) is a genus invariant and is given by the fourier coe cients of a theta series associated to l( as de ned below). weil recast and extended much of this theory in a new language.

journal of high energy physics,. you want to substitute a function in there, so we choose tan ( theta) since it is related to sec ( theta) by tan^ 2 ( theta) + 1 = sec^ 2 ( theta). theta functions 257 for u, v in c" ( column vectors), z in 55( n), one defines the symplectic theta function by loosely speaking, 9 is a siegel modular form of weight 1/ 2; more precisely, for m= ( 1 o) in r. 9, 9 ( m z, m( v) l = x( m) [ det( cz+ d) ] ' ~ 2 ( z, ( uvcf. the indefinite theta functions introduced in this paper generalise zwegers’ s work to the siegel modular setting.

without this simplification we would not have been able to integrate the second term with the knowledge that we currently have. siegel' s re- sults have been generalized and transformation laws for theta functions of higher degree have been established. indefinite theta functions for counting attractor backgrounds. theta functions 257 for u, v in c” ( column vectors), 2 in bcfl), one defines the symplectic theta function by 9 c z, 01 tl v = ~ ~ z~ e{ z( m+ u) - 2tmu- tuu}. csc \ left( \ theta \ right) \ sin \ left( \ theta \ right) = 1\ ] doing this allows us to greatly simplify the integrand and, in fact, allows us to actually do the integral. 12 and arxiv 1207. the theory of theta series for positive deﬁnite quadratic forms was highly advanced in the 19th century. basic facts on jacobi forms 16 2. for the dual pair sp ( n ) × o ( m) with m ≤ n, we prove an identity between a special value of a certain eisenstein series and indefinite siegel theta function the regularized integral of a theta function.

indefinite theta series go back to siegel ( and of course hecke) and originally arose in a more geometric setting which we now indefinite siegel theta function describe. we introduce the inde nite zeta function, de ned from the inde nite theta function using a mellin transform,. a structure theorem for j k, m 26 2. zwegers’ s work, and subsequent work of kathrin bringmann andken ono, led to a renaissance in mock modular forms. they have also been applied to soliton theory. if the quadratic form is indefinite, then siegel' s definition of the theta function also depends on a majorant of the quadratic form, an idea that siegel credits to hermite. assuming a few small ( mainly convergence- related) conditions, if. with respect to one of the complex variables, a theta function has a pr. theta functions and holomorphic jacobi forms 13 2. indefinite theta functions play an important indefinite siegel theta function role in the construction of mixed mock modu- lar forms and their completions.

1 and the de nitions in section 4). the inde nite zeta function is de ned as a mellin transform of an inde nite theta function ( literally, an inde nite theta null with real characteristics, see de nition 6. , in the same way that the principal branch of the log- gamma function is defined. we use siegel’ s theta series together with elementary geometric algebra and the. in mathematics, theta functions are special functions of several complex variables. h- harmonic maass- jacobi forms of degree 1: the analytic theory of some indefinite theta indefinite siegel theta function series. weierstrass ℘ - function 25 2. they are important in many areas, including the theories of abelian varieties and moduli spaces, and of quadratic forms. examples of jacobi forms 20 2.

in which it is visually represented as an integral symbol. we also have a jacobi form version of a; b, given by ( z; ˝ ) = a; c 1; c 2 ( z; ˝ ) : = x n2zr ˆ( n+ a; ˝ ) e( b( n; z) ) qq( n) : theorem ( zwegers). indefinite theta functions were introduced by sander zwegers in indefinite siegel theta function his phd thesis. jacobi theta functions 13 2. in particular, we discussed how to form convergent, modular functions by summing terms of the form p( n) qq( n) ( where pis a function possibly depending on v= im( ˝ ) ) as nranges over an r- dimensional lattice. when generalized to a grassmann algebra, they also appear in quantum field theory. the indefinite zeta function is defined as a mellin transform of an indefinite theta function ( literally, an indefinite theta null with real characteristics, see definition 5. the most common form of theta function is that occurring in the theory of elliptic functions. order of operations factors & primes fractions long arithmetic decimals exponents & radicals ratios & proportions percent modulo mean, median & mode scientific notation arithmetics. we derive explicit formulas for the action of the hecke operator t( p) on the genus theta series of a positive definite integral quadratic form and prove a theorem on the generation of spaces of eisenstein series by genus theta series.

as an application, we prove similar symmetries for borcherds lifts and automorphic green functions, both of which are closely related to siegel theta functions. indeﬁnite theta functions were introduced in the phd thesis ofsander zwegers in. theta functions probably use the letter \$ \ theta\$ because they generalize the value at zero of a jacobi theta function, which is a theta function of a rank- 1 lattice. the inde nite theta functions introduced in this paper. here the argument is chosen in such a way that a continuous function is obtained and holds, i. in, zwegers constructed mock theta functions of weight ( p+ 1) / 2 associated to quadratic spaces of signature ( p, 1) whose non- holomorphic.

we let be the special orthogonal group, and let k be the compact subgroup of g stabilizing the oriented negative q - plane. by converting this theta function into a symplectic theta function, friedberg proves that his theta function is indeed. inde nite theta functions were introduced by sander zwegers in his phd thesis [ 20]. we will show that siegel theta functions attached to integral quadratic forms of signature ( 2, m + 2) satisfy certain symmetries. - special values of elliptick and elliptice. these pages list thousands of expressions like products, sums, relations and limits shown in the following sections: - infinite products. higher- type indefinite theta functions: motivation and applications in the last two lectures, we described several types of theta functions. the modular transformation behavior of theta series for indefinite quadratic forms is well understood in the case of elliptic modular forms due to vignéras, who deduced that solving a differential equation of second order serves as a criterion for modularity. we’ ll start off with some of the basic indefinite integrals. ∫ xndx = xn + 1 n + 1 + c, n ≠ − 1. ∫ x n d x = x n + 1 n + 1 + c, n ≠ − 1.

in this paper we will give a generalization of this result for siegel theta series. andrianov and maloletkin [ 1]. relationship with half- integral weight. following siegel [ 5] and [ 6], friedberg [ 3] de nes a theta function of inde nite quadratic forms overc by using the majorants of the quadratic forms to guarantee that the theta function will converge. download full pdf package. in mathematics, the riemann– siegel theta function is defined in terms of the gamma function as for real values of t. loosely speaking, 8 is a siegel modular form of weight l/ 2; more precisely, for m= ( c i) in rsg, = x( m) [ det( cz+ d) ] 1’ 2\$ ( 2) ( cf. we prove an analytic continuation and functional equation for indefinite zeta functions. this connection can be worked backwards in theories like the stu or fhsv models, where there is a strong suggestion [ 24, 25] that the wall crossing phenomena are encoded in terms of indefinite theta functions, to be able to extract mock modular forms and hence, partition functions for counting single- center black holes. indefinite integral: in calculus, the indefinite integral is a mathematical function that takes the antiderivative from another function.

the first integral that we’ ll look at is the integral of a power of x. its extension to the case of indeﬁnite quadratic forms was developed by hecke, siegel, maass and others in the ﬁrst half of the 20th century. the proof uses the functional equation of the eisenstein series and the regularized siegel– weil formula for sp ( n ) × o ( 2 n + 2− m ). attached to quadratic forms. indefinite theta functions of type ( n; 1) i: definitions and examples 5 where e( z) : = 2 r z 0 e ˇt2dt.

we also discuss connections of our results with kudla' s matching principle for theta integrals. he used them to build harmonic weak maass forms whoseholomorphic parts are ramanujan’ s mock theta functions. the indefinite integrals of the jacobi theta functions,,, and, and their derivatives,,, and with respect to variable can be expressed by the following formulas: the first four sums cannot be expressed in closed form through the named functions. - products involving theta functions. this connection can be worked backwards in theories like the stu or fhsv models, where there is a strong suggestion [ 24, 25] that the wall crossing phenomena are encoded in terms of indefinite theta functions, to be able to extract mock modular forms and hence, partition functions for counting single- center black holes. we define the siegel theta series associated with p as & thetasym; p ( z) = det y − α/ 2 summationdisplay u ∈ z m × n p ( uy 1 / 2) exp ( πi tr( u t auz) ) ( h n ∋ z = x + iy), where h n is the siegel upper half- space and y 1 / 2 is the square root of the positive definite matrix y. we de ne an inde nite theta function in dimension gand index 1 whose modular parameter transforms by a symplectic group, generalizing a construction of sander zwegers used in the theory of mock modular forms. jacobi- eisenstein series 21 2. in this section we need to start thinking about how we actually compute indefinite integrals.

in general, we obtain non- holomorphic functions, so we will also. we generalize this result to siegel theta series of arbitrary genus n. joint with ozlem imamoglu and olav richter. the general rule when. - other formulae and curiosities including sums of hyperbolic and inverse tangent ( arctan) functions and q - series. the jacobi theta function 21 2. indefinite integrals. 1 and the definitions in sect.

we use vignéras’ theta functions to create examples of non- holomorphic jacobi forms associated to indefinite theta series by two different methods. in order to do so, we construct siegel theta series for indefinite quadratic forms of signature ( r, s) by considering polynomials with a certain homogeneity property. so, in order for this substitution to work out okay, you' re letting x= a* tan ( theta) so that when you write it out, you will end up with a^ 2+ ( a* tan ( theta) ) ^ 2 in your denominator. i refrain from giving here a formula for a jacobi theta function, because there is such a bewildering variety of notations that there is a wikipedia page devoted just to “ jacobi.

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