By Nikolai Proskurin

The booklet is an creation to the speculation of cubic metaplectic varieties at the third-dimensional hyperbolic house and the author's examine on cubic metaplectic kinds on precise linear and symplectic teams of rank 2. the themes comprise: Kubota and Bass-Milnor-Serre homomorphisms, cubic metaplectic Eisenstein sequence, cubic theta features, Whittaker capabilities. a unique process is built and utilized to discover Fourier coefficients of the Eisenstein sequence and cubic theta services. The booklet is meant for readers, with starting graduate-level history, attracted to extra learn within the conception of metaplectic types and in attainable applications.

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**Extra info for Cubic Metaplectic Forms and Theta Functions**

**Example text**

4. 5. Let l E Ic\{a} and PE Ia \{C}. The action ofCent(C, a) on Iz\{ al, C} is transitive if and only if the action on Ip\{ C P, a} is transitive. In this case, the action of Cent( C, a) on either set is sharply transitive. Proof. If P f l, then the map Q ---+ PQ is a bijection from Iz\{ al, C} to Ip\{ GP, a}, which commutes with the actions of Cent(C, a) since

Let 'Pt be the map fixing (oo), (n), [oo], [a] and with (x,y+t), [m,b+t]. Then (7, +, 0) is a group andt---+ 'Pt is an isomorphism 7 ---+Cent((oo), [oo]), the group of all ((oo), [oo])-transvections in Q(C). Moreover, the following are equivalent: (a) the projective plane Q(C) is ((oo), [oo])-transitive, (b) 7 = c, (c) (C, +, 0) is a group and -r(m, x, b) = mx + b for all m, x, b EC. Proof. Clearly, 0 E 7, since 0 is an additive identity. and t E 7, we have x + (b + t) -r(l,x,b+t) -r(l,x,b)+t = (x+b)+t.

Finally, we observe that the right distributive condition for C is equivalent to la + lb = la+b· In fact, it suffices to have la + lb = le for some c E C, for then c = le(l) = la(l) + h(l) =a+ b. Thus, C is right distributive if and only if le = {le : c EC} is closed under +. Since l;:; = l~ for mi- 0, we see that l~'In = {l;;- 1 : a i- O} U {lo}, so a3(cin) {::::::::? a4(C). 9" can be simplified to "C is a Cartesian group". 9. We first look at some consequences of left distributivity. 10. Let (C, +, 0) be a group having a left distributive product mx.