@ José Hdz. Stgo.:
Thank you for this great answer. I think I should add a few comments on the related materials I collected from different books. (Since I am neither a native speaker of German language nor well-versed in it, sometimes I need google translation or deepL translation for a better understanding on all these mysteries)
I believe Biermann used some hypotheses originated from L. Schlesinger on Gauss' work of AGM and elliptic functions(see Fragmente zur Theorie des arithmetrisch-geometrischen Mittels aus den Jahren 1797-1799, L. Schlesinger, 1911). Schlesinger made a great effort on dating some fragments on elliptic functions in Gauss' Leistenotizen(in which most of the records are dated no later than 1798), which is extremely important for understanding the development of Gauss' theory on elliptic functions. Schlesinger pointed out that the fragments in the Leistenotizen containing AGM are consisted of:
a)AGM and the rectification of ellipses;
b)AGM and the series whose exponents are quadratic functions(i.e., theta functions);
c)Series expansion of elliptic integrals as well as linear differential equation for elliptic integral of the first kind and the second kind.
In 1911 Schlesinger dated these materials no later than 1798 as well as other records in the Leistenotizen. It is very likely that this hypothesis contradicts what Gauss wrote in his dairy on May, 30th 1799(which implies that Gauss did not know the relation between AGM and elliptic integral before 1799), so later Schlesinger changed it to the summer of 1799(Gauss's Werke, X-1, pp. 273).
In the same year(Nov. 1799), Gauss started his Scheda Ac, another important record on Gauss' own development of elliptic function theory. In the Scheda Ac one can see the encrypted word GALEN(Gauss's Werke, X-1, pp. 273). I trust Biermann's interpretation of the GALEN that Gauss realized the importance of the reciprocal of AGM(which is equal to some elliptic integral of the first kind), and I do not reject Biermann's hypothesis that "Vicimus GEGAN" is a discovery on AGM. but I do not think "Vicimus GEGAN" is a discovery on the AGM and elliptic integrals/thetafunctions:
I)The date of the record "Vicimus GEGAN" is Oct. 21, 1796. Gauss started his research on lemniscate months after that(Jan. 1797). It is highly improbable that Gauss had already known the relation between the general AGM and the elliptic integral of the first kind when he had not even started his research on a special case(the lemniscatic integral).
II)I doubt whether the oral tradition that Schering recorded is reliable. Gauss already know that
in 1798(Scheda Aa, see Gauss's Werke, III, pp. 418), where $\varpi$ is the lemniscatic constant
If he did know the relation between AGM and the theta functions in 1794, he can prove that $$AGM((\theta_3(q))^2,(\theta_4(q))^2)=1$$ without any effort, which would definitely answer Gauss' question on May 30th, 1799. Pfaff's letter(Gauss's Werke, X-1, pp. 273, which is quoted in Hidden Harmony—Geometric Fantasies: The Rise of Complex Function Theory by Jeremy Gray) suggests that the proof was elusive to Gauss even in Nov. 1799.
III)If Gauss had discovered anything about AGM in 1796, it might be something else, e.g. the asymptotic formula for AGM in Scheda Ac(Gauss's Werke, X-1, pp. 186, Gauss also said he found that AGM can be written as the quotient of two transcendental functions long ago in the 101st entry of his diary). But it is likely that we can never know what Gauss discovered on Oct. 21, 1796.